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| [[File:Shapley–Folkman lemma.svg|thumb|300px|alt=The Shapley–Folkman lemma depicted by a diagram with two panes, one on the left and the other on the right. The left-hand pane displays four sets, which are displayed in a two-by-two array. Each of the sets contains exactly two points, which are displayed in red. In each set, the two points are joined by a pink line-segment, which is the convex hull of the original set. Each set has exactly one point that is indicated with a plus-symbol. In the top row of the two-by-two array, the plus-symbol lies in the interior of the line segment; in the bottom row, the plus-symbol coincides with one of the red-points. This completes the description of the left-hand pane of the diagram. The right-hand pane displays the Minkowski sum of the sets, which is the union of the sums having exactly one point from each summand-set; for the displayed sets, the sixteen sums are distinct points, which are displayed in red: The right-hand red sum-points are the sums of the left-hand red summand-points. The convex hull of the sixteen red-points is shaded in pink. In the pink interior of the right-hand sumset lies exactly one plus-symbol, which is the (unique) sum of the plus-symbols from the right-hand side. Comparing the left array and the right pane, one confirms that the right-hand plus-symbol is indeed the sum of the four plus-symbols from the left-hand sets, precisely two points from the original non-convex summand-sets and two points from the convex hulls of the remaining summand-sets.|
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| |The Shapley–Folkman lemma is illustrated by the [[Minkowski addition]] of four sets. The point (+) in the [[convex hull]] of the Minkowski sum of the four [[convex set|non-convex set]]s (''right'') is the sum of four points (+) from the (left-hand) sets—two points in two non-convex sets plus two points in the convex hulls of two sets. The convex hulls are shaded pink. The original sets each have exactly two points (shown as red dots).<ref name="s69"/>]]
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| The '''Shapley–Folkman [[lemma (mathematics)|lemma]]''' is a result in [[convex geometry]] with applications in [[mathematical economics]] that describes the [[Minkowski addition]] of [[set (mathematics)|set]]s in a [[vector space]]. ''Minkowski addition'' is defined as the addition of the sets' [[element (mathematics)|member]]s: for example, adding the set consisting of the [[integer]]s zero and one to itself yields the set consisting of zero, one, and two:
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| : {0, 1} + {0, 1} = {0 + 0, 0 + 1, 1 + 0, 1 + 1} = {0, 1, 2}.
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| The Shapley–Folkman lemma and related results provide an affirmative answer to the question, "Is the sum of many sets close to being [[convex set|convex]]?"<ref name="Howe" >{{harvtxt|Howe|1979|p=1}}: {{citation|title=On the tendency toward convexity of the vector sum of sets|authorlink=Roger Evans Howe|last=Howe|first=Roger |date=3 November 1979 |publisher=[[Cowles Foundation|Cowles Foundation for Research in Economics]], Yale University|series=Cowles Foundation discussion papers|location=Box 2125 Yale Station, New Haven,CT 06520|volume=538 |url=http://cowles.econ.yale.edu/P/cd/d05a/d0538.pdf|<!-- url-2=http://econpapers.repec.org/RePEc:cwl:cwldpp:538 -->|accessdate=1 January 2011}}
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| </ref> A set is defined to be ''convex'' if every [[line segment]] joining two of its points is a [[subset]] in the set: For example, the solid [[unit disk|disk]] <big><math>\bullet</math></big> is a convex set but the [[unit circle|circle]] <big><math>\circ</math></big> is not, because the line segment joining two distinct points <math>\oslash</math> is not a subset of the circle. The Shapley–Folkman lemma suggests that if the number of summed sets exceeds the [[dimension (linear algebra)|dimension]] of the vector space, then their Minkowski sum is approximately convex.<ref name="s69">{{harvtxt|Starr|1969}}</ref> | |
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| The Shapley–Folkman lemma was introduced as a step in the [[mathematical proof|proof]] of the '''Shapley–Folkman [[theorem]]''', which states an [[upper bound]] on the [[Euclidean distance|distance]] between the Minkowski sum and its [[convex hull]]. The ''convex hull'' of a set ''Q'' is the smallest convex set that contains ''Q''. This distance is zero [[if and only if]] the sum is convex.
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| The theorem's bound on the distance depends on the dimension ''D'' and on the shapes of the summand-sets, but ''not'' on the number of summand-sets ''N'', {{nowrap|when ''N'' > ''D''.}}
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| The shapes of a subcollection of only ''D'' summand-sets determine the bound on the distance between the Minkowski ''[[arithmetic mean|average]]'' of ''N'' sets
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| : {{frac|1|''N''}} (''Q''<sub>1</sub> + ''Q''<sub>2</sub> + ... + ''Q''<sub>''N''</sub>)
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| and its convex hull. As ''N'' increases to [[infinity]], the bound [[limit of a sequence|decreases to zero]] (for summand-sets of uniformly bounded size).<ref name="Starr08"/> The Shapley–Folkman theorem's upper bound was decreased by '''Starr's [[corollary]]''' (alternatively, the '''Shapley–Folkman–Starr theorem''').
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| The lemma of [[Lloyd Shapley]] and [[Jon Folkman]] was first published by the economist [[Ross Starr|Ross M. Starr]], who was investigating the existence of [[general equilibrium theory#Nonconvexities in large economies|economic equilibria]] while studying with [[Kenneth Arrow]].<ref name="s69"/> In his paper, Starr studied a ''convexified'' economy, in which non-convex sets were replaced by their convex hulls; Starr proved that the convexified economy has equilibria that are closely approximated by "quasi-equilibria" of the original economy; moreover, he proved that every quasi-equilbrium has many of the optimal properties of true equilibria, which are proved to exist for convex economies. Following Starr's 1969 paper, the Shapley–Folkman–Starr results have been widely used to show that central results of (convex) economic theory are good approximations to large economies with non-convexities; for example, quasi-equilibria closely approximate equilibria of a convexified economy. "The derivation of these results in general form has been one of the major achievements of postwar economic theory", wrote [[Roger Guesnerie]].<ref name="g89-p138">{{harvtxt|Guesnerie|1989|p=138}}</ref> The topic of [[non-convexity (economics)|non-convex sets in economics]] has been studied by many [[Nobel Prize in Economics|Nobel laureates]], besides Lloyd Shapley who won the prize in 2012: Arrow (1972), [[Robert Aumann]] (2005), [[Gérard Debreu]] (1983), [[Tjalling Koopmans]] (1975), [[Paul Krugman]] (2008), and [[Paul Samuelson]] (1970); the complementary topic of [[convexity in economics|convex sets in economics]] has been emphasized by these laureates, along with [[Leonid Hurwicz]], [[Leonid Kantorovich]] (1975), and [[Robert Solow]] (1987).
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| The Shapley–Folkman lemma has applications also in [[mathematical optimization|optimization]] and [[probability theory]].<ref name="Starr08" >{{harvtxt|Starr|2008}}</ref> In optimization theory, the Shapley–Folkman lemma has been used to explain the successful solution of minimization problems that are sums of many [[function (mathematics)|function]]s.<ref name="Ekeland76"/><ref name="Bertsekas82"/> The Shapley–Folkman lemma has also been used in [[mathematical proof|proofs]] of the [[law of large numbers|"law of averages"]] for [[stochastic geometry|random sets]], a theorem that had been proved <!-- to hold --> for only convex sets.<ref name="ArtsteinVitale"/>
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| {{TOC limit|3}}
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| ==Introductory example==
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| For example, the subset of the integers {0, 1, 2} is contained in the [[interval (mathematics)|interval]] of [[real number]]s [0, 2], which is convex. The Shapley–Folkman lemma implies that every point in [0, 2] is the sum of an integer from {0, 1} and a real number from [0, 1].<ref name="Carter94" >{{harvtxt|Carter|2001|p=94|}}</ref>
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| The distance between the convex interval [0, 2] and the non-convex set {0, 1, 2} equals one-half
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| : 1/2 = |1 − 1/2| = |0 − 1/2| = |2 − 3/2| = |1 − 3/2|.
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| However, the distance between the ''[[arithmetic mean|average]]'' Minkowski sum
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| : 1/2 ( {0, 1} + {0, 1} ) = {0, 1/2, 1}
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| and its convex hull [0, 1] is only 1/4, which is half the distance (1/2) between its summand {0, 1} and [0, 1]. As more sets are added together, the average of their sum "fills out" its convex hull: The maximum distance between the average and its convex hull approaches zero as the average includes more [[addition#summand|summand]]s.<ref name="Carter94"/>
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| ==Preliminaries==
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| The Shapley–Folkman lemma depends upon the following definitions and results from [[convex geometry]].
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| ===Real vector spaces===
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| A [[real number|real]] [[vector space]] of two [[dimension (vector space)|dimension]]s can be given a [[Cartesian coordinate system]] in which every point is identified by an [[ordered pair]] of real numbers, called "coordinates", which are conventionally denoted by ''x'' and ''y''. Two points in the Cartesian plane can be ''[[Euclidean vector#Addition and subtraction|added]]'' coordinate-wise
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| : (''x''<sub>1</sub>, ''y''<sub>1</sub>) + (''x''<sub>2</sub>, ''y''<sub>2</sub>) = (''x''<sub>1</sub>+''x''<sub>2</sub>, ''y''<sub>1</sub>+''y''<sub>2</sub>);
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| further, a point can be ''[[scalar multiplication|multiplied]]'' by each real number ''λ'' coordinate-wise
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| : ''λ'' (''x'', ''y'') = (''λx'', ''λy'').
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| More generally, any real vector space of (finite) dimension ''D'' can be viewed as the [[set (mathematics)|set]] of all [[tuple|''D''-tuple]]s of ''D'' real numbers {{nowrap|{ (''v''<sub>1</sub>, ''v''<sub>2</sub>, . . . , ''v''<sub>D</sub>)}} } on which two [[operation (mathematics)|operation]]s are defined: [[Euclidean vector#Addition and subtraction|vector addition]] and [[scalar multiplication|multiplication by a real number]]. For finite-dimensional vector spaces, the operations of vector addition and real-number multiplication can each be defined coordinate-wise, following the example of the Cartesian plane.<ref>{{harvtxt|Arrow|Hahn|1980|p=375}}</ref>
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| ===Convex sets===
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| {{multiple image
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| | width =155
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| | footer = [[Line segment]]s test whether a subset be [[convex set|convex]].
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| | image1 = Convex polygon illustration1.png
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| | alt1 = Illustration of a convex set, which looks somewhat like a disk: A (green) convex set contains the (black) line-segment joining the points x and y. The entire line-segment is a subset of the convex set.
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| | caption1 =In a [[convex set]] ''Q'', the [[line segment]] connecting any two of its points is a subset of ''Q''.
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| | image2 = Convex polygon illustration2.png
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| | alt2 = Illustration of a green non-convex set, which looks somewhat like a [[boomerang]] or [[cashew]] nut. The black line-segment joins the points ''x'' and ''y'' of the green non-convex set. Part of the line segment is not contained in the green non-convex set.
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| | caption2 =In a [[convex set|non-convex set]] ''Q'', a point in some [[line segment|line-segment]] joining two of its points is not a member of ''Q''.
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| }}
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| In a real vector space, a [[empty set|non-empty]] set ''Q'' is defined to be ''[[convex set|convex]]'' if, for each pair of its points, every point on the [[line segment]] that joins them is a [[subset]] of ''Q''. For example, a solid [[unit disk|disk]] <big><math>\bullet</math></big> is convex but a [[unit circle|circle]] <big><math>\circ</math></big> is not, because it does not contain a line segment joining its points <math>\oslash</math>; the non-convex set of three integers {0, 1, 2} is contained in the interval [0, 2], which is convex. For example, a solid [[cube (geometry)|cube]] is convex; however, anything that is hollow or dented, for example, a [[crescent]] shape, is non-convex. The [[empty set]] is convex, either by definition<ref name="Rock10" /> or [[vacuous truth|vacuously]], depending on the author.
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| More formally, a set ''Q'' is convex if, for all points ''v''<sub>0</sub> and ''v''<sub>1</sub> in ''Q'' and for every real number ''λ'' in the [[unit interval]] [0,1], the point
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| : (1 − ''λ'') ''v''<sub>0</sub> + ''λv''<sub>1</sub>
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| is a [[element (mathematics)|member]] of ''Q''.
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| By [[mathematical induction]], a set ''Q'' is convex if and only if every [[convex combination]] of members of ''Q'' also belongs to ''Q''. By definition, a ''convex combination'' of an indexed subset {''v''<sub>0</sub>, ''v''<sub>1</sub>, . . . , ''v''<sub>D</sub>} of a vector space is any weighted average {{nowrap|''λ''<sub>0</sub>''v''<sub>0</sub> + ''λ''<sub>1</sub>''v''<sub>1</sub> + . . . + ''λ''<sub>D</sub>''v''<sub>D</sub>,}} for some indexed set of non-negative real numbers {''λ''<sub>d</sub>} satisfying the equation {{nowrap|''λ''<sub>0</sub> + ''λ''<sub>1</sub> + . . . + ''λ''<sub>D</sub>}} = 1.<ref>{{harvtxt|Arrow|Hahn|1980|p=376}}, {{harvtxt|Rockafellar|1997|pp=10–11}}, and {{harvtxt|Green|Heller|1981|p=37}}</ref>
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| The definition of a convex set implies that the ''[[intersection (set theory)|intersection]]'' of two convex sets is a convex set. More generally, the intersection of a family of convex sets is a convex set. In particular, the intersection of two [[disjoint sets]] is the empty set, which is convex.<ref name="Rock10" >{{harvtxt|Rockafellar|1997|p=10}}</ref><!-- In this proposition, the family can be empty, finite, countably infinite, or uncountably infinite. -->
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| ===Convex hull===
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| [[File:Extreme points illustration.png|thumb|right|alt=A picture of a smoothed triangle, like a triangular (Mexican) tortilla-chip or a triangular road-sign. Each of the three rounded corners is drawn with a red curve. The remaining interior points of the triangular shape are shaded with blue.|In the [[convex hull]] of the red set, each blue point is a [[convex combination]] of some red points.]]
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| For every subset ''Q'' of a real vector space, its {{nowrap|''[[convex hull]]'' Conv(''Q'')}} is the [[minimal element|minimal]] convex set that contains ''Q''. Thus Conv(''Q'') is the intersection of all the convex sets that [[cover (mathematics)|cover]] ''Q''. The convex hull of a set can be equivalently defined to be the set of all convex combinations of points in ''Q''.<ref>{{harvtxt|Arrow|Hahn|1980|p=385}} and {{harvtxt|Rockafellar|1997|pp=11–12}}</ref> For example, the convex hull of the set of [[integer]]s {0,1} is the closed [[interval (mathematics)|interval]] of [[real number]]s [0,1], which contains the integer end-points.<ref name="Carter94" /> The convex hull of the [[unit circle]] is the closed [[unit disk]], which contains the unit circle.
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| ===Minkowski addition===
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| [[File:Minkowski sum.png|thumb|alt=Three squares are shown in the non-negative quadrant of the Cartesian plane. The square Q<sub>1</sub>=[0,1]×[0,1] is green. The square Q<sub>2</sub>=[1,2]×[1,2] is brown, and it sits inside the turquoise square Q<sub>1</sub>+Q<sub>2</sub>=[1,3]×[1,3].|[[Minkowski addition]] of sets. The <!-- [[Minkowski addition|Minkowski]] -->sum of the squares ''Q''<sub>1</sub>=[0,1]<sup>2</sup> and ''Q''<sub>2</sub>=[1,2]<sup>2</sup> is the square ''Q''<sub>1</sub>+''Q''<sub>2</sub>=[1,3]<sup>2</sup>.]]
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| In a real vector space, the ''[[Minkowski addition|Minkowski sum]]'' of two (non-empty) sets ''Q''<sub>1</sub> and ''Q''<sub>2</sub> is defined to be the [[sumset|set]] ''Q''<sub>1</sub> + ''Q''<sub>2</sub> formed by the addition of vectors element-wise from the summand sets
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| : ''Q''<sub>1</sub> + ''Q''<sub>2</sub> = { ''q''<sub>1</sub> + ''q''<sub>2</sub> : ''q''<sub>1</sub> ∈ ''Q''<sub>1</sub> and ''q''<sub>2</sub> ∈ ''Q''<sub>2</sub> }.<ref>{{harvtxt|Schneider|1993|p=xi}} and {{harvtxt|Rockafellar|1997|p=16}}</ref>
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| For example
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| : {0, 1} + {0, 1} = {0+0, 0+1, 1+0, 1+1} = {0, 1, 2}.<ref name="Carter94" />
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| By the principle of mathematical induction, the ''Minkowski sum'' of a finite family of (non-empty) sets
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| : {''Q''<sub>n</sub> : ''Q''<sub>n</sub> ≠ Ø and 1 ≤ ''n'' ≤ ''N'' }
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| is <!-- defined to be --> the
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| set <!-- of vectors -->
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| formed by element-wise addition of vectors <!-- from the summand-sets -->
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| : ∑ ''Q''<sub>n</sub> = {∑ ''q''<sub>n</sub> : ''q''<sub>n</sub> ∈ ''Q''<sub>n</sub>}.<ref>{{harvtxt|Rockafellar|1997|p=17}} and {{harvtxt|Starr|1997|p=78}}</ref>
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| ===Convex hulls of Minkowski sums===
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| Minkowski addition behaves well with respect to "''convexification''"—the operation of taking convex hulls. Specifically, for all subsets ''Q''<sub>1</sub> and ''Q''<sub>2</sub> of a real vector space, the [[convex hull]] of their Minkowski sum is the Minkowski sum of their convex hulls. That is,
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| :Conv( ''Q''<sub>1</sub> + ''Q''<sub>2</sub> ) = Conv( ''Q''<sub>1</sub> ) + Conv( ''Q''<sub>2</sub> ).
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| This result holds more generally, as a consequence of the principle of mathematical induction. For each [[finite set|finite]] collection of sets,
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| : Conv( ∑ ''Q''<sub>n</sub> ) = ∑ Conv( ''Q''<sub>n</sub> ).<ref name="Schneider">{{harvtxt|Schneider|1993|pp=2–3}}</ref><ref>{{harvtxt|Arrow|Hahn|1980|p=387}}</ref>
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| ==Statements==
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| [[File:Shapley–Folkman lemma.svg|thumb|300px|alt=The Shapley–Folkman lemma depicted by a diagram with two panes, one on the left and the other on the right. The left-hand pane displays four sets, which are displayed in a two-by-two array. Each of the sets contains exactly two points, which are displayed in red. In each set, the two points are joined by a pink line-segment, which is the convex hull of the original set. Each set has exactly one point that is indicated with a plus-symbol. In the top row of the two-by-two array, the plus-symbol lies in the interior of the line segment; in the bottom row, the plus-symbol coincides with one of the red-points. This completes the description of the left-hand pane of the diagram. The right-hand pane displays the Minkowski sum of the sets, which is the union of the sums having exactly one point from each summand-set; for the displayed sets, the sixteen sums are distinct points, which are displayed in red: The right-hand red sum-points are the sums of the left-hand red summand-points. The convex hull of the sixteen red-points is shaded in pink. In the pink interior of the right-hand sumset lies exactly one plus-symbol, which is the (unique) sum of the plus-symbols from the right-hand side. The right-hand plus-symbol is indeed the sum of the four plus-symbols from the left-hand sets, precisely two points from the original non-convex summand-sets and two points from the convex hulls of the remaining summand-sets.|
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| |Minkowski addition and convex hulls. The sixteen dark-red points (on the right) form the [[Minkowski addition|Minkowski sum]] of the four non-convex sets (on the left), each of which consists of a pair of red points. Their convex hulls (shaded pink) contain plus-signs (+): The right plus-sign is the sum of the left plus-signs.]]
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| The preceding identity
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| Conv( ∑ ''Q''<sub>n</sub> ) = ∑ Conv( ''Q''<sub>n</sub> )
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| implies that
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| if a point ''x'' lies in the convex hull of the Minkowski sum of ''N'' sets
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| : ''x'' ∈ Conv( ∑ ''Q''<sub>n</sub> )
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| then ''x'' lies in the sum of the convex hulls of the summand-sets
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| : ''x'' ∈ ∑ Conv( ''Q''<sub>n</sub> ).
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| By the definition of Minkowski addition, this last expression means that ''x'' = ∑ ''q''<sub>n</sub> for some selection of points ''q''<sub>n</sub> in the convex hulls of the summand-sets, that is, where each ''q''<sub>n</sub> ∈ Conv(''Q''<sub>n</sub>). In this representation, the selection of the summand-points ''q''<sub>n</sub> depends on the chosen sum-point ''x''.
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| ===Lemma of Shapley and Folkman===
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| [[File:Shapley, Lloyd (1923).jpeg|thumb|alt=Picture of Lloyd Shapley|A Winner of the 2012 Nobel Award in Economics, [[Lloyd Shapley]] proved the Shapley–Folkman lemma with [[Jon Folkman]].<ref name="s69"/>]]
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| For this representation of the point ''x'', the '''Shapley–Folkman lemma''' states that if the dimension ''D'' is less than the number of summands
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| : {{nowrap|''D'' < ''N''}}
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| then convexification is needed for only ''D'' summand-sets, whose choice depends on ''x'': The point has a representation <!-- : ''x'' = -->
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| : <math> x = \sum_{1\leq{d}\leq{D}}{q_d} + \sum_{D+1\leq{n}\leq{N}}{q_n} </math> <!-- : ∑<sub>1≤''d''≤''D''</sub> ''x''<sub>''d''</sub> + ∑<sub>''D''+1≤''n''≤''N''</sub> ''x''<sub>''n''</sub>
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| -->
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| where ''q''<sub>d</sub> belongs to the convex hull of ''Q''<sub>d</sub> for ''D'' (or fewer) summand-sets and ''q''<sub>n</sub> belongs to ''Q''<sub>n</sub> itself for the remaining <!-- summand- -->sets. That is,
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| : <math> x \in{ \sum_{1\leq{d}\leq{D}}{\operatorname{Conv}{(Q_d)}} + \sum_{D+1\leq{n}\leq{N}}{Q_n} }</math>
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| <!-- : x ∈ ∑<sub>1≤''d''≤''D''</sub> Conv(S<sub>''d''</sub>) + ∑<sub>''D''+1≤''n''≤''N''</sub> S<sub>''n''</sub>,
| |
| -->
| |
| | |
| for some re-indexing of the summand sets; this re-indexing depends on the particular point ''x'' being represented.<ref>{{harvtxt|Starr|1969|pp=35–36}}</ref>
| |
| | |
| The Shapley–Folkman lemma implies, for example, that every point in [0, 2] is the sum of an [[integer]] from {0, 1} and a [[real number]] from [0, 1].<ref name="Carter94" />
| |
| | |
| ====Dimension of a real vector space====
| |
| Conversely, the Shapley–Folkman lemma characterizes the [[dimension (vector space)|dimension]] of finite-dimensional, real vector spaces. That is, if a vector space obeys the Shapley–Folkman lemma for a [[natural number]] ''D'', and for no number less than ''D'', then its dimension is exactly ''D'';<ref>{{harvtxt|Schneider|1993|p=131}}</ref> the Shapley–Folkman lemma holds for only ''finite-dimensional'' vector spaces.<ref>{{harvtxt|Schneider|1993|p=140}} credits this result to {{harvtxt|Borwein|O'Brien|1978}}: {{cite journal|last1=Borwein|first1=J. M.|authorlink=Jonathan Borwein|last2=O'Brien|first2=R. C.|title=Cancellation characterizes convexity|journal=Nanta Mathematica (Nanyang University)|issn=0077-2739|volume=11|year=1978|pages=100–102|mr=510842|ref=harv}}</ref>
| |
| | |
| ===Shapley–Folkman theorem and Starr's corollary===
| |
| [[File:Inner radius.svg|thumb|240px|alt=A blue disk contains red points. A smaller green disk sits in the largest concavity in among these red points.|The circumradius (blue) and inner radius (green) of a point set (dark red, with its convex hull shown as the lighter red dashed lines). The inner radius is smaller than the circumradius except for subsets of a single circle, for which they are equal.]]
| |
| | |
| Shapley and Folkman used their lemma <!-- , which is purely [[discrete geometry|combinatorial]], --> to prove their <!-- [[metric space|metric]] --> theorem, which bounds the distance between a Minkowski sum and its convex hull, the "''convexified''" sum:
| |
| * The ''Shapley–Folkman theorem'' states that the squared [[Euclidean distance]] from any point in the convexified sum {{nowrap|Conv( ∑ ''Q''<sub>''n''</sub> )}} to the original (unconvexified) sum {{nowrap|∑ ''Q''<sub>''n''</sub>}} is bounded by the sum of the squares of the ''D'' largest circumradii of the sets ''Q''<sub>''n''</sub> (the radii of the [[Smallest circle problem|smallest spheres enclosing these sets]]).<ref>{{harvtxt|Schneider|1993|p=129}}</ref> This bound is independent of the number of summand-sets ''N'' (if {{nowrap|''N'' > ''D'').}}<ref>{{harvtxt|Starr|1969|p=36}}</ref>
| |
| The Shapley–Folkman theorem states a bound on the distance between the Minkowski sum and its convex hull; this distance is zero [[if and only if]] the sum is convex. Their bound on the distance depends on the dimension ''D'' and on the shapes of the summand-sets, but ''not'' on the number of summand-sets ''N'', {{nowrap|when ''N'' > ''D''.}}<ref name="Starr08"/>
| |
| | |
| The circumradius often exceeds (and cannot be less than) the ''inner radius'':<ref name="Starr 1969 37">{{harvtxt|Starr|1969|p=37}}</ref>
| |
| | |
| * The ''inner radius'' of a <!-- non–convex --> set ''Q''<sub>''n''</sub> is defined to be the smallest number ''r'' such that, for any point ''q'' in the convex hull of ''Q''<sub>''n''</sub>, there is a [[sphere]] of radius ''r'' that contains a subset of ''Q''<sub>''n''</sub> whose convex hull contains ''q''.
| |
| Starr used the inner radius to reduce the upper bound stated in the Shapley–Folkman theorem:
| |
| * ''Starr's corollary to the Shapley–Folkman theorem'' states that the squared Euclidean distance from any point ''x'' in the convexified sum {{nowrap|Conv( ∑ ''Q''<sub>''n''</sub> )}} to the original (unconvexified) sum {{nowrap|∑ ''Q''<sub>''n''</sub>}} is bounded by the sum of the squares of the ''D'' largest inner-radii of the sets ''Q''<sub>''n''</sub>.<ref name="Starr 1969 37"/><ref>{{harvtxt|Schneider|1993|pp=129–130}}
| |
| </ref>
| |
| Starr's corollary <!-- to the Shapley–Folkman theorem --> states an [[upper and lower bounds|upper bound]] on the Euclidean distance between the Minkowski sum of ''N'' sets and the convex hull of the Minkowski sum; this distance between the sum and its convex hull is a measurement of the non-convexity of the set. For [[abuse of notation|simplicity]], this distance is called the "''non-convexity''" of the set (with respect to Starr's measurement). Thus, Starr's bound on the non-convexity of the sum depends on only the ''D'' largest inner radii of the summand-sets; however, Starr's bound does not depend on the number of summand-sets ''N'', when {{nowrap|''N'' > ''D''}}.
| |
| For example, the distance between the convex interval [0, 2] and the non-convex set {0, 1, 2} equals one-half
| |
| : 1/2 = |1 − 1/2| = |0 − 1/2| = |2 − 3/2| = |1 − 3/2|.
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| Thus, Starr's bound on the non-convexity of the ''average''<!-- -->
| |
| : {{frac|1|''N''}} ∑ ''Q''<sub>''n''</sub>
| |
| decreases as the number of summands ''N'' increases.
| |
| For example, the distance between the ''averaged'' set
| |
| : 1/2 ( {0, 1} + {0, 1} ) = {0, 1/2, 1}
| |
| and its convex hull [0, 1] is only 1/4, which is half the distance (1/2) between its summand {0, 1} and [0, 1].
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| The shapes of a subcollection of only ''D'' summand-sets determine the bound on the distance between the ''average set'' <!-- : {{frac|1|''N''}} ∑ ''Q''<sub>''n''</sub> --> and its convex hull; thus, as the number of summands increases to [[infinity]], the bound [[limit of a sequence|decreases to zero]] (for summand-sets of uniformly bounded size).<ref name="Starr08"/> In fact, Starr's bound on the non-convexity of this average set [[limit of a sequence|decreases to zero]] as the number of summands ''N'' increases to [[infinity]] (when the inner radii of all the summands are bounded by the same number).<ref name="Starr08"/>
| |
| | |
| ===Proofs and computations===
| |
| The original proof of the Shapley–Folkman lemma established only the [[existence theorem|existence]] of the representation, but did not provide an [[algorithm]] for computing the representation: Similar proofs have been given by [[Kenneth Arrow|Arrow]] and [[Frank Hahn|Hahn]],<ref>{{harvtxt|Arrow|Hahn|1980|pp=392–395}}</ref> [[J. W. S. Cassels|Cassels]],<ref>{{harvtxt|Cassels|1975|pp=435–436}}</ref> and Schneider,<ref>{{harvtxt|Schneider|1993|p=128}}</ref> among others. An abstract and elegant proof by [[Ivar Ekeland|Ekeland]] has been extended by Artstein.<ref>{{harvtxt|Ekeland|1999|pp=357–359}}</ref><ref>{{harvtxt|Artstein|1980|p=180}}</ref> Different proofs have appeared in unpublished papers, also.<ref name="Howe"/><ref>{{citation|title=Economics 201B: Nonconvex preferences and approximate equilibria|chapter=1 The Shapley–Folkman theorem|pages=1–5|<!-- date=2005–03–14 -->|date=14 March 2005|first=Robert M.|last=Anderson|authorlink=<!-- NOT WP's Robert M. Anderson -->|location=Berkeley, CA|publisher=Economics Department, University of California, Berkeley|url=http://elsa.berkeley.edu/users/anderson/Econ201B/NonconvexHandout.pdf|accessdate=1 January 2011}}</ref> In 1981, Starr published an [[iterative method]] for computing a representation of a given sum-point; however, his computational proof provides a weaker bound than does the original result.<ref>{{cite journal|mr=640201|last=Starr|first=Ross M.|authorlink=Ross M. Starr|title=Approximation of points of convex hull of a sum of sets by points of the sum: An elementary approach|journal=Journal of Economic Theory|volume=25|year=1981|issue=2|pages=314–317
| |
| |doi=10.1016/0022-0531(81)90010-7
| |
| |url=http://www.sciencedirect.com/science/article/B6WJ3-4CYGB4B-FB/2/9e65178b1c246365bee61dc19127175d|ref=harv}}</ref>
| |
| | |
| ==Applications==
| |
| The Shapley–Folkman lemma enables researchers to extend results for Minkowski sums of convex sets to sums of general sets, which need not be convex. Such sums of sets arise in [[economics]], in [[mathematical optimization]], and in [[probability theory]]; in each of these three mathematical sciences, non-convexity is an important feature of applications.
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| | |
| ===Economics===
| |
| [[File:Indifference curves showing budget line.svg|thumb|right|alt=The nonnegative quadrant of the Cartesian plane appears. A blue straight-line slopes downward as a secant joining two points, one on each of the axes. This blue line is tangent to a red curve that touches it at a marked point, whose coordinates are labeled ''Qx'' and ''Qy''.|The consumer [[preference (economics)|prefers]] every basket of goods on the [[indifference curve]] ''I''<sub>3</sub> over each basket on ''I''<sub>2</sub>.
| |
| The basket (''Q''<sub>x</sub>, ''Q''<sub>y</sub>), where the budget line (''shown in blue'') [[supporting hyperplane|supports]] ''I''<sub>2</sub>, is optimal and also feasible, unlike any basket lying on ''I''<sub>3</sub> which is preferred but unfeasible.]]
| |
| {{See also|Convexity in economics}}
| |
| In [[microeconomics|economics]], a consumer's [[Preference (economics)|preferences]] are defined over all "baskets" of goods. Each basket is represented as a non-negative vector, whose coordinates represent the quantities of the goods. On this set of baskets, an ''[[indifference curve]]'' is defined for each consumer; a consumer's indifference curve contains all the baskets of commodities that the consumer regards as equivalent: That is, for every pair of baskets on the same indifference curve, the consumer does not prefer one basket over another. Through each basket of commodities passes one indifference curve. A consumer's ''preference set'' (relative to an indifference curve) is the [[union (set theory)|union]] of the indifference curve and all the commodity baskets that the consumer prefers over the indifference curve. A consumer's ''preferences'' are ''convex'' if all such preference sets are convex.<ref>{{harvtxt|Mas-Colell|1985|pp=58–61}} and {{harvtxt|Arrow|Hahn|1980|pp=76–79}}</ref>
| |
| | |
| An optimal basket of goods occurs where the budget-line [[supporting hyperplane|supports]] a consumer's preference set, as shown in the diagram. This means that an optimal basket is on the highest possible indifference curve given the budget-line, which is defined in terms of a price vector and the consumer's income (endowment vector). Thus, the set of optimal baskets is a <!-- set valued function, or multifunction, or correspondence or relation-->[[function (mathematics)|function]] of the <!-- relative --> prices, and this function is called the consumer's ''[[demand]]''. If the preference set is convex, then at every price the consumer's demand is a convex set, for example, a unique optimal basket or a line-segment of baskets.<ref>{{harvtxt|Arrow|Hahn|1980|pp=79–81}}</ref>
| |
| | |
| ====Non-convex preferences====
| |
| [[File:NonConvex.gif|right|thumb|300px|alt=Image of a non-convex preference set with a concavity un-supported by the budget line|When the consumer's preferences have concavities, the consumer may jump between two separate optimal baskets.]]
| |
| {{See also|Non-convexity (economics)}}
| |
| However, if a preference set is ''non-convex'', then some prices determine a budget-line that supports two ''separate'' optimal-baskets. For example, we can imagine that, for zoos, a lion costs as much as an eagle, and further that a zoo's budget suffices for one eagle or one lion. We can suppose also that a zoo-keeper views either animal as equally valuable. In this case, the zoo would purchase either one lion or one eagle. Of course, a contemporary zoo-keeper does not want to purchase half of an eagle and half of a lion (or a [[griffin]])! Thus, the zoo-keeper's preferences are non-convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both.<ref>{{harvtxt|Starr|1969|p=26}}: "After all,
| |
| one may be indifferent between an automobile and a boat, but in most cases one can neither drive nor sail the combination of half boat, half car."</ref>
| |
| | |
| When the consumer's preference set is non-convex, then (for some prices) the consumer's demand is not [[connected space|connected]]; a disconnected demand implies some discontinuous behavior by the consumer, as discussed by [[Harold Hotelling]]:
| |
| <blockquote>
| |
| If indifference curves for purchases be thought of as possessing a wavy character, convex to the origin in some regions and concave in others, we are forced to the conclusion that it is only the portions convex to the origin that can be regarded as possessing any importance, since the others are essentially unobservable. They can be detected only by the discontinuities that may occur in demand with variation in price-ratios, leading to an abrupt jumping of a point of tangency across a chasm when the straight line is rotated. But, while such discontinuities may reveal the existence of chasms, they can never measure their depth. The concave portions of the indifference curves and their many-dimensional generalizations, if they exist, must forever remain in
| |
| unmeasurable obscurity.<ref>
| |
| {{harvtxt|Hotelling|1935|p=74}}:
| |
| {{cite journal|first=Harold|last=Hotelling|authorlink=Harold Hotelling
| |
| |title=Demand functions with limited budgets|journal=Econometrica|volume=3|issue=1|date=January 1935|pages=66–78|jstor=1907346}}
| |
| </ref>
| |
| </blockquote>
| |
| <!-- [[File:Convex.gif|right|300px|alt=An image of a convex preference set being supported by a budget line.|With "quasi-equilibrium" prices, the budget-line [[supporting hyperplane|supports]] the convex hull of the [[indifference curve]].]] -->
| |
| The difficulties of studying non-convex preferences were emphasized by [[Herman Wold]]<ref>{{harvtxt|Wold|1943b|pp=231 and 239–240}}: {{cite journal|last=Wold|first=Herman|authorlink=Herman Wold|year=1943b|title=A synthesis of pure demand analysis '''II'''|journal=Skandinavisk Aktuarietidskrift [Scandinavian Actuarial Journal]|volume=26|pages=220–263<!-- Diewert gives wrong pages, according to Math Rev and my inspection of the article and of Wold's book with Jureen -->|mr=11939|ref=harv}}<p>{{harvtxt|Wold|Juréen|1953|p=146}}: {{cite book|last1=Wold|first1=Herman|authorlink1=Herman Wold|last2=Juréen|first2=Lars (in association with Wold)|chapter=8 Some further applications of preference fields (pp. 129–148)|title=Demand analysis: A study in econometrics|location=New York|publisher=John Wiley and Sons, Inc|series=Wiley publications in statistics|year=1953|pages=xvi+358|mr=64385|ref=harv}}<p/></ref> and again by [[Paul Samuelson]], who wrote that non-convexities are "shrouded in eternal {{nowrap|darkness ...",}}<ref>{{harvtxt|Samuelson|1950|pp=359–360}}:<blockquote>It will be noted that any point where the indifference curves are convex rather than concave cannot be observed in a competitive market. Such points are shrouded in eternal darkness—unless we make our consumer a monopsonist and let him choose between goods lying on a very convex "budget curve" (along which he is affecting the price of what he buys). In this monopsony case, we could still deduce the slope of the man's indifference curve from the slope of the observed constraint at the equilibrium point.</blockquote>{{cite journal|last=Samuelson|first=Paul A.|authorlink=Paul Samuelson|title=The problem of integrability in utility theory|journal=Economica|series=New Series|volume=17|issue=68|date=November 1950|pages=355–385|mr=43436|jstor=2549499|ref=harv}}<p>"Eternal darkness" describes the Hell of [[John Milton]]'s ''[[Paradise Lost]]'', whose concavity is compared to the [[Serbonian Bog]] in [[wikisource:Paradise Lost (1674)/Book II|Book II, lines 592–594]]:</p><blockquote>A gulf profound as that Serbonian Bog<br />Betwixt Damiata and <!-- correcting failed-capitalization in Arrow Hahn "m" (sic) -->Mount Casius old,<br />Where Armies whole have sunk.</blockquote>Milton's description of concavity serves as the [[epigraph (literature)|literary epigraph]] prefacing chapter seven of {{harvtxt|Arrow|Hahn|1971|p=169}}, "Markets with non-convex preferences and production", which presents the results of {{harvtxt|Starr|1969}}.</ref> according to Diewert.<ref name="Diewert" >{{harvtxt|Diewert|1982|pp=552–553}}</ref>
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| | |
| Nonetheless, non-convex preferences were illuminated from 1959 to 1961 by a sequence of papers in ''[[The Journal of Political Economy]]'' (''JPE''). The main contributors were <!-- M. J. -->Farrell,<ref>{{cite journal
| |
| |title=The Convexity assumption in the theory of competitive markets
| |
| |last=Farrell
| |
| |first=M. J.
| |
| |journal=[[The Journal of Political Economy]]
| |
| |volume=67
| |
| |issue =4
| |
| |month=August
| |
| |pages=371–391
| |
| |year=1959
| |
| |jstor=1825163
| |
| }}
| |
| {{cite journal
| |
| |title=On Convexity, efficiency, and markets: A Reply
| |
| |last=Farrell
| |
| |first=M. J.
| |
| <!-- |journal=The Journal of Political Economy -->
| |
| |volume=69
| |
| |issue=5
| |
| |month=October
| |
| |pages=484–489
| |
| |year=1961a
| |
| |jstor=1828538
| |
| }}
| |
| {{cite journal
| |
| |title=The Convexity assumption in the theory of competitive markets: Rejoinder
| |
| |last=Farrell
| |
| |first=M. J.
| |
| <!-- |journal=The Journal of Political Economy -->
| |
| |volume=69
| |
| |issue=5
| |
| |month=October
| |
| |pages=493
| |
| |year=1961b
| |
| |jstor=1828541
| |
| }}</ref><!-- F. M. --> Bator,<ref>{{cite journal|title=On convexity, efficiency, and markets|last=Bator|first=Francis M.|journal=The Journal of Political Economy|volume=69|issue =5|month=October|pages=480–483|year=1961a|jstor=1828537}} {{cite journal|title=On convexity, efficiency, and markets: Rejoinder|last=Bator|first=Francis M.|<!-- journal=The Journal of Political Economy -->|volume=69|issue =5|month=October|pages=489|year=1961b|jstor=1828539}}</ref> [[Tjalling Koopmans|<!-- T. C. -->Koopmans]],<ref>{{cite journal|title=Convexity assumptions, allocative efficiency, and competitive equilibrium
| |
| |last=Koopmans
| |
| |first=Tjalling C.
| |
| |authorlink=Tjalling Koopmans
| |
| |journal=The Journal of Political Economy
| |
| |volume=69
| |
| |issue=5
| |
| |month=October
| |
| |pages=478–479
| |
| |year=1961
| |
| |jstor=1828536
| |
| |ref=harv}}<p>{{harvtxt|Koopmans|1961|p=478}} and others—for example, {{harvtxt|Farrell|1959|pp=390–391}} and {{harvtxt|Farrell|1961a|p=484}}, {{harvtxt|Bator|1961|pp=482–483}}, {{harvtxt|Rothenberg|1960|p=438}}, and {{harvtxt|Starr|1969|p=26}}—commented on {{harvtxt|Koopmans|1957|pp=1–126, especially 9–16 [1.3 Summation of opportunity sets], 23–35 [1.6 Convex sets and the price implications of optimality], and 35–37 [1.7 The role of convexity assumptions in the analysis]}}:<p>{{cite book|last=Tjalling C.|first=Koopmans|authorlink=Tjalling Koopmans|chapter=Allocation of resources and the price system|editor-last=Koopmans|editor-first=Tjalling C|editor-link=Tjalling Koopmans|title=Three essays on the state of economic science|publisher=McGraw–Hill Book Company|location=New York|pages=1–126|year=1957|isbn=0-07-035337-9}}<p/>
| |
| </ref> and <!-- J --><!-- urban economist Jerome, not structural econometrician T.J. --><!-- . -->Rothenberg.<ref name="Rothenberg" >{{harvtxt|Rothenberg|1960|p=447}}: {{cite journal
| |
| |title=Non-convexity, aggregation, and Pareto optimality
| |
| |last=Rothenberg
| |
| |first=Jerome
| |
| |journal=The Journal of Political Economy
| |
| |volume=68
| |
| |issue=5
| |
| |month=October
| |
| |pages=435–468
| |
| |year=1960
| |
| |jstor=1830308
| |
| }} ({{cite journal
| |
| |title=Comments on non-convexity
| |
| |last=Rothenberg
| |
| |first=Jerome
| |
| |authorlink=<!-- |journal=The Journal of Political Economy -->
| |
| |volume=69
| |
| |issue=5
| |
| |month=October
| |
| |pages=490–492
| |
| |year=1961
| |
| |jstor=1828540
| |
| }})
| |
| </ref> In particular, Rothenberg's paper discussed the approximate convexity of sums of non-convex sets.<ref name="ArrowHahn182" >{{harvtxt|Arrow|Hahn|1980|p=182}}</ref> These <!-- ''Journal of Political Economy'' --> ''JPE''-papers stimulated a paper by [[Lloyd Shapley]] and [[Martin Shubik]], which considered convexified consumer-preferences and introduced the concept of an "approximate equilibrium".<ref>{{harvtxt|Shapley|Shubik|1966|p=806}}: {{cite journal|authorlink1=Lloyd Shapley|first1=L. S.| last1=Shapley|authorlink2=Martin Shubik|first2=M.|last2=Shubik|title=Quasi-cores in a monetary economy with nonconvex preferences|journal=Econometrica|volume=34|issue=4|date=October 1966|pages=805–827|jstor=1910101|zbl=154.45303|ref=harv|doi=10.2307/1910101}}</ref> The <!-- ''Journal of Political Economy'' -->''JPE''-papers and the Shapley–Shubik paper influenced another notion of "quasi-equilibria", due to [[Robert Aumann]].<ref name="Aumann" >{{harvtxt|Aumann|1966|pp=1–2}}: {{cite journal|authorlink=Robert Aumann|first=Robert J.|last=Aumann|title=Existence of competitive equilibrium in markets with a continuum of traders|journal=Econometrica|volume=34|issue=1|date=January 1966|pages=1–17|jstor=1909854|mr=191623|ref=harv}} {{harvtxt|Aumann|1966}} uses results from
| |
| {{harvs|txt|last=Aumann|year1=1964|year2=1965}}:
| |
| <!-- NOT original research, this comment appears often, e.g. in "What is Bob Aumann trying to accomplish" in the CORE 20th anniversary volume, in which Guesnerie appears --><p>{{cite journal|authorlink=Robert Aumann|first=Robert J.|last=Aumann|title=Markets with a continuum of traders|journal=Econometrica|volume=32|issue=1–2|date=January–April 1964|pages=39–50|jstor=1913732|mr=172689|ref=harv}}
| |
| <p>{{cite journal|authorlink=Robert Aumann|first=Robert J.|last=Aumann|title=Integrals of set-valued functions|journal=Journal of Mathematical Analysis and Applications|volume=12|issue=1|date=August 1965|pages=1–12|url=http://www.sciencedirect.com/science/article/B6WK2-4CRJ2XG-1D4/2/761eda1b7acffb52fde213d766059f3c|doi=10.1016/0022-247X(65)90049-1|MR=185073|ref=harv}}</ref><ref>Taking the convex hull of non-convex preferences had been discussed earlier by {{harvtxt|Wold|1943b|p=243}} and by {{harvtxt|Wold|Juréen|1953|p=146}}, according to {{harvtxt|Diewert|1982|p=552}}.</ref>
| |
| | |
| ==== Starr's 1969 paper and contemporary economics ====
| |
| <!-- [[File:Price of market balance.gif|thumb|right|alt=Diagram of an increasing supply curve and a decreasing demand curve, which intersect at the equilibrium.|At an [[economic equilibrium|equilibrium price]] ''P''0, the [[Supply and demand|quantity supplied ''S''(''P''0) equals the quantity demanded ''D''(''P''0)]].]] -->
| |
| [[File:Kenneth Arrow, Stanford University.jpg|thumb|alt=Picture of Kenneth Arrow|[[Kenneth Arrow]] (1972 [[Nobel Prize in Economics|Nobel laureate]]) helped [[Ross M. Starr]] to study [[convex set|non-convex]] [[convex preferences|economies]].<ref name="StarrArrow"/>]]
| |
| | |
| Previous publications on [[non-convexity (economics)|non-convexity and economics]] were collected in an annotated bibliography by [[Kenneth Arrow]]. He gave the bibliography to [[Ross Starr|Starr]], who was then a<!-- [[Stanford University|Stanford]] -->n undergraduate enrolled in Arrow's (graduate) advanced mathematical-economics course.<ref name="StarrArrow" >{{harvtxt|Starr|Stinchcombe|1999|pp=217–218}}: {{cite book|chapter=Exchange in a network of trading posts|last1=Starr|first1=R. M.|authorlink1=Ross Starr|last2=Stinchcombe|first2=M. B.|title=Markets, information and uncertainty: Essays in economic theory in honor of Kenneth J. Arrow|editor-first=Graciela|editor-last=Chichilnisky|editor-link=Graciela Chichilnisky|pages=217–234|publisher=Cambridge University Press|location=Cambridge|year=1999|doi=10.2277/0521553555|isbn=978-0-521-08288-4|ref=harv}}
| |
| </ref> In his term-paper, Starr studied the general equilibria of an artificial economy in which non-convex preferences were replaced by their convex hulls. In the convexified economy, at each price, the <!-- not necessarily closed -->[[aggregate demand]] was the sum of convex hulls of the consumers' demands. Starr's ideas interested the mathematicians [[Lloyd Shapley]] and [[Jon Folkman]], who proved their [[eponym]]ous <!-- Shapley–Folkman --> lemma and <!-- the Shapley–Folkman --> theorem in "private correspondence", <!-- entitled "Starr's problem" (1966), --> which was reported by Starr's published paper of 1969.<ref name="s69"/>
| |
| | |
| In his 1969 publication, Starr applied the Shapley–Folkman–Starr theorem. Starr proved that the "convexified" economy has general equilibria that can be closely approximated by "''quasi-equilbria''" of the original economy, when the number of agents exceeds the dimension of the goods: Concretely, Starr proved that there exists at least one quasi-equilibrium of prices ''p''<sub>opt</sub> with the following properties:
| |
| | |
| * For each quasi-equilibrium's prices ''p''<sub>opt</sub>, all consumers can choose optimal baskets (maximally preferred and meeting their budget constraints).
| |
| | |
| * At quasi-equilibrium prices ''p''<sub>opt</sub> in the convexified economy, every good's market is in equilibrium: Its supply equals its demand.
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| | |
| * For each quasi-equilibrium, the prices "nearly clear" the markets for the original economy: an [[upper bound]] on the [[Hausdorff distance|distance]] between the set of equilibria of the "convexified" economy and the set of quasi-equilibria of the original economy followed from Starr's corollary to the Shapley–Folkman theorem.<ref>{{harvtxt|Arrow|Hahn|1980|pp=169–182}}. {{harvtxt|Starr|1969|pp=27–33}}
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| </ref>
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| Starr established that
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| <blockquote>
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| "in the aggregate, the discrepancy between an allocation in the fictitious economy generated by [taking the convex hulls of all of the consumption and production sets] and some allocation in the real economy is bounded in a way that is independent of the number of economic agents. Therefore, the average agent experiences a deviation from intended actions that vanishes in significance as the number of agents goes to infinity".<ref>{{harvtxt|Green|Heller|1981|p=44}}</ref>
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| </blockquote>
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| Following Starr's 1969 paper, the Shapley–Folkman–Starr results have been widely used in economic theory. [[Roger Guesnerie]] summarized their economic implications: "<!-- [s] -->Some key results obtained under the convexity assumption remain (approximately) relevant in circumstances where convexity fails. For example, in economies with a large consumption side, preference nonconvexities do not destroy the standard results".<ref>{{harvtxt|Guesnerie|1989|pp=99}}</ref> "The derivation of these results in general form has been one of the major achievements of postwar economic theory", wrote Guesnerie.<ref name="g89-p138"/> The topic of [[non-convexity (economics)|non-convex sets in economics]] has been studied by many [[Nobel Prize in Economics|Nobel laureates]]: Arrow (1972), [[Robert Aumann]] (2005), [[Gérard Debreu]] (1983), [[Tjalling Koopmans]] (1975), [[Paul Krugman]] (2008), and [[Paul Samuelson]] (1970); the complementary topic of [[convexity in economics|convex sets in economics]] has been emphasized by these laureates, along with [[Leonid Hurwicz]], [[Leonid Kantorovich]] (1975), and [[Robert Solow]] (1987).<ref name="MasColell87">{{harvtxt|Mas-Colell|1987}}</ref> The Shapley–Folkman–Starr results have been featured in the economics literature: in [[microeconomics]],<ref>{{harvtxt|Varian|1992|pp=393–394}}: {{cite book|authorlink=Hal Varian|last=Varian|first=Hal R.|chapter=21.2 Convexity and size|title=Microeconomic Analysis|publisher=W. W. Norton & Company|edition=3rd|year=1992|isbn=978-0-393-95735-8|mr=1036734}}<p>{{harvtxt|Mas-Colell|Whinston|Green|1995|pp=627–630}}: {{cite book|last1=Mas-Colell|first1=Andreu|authorlink=Andreu Mas-Colell|last2=Whinston|first2=Michael D.|first3=Jerry R.|last3=Green|chapter=17.1 Large economies and nonconvexities|title=Microeconomic theory|publisher=Oxford University Press|year=1995|isbn=978-0-19-507340-9}}</ref> in general-equilibrium theory,<ref>{{harvtxt|Arrow|Hahn|1980|pp=169–182}}<p>{{harvtxt|Mas-Colell|1985|pp=52–55, 145–146, 152–153, and 274–275}}: {{cite book|last=Mas-Colell|first=Andreu|authorlink=Andreu Mas-Colell|year=1985|chapter=1.L Averages of sets|title=The Theory of general economic equilibrium: A ''differentiable'' approach|series=Econometric Society monographs|volume=9|publisher=Cambridge University Press|isbn=0-521-26514-2|mr=1113262|ref=harv}}</p><p>{{harvtxt|Hildenbrand|1974|pp=37, 115–116, 122, and 168}}: {{cite book|last=Hildenbrand|first=Werner|authorlink=Werner Hildenbrand|title=Core and equilibria of a large economy|series=Princeton studies in mathematical economics|volume=5|publisher=Princeton University Press|location=Princeton, N.J.|year=1974|pages=viii+251|isbn=978-0-691-04189-6|mr=389160}}</p></ref><ref>{{harvtxt|Starr|1997|p=169}}: {{cite book|last=Starr|first=Ross M.|chapter=8 Convex sets, separation theorems, and non-convex sets in '''R'''<sup>''N''</sup> (new chapters 22 and 25–26 in (2011) second ed.)|title=General equilibrium theory: An introduction|edition=First|publisher=Cambridge University Press|location=Cambridge|year=1997|pages=xxiii+250|isbn=0-521-56473-5|mr=1462618|ref=harv}}<p>{{harvtxt|Ellickson|1994|pp=xviii, 306–310, 312, 328–329, 347, and 352}}: {{cite book|title=Competitive equilibrium: Theory and applications|first=Bryan|last=Ellickson |publisher=Cambridge University Press|isbn=978-0-521-31988-1|doi=10.2277/0521319889|year=1994|pages=|ref=harv}}</p></ref> in [[public economics]]<ref>{{harvtxt|Laffont|1988|pp=63–65}}: {{cite book|last=Laffont|first=Jean-Jacques|authorlink=Jean-Jacques Laffont|year=1988|chapter=3 Nonconvexities <!-- Not "Non–convexities" -->|title=Fundamentals of public economics|url=http://books.google.com/books?q=editions:ISBN 0-262-12127-1&id=O5MnAQAAIAAJ|publisher=[http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=7534 MIT]
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| |isbn=0-262-12127-1|ref=harv}}</ref> (including [[market failure]]s),<ref>{{harvtxt|Salanié|2000|pp=112–113 and 107–115}}: {{cite book|last=Salanié|first=Bernard|chapter=7 Nonconvexities <!-- Not "Non–convexities" -->|title=Microeconomics of market failures|edition=English translation of the (1998) French ''Microéconomie: Les défaillances du marché'' (Economica, Paris)|year=2000|publisher=MIT Press|location=Cambridge, MA|pages=107–125|isbn=0-262-19443-0|ref=harv}}</ref> as well as in [[game theory]],<ref>{{harvtxt|Ichiishi|1983|pp=24–25}}: {{cite book|last=Ichiishi|first=Tatsuro|title=Game theory for economic analysis|series=Economic theory, econometrics, and mathematical economics|publisher=Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]|location=New York|year=1983|pages=x+164|isbn=0-12-370180-5|mr=700688|ref=harv}}</ref> in [[mathematical economics]],<ref>{{harvtxt|Cassels|1981|pp=127 and 33–34}}: {{cite book|last=Cassels|first=J. W. S.|authorlink=J. W. S. Cassels|chapter=Appendix A Convex sets|title=Economics for mathematicians|series=London Mathematical Society lecture note series|volume=62|publisher=Cambridge University Press|location=Cambridge, New York|year=1981|pages=xi+145|isbn=0-521-28614-X|mr=657578|ref=harv}}</ref> and in [[applied mathematics#Mathematics for economists|applied mathematics]] (for economists).<ref name="Aubin"/><ref name="Carter" >{{harvtxt|Carter|2001|pp=93–94, 143, 318–319, 375–377, and 416}}</ref><!-- <ref name="Moore">{{harvtxt|Moore|1999|p=309}}: {{cite book|last=Moore|first=James C.|title=Mathematical methods for economic theory: Volume '''I'''
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| |series=Studies in economic theory|volume=9|publisher=Springer-Verlag|location=Berlin|year=1999|pages=xii+414|isbn=3-540-66235-9 |MR=1727000|ref=harv}}</ref><ref>{{harvtxt|Florenzano|Le Van|2001|pp=47–48}}: {{cite book|MR=1878374|last1=Florenzano|first1=Monique|last2=Le Van|first2=Cuong|title=Finite dimensional convexity and optimization|author3=in cooperation with Pascal Gourdel|series=Studies in economic theory|volume=13|publisher=Springer-Verlag|location=Berlin|year=2001|pages=xii+154|isbn=3-540-41516-5|ref=harv}} </ref> --> The Shapley–Folkman–Starr results have also influenced economics research using [[measure (mathematics)|measure]] and [[integral|integration theory]].<ref>{{harvtxt|Trockel|1984|p=30}}: {{cite book|last=Trockel|first=Walter|title=Market demand: An analysis of large economies with nonconvex preferences|series=Lecture notes in economics and mathematical systems|volume=223|publisher=Springer-Verlag|location=Berlin|year=1984|pages=viii+205|isbn=3-540-12881-6|mr=737006}}</ref>
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| ===Mathematical optimization===
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| [[File:Epigraph convex.svg|right|thumb|300px|alt=A graph of a convex function, which is drawn in black. Its epigraph, the area above its graph, is solid green.|A [[function (mathematics)|function]] is [[convex function|convex]] if the region above its [[graph of a function|graph]] is a [[convex set]].]]
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| The Shapley–Folkman lemma has been used to explain why large [[nonlinear programming|minimization]] problems with [[convex function|non-convexities]] can be nearly solved (with [[iterative methods]] whose convergence proofs are stated for only [[convex optimization|convex problems]]). The Shapley–Folkman lemma has encouraged the use of methods of convex minimization on other applications with sums of many functions.<ref name="Bertsekas99"/>
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| | |
| ====Preliminaries of optimization theory====
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| [[Nonlinear programming|Nonlinear optimization]] relies on the following definitions for [[function (mathematics)|function]]s:
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| *The [[graph of a function|''graph'']] of a function ''f'' is the set of the pairs of [[domain of a function|argument]]s ''x'' and function evaluations ''f''(''x'')
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| : Graph(''f'') = <big><big>{</big></big> <big>(</big>''x'', ''f''(''x'') <big>)</big> <big><big>}</big></big>
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| * The ''[[epigraph (mathematics)|epigraph]]'' of a [[real-valued function]] ''f'' is the set of points ''above'' the graph
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| [[File:Sine.svg|right|thumb|alt=A graph of the sine function, which periodically oscillates up and down between −1 and +1, with the period 2π.|The [[sine|sine function]] is [[convex function|non-convex]]<!-- on the [[interval_(mathematics)#Terminology|interval]] (0, π) -->.]]
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| : Epi(''f'') = <big>{</big> (''x'', ''u'') : ''f''(''x'') ≤ ''u'' <big>}</big>.
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| *A real-valued function is defined to be a ''[[convex function]]'' if its epigraph is a convex set.<ref name="Rock23" >{{harvtxt|Rockafellar|1997|p=23}}</ref>
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| For example, the [[quadratic function]] ''f''(''x'') = ''x''<sup>2</sup> is convex, as is the [[absolute value]] function ''g''(''x'') = |''x''|. However, the [[sine|sine function]] (pictured) is non-convex on the [[interval (mathematics)#Terminology|interval]] (0, π).
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| ====Additive optimization problems====
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| In many optimization problems, the [[optimization (mathematics)#objective function|objective function]] f is ''separable'': that is, ''f'' is the sum of ''many'' summand-functions, each of which has its own argument:
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| : ''f''(''x'') = ''f''<big>(</big> (''x''<sub>1</sub>, ..., ''x''<sub>''N''</sub>)<big> )</big> = <big>∑</big> ''f''<sub>''n''</sub>(''x''<sub>''n''</sub>).
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| For example, problems of [[linear programming|linear optimization]] are separable. Given a separable problem with an optimal solution, we fix an optimal solution
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| : ''x''<sub>min</sub> = (''x''<sub>1</sub>, ..., ''x''<sub>''N''</sub>)<sub>min</sub>
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| with the minimum value {{nowrap|''f''(''x''<sub>min</sub>).}} For this separable problem, we also consider an optimal solution <big>(</big>''x''<sub>min</sub>, ''f''(''x''<sub>min</sub>) <big>)</big>
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| to the "''convexified problem''", where convex hulls are taken of the graphs of the summand functions. Such an optimal solution is the [[limit of a sequence]] of points in the convexified problem
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| : <big>(</big>''x''<sub>''j''</sub>, ''f''(''x''<sub>j</sub>) <big>)</big><big> ∈ </big> <big>∑</big> Conv <big>(</big>Graph( ''f''<sub>''n''</sub> ) <big>)</big>.<ref name="Ekeland76"/><ref>
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| The [[limit of a sequence]] is a member of the [[closure (topology)|closure of the original set]], which is the smallest [[closed set]] that contains the original set. The Minkowski sum of two [[closed set]]s need not be closed, so the following [[subset#inclusion|inclusion]] can be strict
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| : Clos(P) + Clos(Q) ⊆ Clos( Clos(P) + Clos(Q) );
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| the inclusion can be strict even for two ''convex'' closed summand-sets, according to {{harvtxt|Rockafellar|1997|pp=49 and 75}}. Ensuring that the Minkowski sum of sets be closed requires the closure operation, which appends limits of convergent sequences.</ref>
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| Of course, the given optimal-point is a sum of points in the graphs of the original summands and of a small number of convexified summands, by the Shapley–Folkman lemma.
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| This analysis was published by [[Ivar Ekeland]] in 1974 to explain the apparent convexity of separable problems with many summands, despite the non-convexity of the summand problems. In 1973, the young mathematician [[Claude Lemaréchal]] was surprised by his success with [[convex optimization|convex minimization]] [[iterative method|method]]s on problems that were known to be non-convex; for [[nonlinear programming|minimizing nonlinear]] problems, a solution of the [[dual problem]] problem need not provide useful information for solving the primal problem, unless the primal problem be convex and satisfy a [[constraint qualification]]. Lemaréchal's problem was additively separable, and each summand function was non-convex; nonetheless, a solution to the dual problem provided a close approximation to the primal problem's optimal value.<ref>{{harvtxt|Lemaréchal|1973|p=38}}: {{citation|last=Lemaréchal|first=Claude|authorlink=Claude Lemaréchal|title=Utilisation de la dualité dans les problémes non convexes [Use of duality for non–convex problems]|language=French|year=1973|month=''Avril'' [April]|issue=16|location=Domaine de Voluceau, [[Rocquencourt]], 78150 [[Le Chesnay|Le Chesnay]], France|publisher=[[National Institute for Research in Computer Science and Control|IRIA (now INRIA)]], Laboratoire de recherche en informatique et automatique|page=41|ref=harv}}. <!-- Ekeland cites this report in the ''[[Comptes Rendus|CRAS]]'' announcement of the results of his Appendix I -->
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| Lemaréchal's experiments were discussed in later publications: <p>{{harvtxt|Aardal|1995|pp=2–3}}: {{cite journal|first=Karen|last=Aardal|title=''Optima'' interview <!--sic., neither colon nor m-dash appear -->Claude Lemaréchal|journal=Optima: Mathematical Programming Society newsletter|pages=2–4|date=March 1995|volume=45|url=http://www.mathprog.org/Old-Optima-Issues/optima45.pdf|accessdate=2 February 2011|ref=harv}}</p><p>{{harvtxt|Hiriart-Urruty|Lemaréchal|1993|pp=143–145, 151, 153, and 156}}: {{cite book|last1=Hiriart-Urruty|first1=Jean-Baptiste|last2=Lemaréchal|first2=Claude|authorlink2=Claude Lemaréchal|chapter=XII Abstract duality for practitioners|title=Convex analysis and minimization algorithms, Volume '''II''': Advanced theory and bundle methods|series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]|volume=306|publisher=Springer-Verlag|location=Berlin|year=1993|pages=136–193 (and bibliographical comments on pp. 334–335)|isbn=3-540-56852-2|mr=1295240}}</p>
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| </ref><ref name="Ekeland76" >{{harv|Ekeland|1999|pp=357–359}}: Published in the first English edition of 1976, Ekeland's appendix proves the Shapley–Folkman lemma, also acknowledging [[Claude Lemaréchal|Lemaréchal]]'s experiments on page 373.</ref><ref name="Ekeland74" >{{cite journal|last=Ekeland|first=Ivar|<!-- authorlink=Ivar Ekeland -->|title=Une estimation {{nowrap|''a priori''}} en programmation {{nowrap|non convexe}}|journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences|series=Séries A et B|language=French|issn=0151-0509<!-- Not archived on CRAS site, and CRAS's title has changed multiple times -->|volume=279|year=1974|pages=149–151|MR=395844|ref=harv}}<!-- Ekeland sole author; cites Lemaréchal by title whereas the 1976 appendix acknowledges but does not cite Lemaréchal (1973)--></ref> Ekeland's analysis explained the success of methods of convex minimization on ''large'' and ''separable'' problems, despite the non-convexities of the summand functions. Ekeland and later authors argued that additive separability produced an approximately convex aggregate problem, even though the summand functions were non-convex. The crucial step in these publications is the use of the Shapley–Folkman lemma.<ref name="Ekeland76" /><ref name="Ekeland74" /><ref name="AubinEkeland" >{{harvtxt|Aubin|Ekeland|1976|pp=226, 233, 235, 238, and 241}}: {{cite journal|last1=Aubin|first1=J. P.|last2=Ekeland|first2=I.|issue=3|journal=Mathematics of Operations Research|pages=225–245|title=Estimates of the duality gap in nonconvex optimization|volume=1| year = 1976
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| |doi=10.1287/moor.1.3.225|mr=449695|jstor=3689565|ref=harv}}<p>{{harvtxt|Aubin|Ekeland|1976}} and {{harvtxt|Ekeland|1999|pp=362–364}} also considered the ''[[Convex conjugate#Biconjugate convex|convex]]'' [[Convex conjugate#Biconjugate|closure]] of a problem of non-convex minimization—that is, the problem defined as the [[Kuratowski closure axioms|closed]] [[convex hull|convex]] [[closure operator|hull]] of the [[epigraph (mathematics)|epigraph]] of the original problem. Their study of duality gaps was extended by Di Guglielmo to the ''[[quasiconvex function|quasiconvex]]'' closure of a non-convex [[multiobjective optimization|minimization]] problem—that is, the problem defined as the [[Kuratowski closure axioms|closed]] [[convex hull|convex]] [[closure operator|hull]] of the [[semicontinuity#lower|lower]] [[level set|level set]]s:<p/><p>{{harvtxt|Di Guglielmo|1977|pp=287–288}}: {{cite journal|last=Di Guglielmo|first=F.|title=Nonconvex duality in multiobjective optimization|doi=10.1287/moor.2.3.285|volume=2|year=1977|issue=3|pages=285–291|journal=Mathematics of Operations Research|mr=484418|jstor=3689518}}<p/>
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| <!-- <p>{{cite book||last=Di Guglielmo|first=F.|chapter=Estimates of the duality gap for discrete and quasiconvex optimization problems|title=Generalized concavity in optimization and economics: Proceedings of the NATO Advanced Study Institute held at the University of British Columbia, Vancouver, B.C., August 4–15, 1980
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| |editor1-first=Siegfried|editor1-last=Schaible|editor2-first=William T.|editor2-last=Ziemba|publisher=Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers]|location=New York|year=1981|pages=281–298|isbn=0-12-621120-5|MR=652702|}}</p> -->
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| </ref> The Shapley–Folkman lemma has encouraged the use of methods of convex minimization on other applications with sums of many functions.<ref name="Ekeland76" /><ref name="Bertsekas82" >{{harvtxt|Bertsekas|1996|pp=364–381}} acknowledging {{harvtxt|Ekeland|1999}} on page 374 and {{harvtxt|Aubin|Ekeland|1976}} on page 381:<p>
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| {{cite book|last=Bertsekas|first=Dimitri P.|authorlink=Dimitri P. Bertsekas|chapter=5.6 Large scale separable integer programming problems and the exponential method of multipliers|title=Constrained optimization and Lagrange multiplier methods|edition=Reprint of (1982) Academic Press|year=1996|location=Belmont, MA|isbn=1-886529-04-3|pages=xiii+395|publisher=Athena Scientific|mr=690767|ref=harv}}</p>
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| <p>{{harvtxt|Bertsekas|1996|pp=364–381}} describes an application of [[dual problem|Lagrangian dual]] methods to the [[scheduling (production processes)|scheduling]] of [[electricity generation|electrical power plant]]s ("[[power system simulation#Unit commitment|unit commitment problem]]s"), where non-convexity appears because of [[integer programming|integer constraints]]:</p><p>{{cite journal|journal=IEEE Transactions on Automatic Control|volume=AC-28|date=January 1983|title=Optimal short-term scheduling of large-scale power systems|first1=Dimitri P.|last1=Bertsekas|authorlink1=Dimitri Bertsekas|first2=Gregory S.|last2=Lauer|first3=Nils R., Jr.|last3=Sandell|first4=Thomas A.|last4=Posbergh|pages=1–11|
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| issue=Proceedings of 1981 IEEE Conference on Decision and Control, San Diego, CA, December 1981, pp. 432–443|ref=harv|url=http://web.mit.edu/dimitrib/www/Unit_Comm.pdf|accessdate=2 February 2011}}<p/></ref><ref name="Aubin" >{{harvtxt|Aubin|2007|pp=458–476}}: {{cite book|last=Aubin|first=Jean-Pierre|chapter=14.2 Duality in the case of non-convex integral criterion and constraints (especially 14.2.3 The Shapley–Folkman theorem, pages 463–465)|title=Mathematical methods of game and economic theory|edition=Reprint with new preface of 1982 North-Holland revised English|publisher=Dover Publications, Inc|location=Mineola, NY|year=2007|pages=xxxii+616|isbn=978-0-486-46265-3|mr=2449499|ref=harv}}</ref><ref name="Bertsekas99" >{{harvtxt|Bertsekas|1999|p=496}}: {{cite book|last=Bertsekas|first=Dimitri P.|authorlink=Dimitri P. Bertsekas
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| |title=Nonlinear Programming|edition=Second|chapter=5.1.6 Separable problems and their geometry|pages=494–498|publisher=Athena Scientific|year=1999|location=Cambridge, MA.|isbn =1-886529-00-0}}</ref>
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| ===Probability and measure theory===
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| Convex sets are often studied with [[probability theory]]. Each point in the convex hull of a ([[empty set|non-empty]]) subset ''Q'' of a finite-dimensional space is the [[expected value]] of a [[simple function|simple]] [[multivariate random variable|random vector]] that takes its values in ''Q'', as a consequence of [[Carathéodory's theorem (convex hull)|Carathéodory's lemma]].<!-- <ref>
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| This property (representation of points in convex sets via simple random variables) holds for closed and [[bounded set (topological vector space)|bounded set]]s in [[Banach space]]s with the [[Bochner_integral#Radon.E2.80.93Nikodym_property|Radon–Nikodym property]] (by [[Gerald Edgar|Edgar]]'s theorem) and for closed and [[totally bounded space|totally bounded set]]s of a [[locally convex topological vector space]] (by the [[Krein–Milman theorem]]).</ref> --> Thus, for a non-empty set ''Q'', the collection of the expected values of the simple, ''Q''-valued random vectors equals ''Q''{{'s}} convex hull; this equality implies that the Shapley–Folkman–Starr results are useful in probability theory.<ref>{{harvtxt|Schneider|Weil|2008|p=45}}: {{cite book|last1=Schneider|first1=Rolf|last2=Weil|first2=Wolfgang |title=Stochastic and integral geometry |url=http://www.springerlink.com/content/978-3-540-78858-4|series=Probability and its applications|doi=10.1007/978-3-540-78859-1|year=2008|publisher=Springer |isbn=978-3-540-78858-4|mr=2455326}}</ref> In the other direction, probability theory provides tools to examine convex sets generally and the Shapley–Folkman–Starr results specifically.<ref>{{harvtxt|Cassels|1975|pp=433–434}}: {{cite journal|last=Cassels| first=J. W. S.|authorlink=J. W. S. Cassels|title=Measures of the non-convexity of sets and the Shapley–Folkman–Starr theorem|journal=Mathematical Proceedings of the Cambridge Philosophical Society|volume=78|year=1975|issue=3|pages=433–436|doi=10.1017/S0305004100051884
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| |url=http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2075868&fulltextType=RA&fileId=S0305004100051884ER|mr=385711|ref=harv}}</ref> The Shapley–Folkman–Starr results have been widely used in the [[stochastic geometry|probabilistic theory of random sets]],<ref>{{harvtxt|Molchanov|2005|pp=195–198, 218, 232, 237–238 and 407}}: {{cite book|last=Molchanov|first=Ilya|chapter=3 Minkowski addition|title=Theory of random sets|series=Probability and its applications|publisher=Springer-Verlag London Ltd|location=London |year=2005|pages=194–240|isbn=978-1-84996-949-9|doi=10.1007/1-84628-150-4 |url=http://www.springerlink.com/content/978-1-85233-892-3|mr=2132405|ref=harv}}</ref> for example, to prove a [[law of large numbers|law of large numbers]],<ref name="ArtsteinVitale" >{{harvtxt|Artstein|Vitale|1975|pp=881–882}}: {{citation|last1=Artstein|first1=Zvi|last2=Vitale|first2=Richard A.|year=1975|title=A strong law of large numbers for random compact sets|journal=The Annals of Probability|volume=3|issue=5|pages=879–882 |url=http://projecteuclid.org/euclid.aop/1176996275|doi=10.1214/aop/1176996275|mr=385966|jstor=2959130|zbl=0313.60012|id={{Euclid|euclid.ss/1176996275}}|ref=harv}}</ref><ref name="PurRal85" >{{harvtxt|Puri|Ralescu|1985|pp=154–155}}: {{cite journal|last1=Puri|first1=Madan L.|last2=Ralescu|first2=Dan A.|title=Limit theorems for random compact sets in Banach space|url=http://journals.cambridge.org/action/displayAbstract?aid=2087952|journal=Mathematical Proceedings of the Cambridge Philosophical Society|volume=97|year=1985|issue=1|pages=151–158|doi=10.1017/S0305004100062691|mr=764504|ref=harv}}
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| </ref> a [[central limit theorem]],<ref name="PurRal85" /><ref>{{harvtxt|Weil|1982|pp=203, and 205–206}}: {{cite journal|last=Weil|first=Wolfgang|title=An application of the central limit theorem for Banach-space–valued random variables to the theory of random sets|journal=Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete [Probability Theory and Related Fields]|volume=60|year=1982 |issue=2|pages=203–208|doi=10.1007/BF00531823|mr=663901|ref=harv}}</ref> and a [[large deviations theory|large-deviations]] [[rate function|principle]].<ref>{{harvtxt|Cerf|1999|pp=243–244}}: {{cite journal|last=Cerf|first=Raphaël|title=Large deviations for sums of {{nowrap|i.i.d. random}} compact sets |url=http://www.ams.org/journals/proc/1999-127-08/S0002-9939-99-04788-7|journal=Proceedings of the American Mathematical Society|volume=127|year=1999|issue=8|pages=2431–2436|doi=10.1090/S0002-9939-99-04788-7|mr=1487361|ref=harv}} Cerf uses applications of the Shapley–Folkman lemma from {{harvtxt|Puri|Ralescu|1985|pp=154–155}}.</ref> These proofs of [[convergence of random variables|probabilistic limit theorems]] used the Shapley–Folkman–Starr results to avoid the assumption that all the random sets be convex.
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| A [[probability measure]] is a finite [[measure (mathematics)|measure]], and the Shapley–Folkman lemma has applications in non-probabilistic measure theory, such as the theories of [[volume]] and of [[vector measure]]s. The Shapley–Folkman lemma enables a refinement of the [[Brunn–Minkowski theorem|Brunn–Minkowski inequality]], which bounds the volume of sums in terms of the volumes of their summand-sets.<ref>{{harvtxt|Ruzsa|1997|p=345}}: {{cite journal|last=Ruzsa|first=Imre Z.|authorlink=Imre Z. Ruzsa|title=The Brunn–Minkowski inequality and nonconvex sets|journal=Geometriae Dedicata|volume=67|doi=10.1023/A:1004958110076 |year=1997|issue=3|pages=337–348|mr=1475877|ref=harv}}</ref> The volume of a set is defined in terms of the [[Lebesgue <!-- outer -->measure]], which is defined on <!-- measurable ; the, for outer measure -->subsets of [[Euclidean space]]. In advanced measure-theory, the Shapley–Folkman lemma has been used to prove [[Vector measure#Lyapunov's theorem|Lyapunov's theorem]], which states that the [[image (mathematics)|range]] of a <!-- ([[atom (measure theory)|non-atomic]]) -->[[vector measure]] is convex.<ref name="Tardella" >{{harvtxt|Tardella|1990|pp=478–479}}: {{cite journal|last=Tardella|first=Fabio|title=A new proof of the Lyapunov convexity theorem|journal=SIAM Journal on Control and Optimization|volume=28|year=1990|issue=2|pages=478–481 |doi=10.1137/0328026|mr=1040471|ref=harv}}</ref> Here, the traditional term "''range''" (alternatively, "image") is the set of values produced by the function.
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| A ''vector measure'' is a vector-valued generalization of a measure;
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| for example,
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| if ''p''<sub>1</sub> and ''p''<sub>2</sub> are [[probability measure]]s defined on the same [[measure (mathematics)#Measurable space|measurable space]],
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| then the [[product function]] {{nowrap|''p''<sub>1</sub> ''p''<sub>2</sub>}} is a vector measure,
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| where {{nowrap|''p''<sub>1</sub> ''p''<sub>2</sub>}}
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| is defined for every [[event (probability theory)|event]] ''ω''
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| by<!-- the assignment -->
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| :<big>(</big>''p''<sub>1</sub> ''p''<sub>2</sub><big>)</big>(''ω'')=<big>(</big>''p''<sub>1</sub>(''ω''), ''p''<sub>2</sub>(''ω'')<big>)</big>.
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| Lyapunov's theorem has been used in [[mathematical economics|economics]],<ref name="Aumann"/><ref>{{harvtxt|Vind|1964|pp=168 and 175}}: {{cite journal|last=Vind|first=Karl|year=1964|title=Edgeworth-allocations in an exchange economy with many traders|journal=International Economic Review|volume=5|pages=165–77|issue=2|month=May|ref=harv|jstor=2525560}} Vind's article was noted by the winner of the 1983 [[Nobel Prize in Economics]], [[Gérard Debreu]]. {{harvtxt|Debreu|1991|p=4}} wrote:
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| <blockquote>
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| The concept of a convex set (i.e., a set containing the segment connecting any two of its points) had repeatedly been placed at the center of economic theory before 1964. It appeared in a new light with the introduction of integration theory in the study of economic competition: If<!-- original "if" inconsistent with our capitalization --> one associates with every agent of an economy an arbitrary set in the commodity space and ''if one averages those individual sets'' over a collection of insignificant agents, ''then the resulting set is necessarily convex''. [Debreu appends this footnote: "On this direct consequence of a theorem of A. A. Lyapunov, see {{harvtxt|Vind|1964}}."] But explanations of the <!-- three --> ... functions of prices <!-- taken as examples --> ... can be made to rest on the ''convexity of sets derived by that averaging process''. ''Convexity'' in the commodity space ''obtained by aggregation'' over a collection of insignificant agents is an insight that economic theory owes <!-- in its revealing clarity --> ... to integration theory. [''Italics added'']
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| </blockquote>
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| {{cite journal|title=The Mathematization of economic theory|first=Gérard|last=Debreu|authorlink=Gérard Debreu|issue=Presidential address delivered at the 103rd meeting of the American Economic Association, 29 December 1990, Washington, DC|journal=The American Economic Review|volume=81|date=March 1991|pages=1–7|jstor=2006785|ref=harv}}</ref> in ([[bang–bang control|"bang-bang"]]) [[control theory]], and in [[statistical theory]].<ref name="Artstein" >{{harvtxt|Artstein|1980|pp=172–183}} {{harvtxt|Artstein|1980}} was republished in a [[festschrift]] for [[Robert Aumann|Robert J. Aumann]], winner of the 2008 [[Nobel Prize in Economics]]: {{cite book|first1=Zvi|last1=Artstein|chapter=22 Discrete and continuous bang–bang and facial spaces or: Look for the extreme points|pages=449–462|title=Game and economic theory: Selected contributions in honor of Robert J. Aumann |url=http://www.press.umich.edu/titleDetailDesc.do?id=14414|editor1-first=Sergiu|editor1-last=Hart|editor2-first=Abraham|editor2-last=Neyman|publisher=University of Michigan Press|location=Ann Arbor, MI|year=1995|isbn=0-472-10673-2|ref=harv}}</ref> Lyapunov's theorem has been called a [[discretization|continuous]] counterpart of the Shapley–Folkman lemma,<ref name="Starr08" /> which has itself been called a [[discrete mathematics#Discrete analogues of continuous mathematics|discrete analogue]] of Lyapunov's theorem.<ref name="MCBlock78" >{{harvtxt|Mas-Colell|1978|p=210}}: {{cite journal|last=Mas-Colell|first=Andreu|authorlink=Andreu Mas-Colell|title=A note on the core equivalence theorem: How many blocking coalitions are there?|journal=Journal of Mathematical Economics|volume=5|year=1978|issue=3|pages=207–215|doi=10.1016/0304-4068(78)90010-1|url=http://www.sciencedirect.com/science/article/B6VBY-4582G5H-2G/2/576b6893a9a730c3557fde0f52d3a9c2|mr=514468|ref=harv}}</ref>
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| {{clear}}
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| ==Notes==
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| {{Reflist|colwidth=30em}}
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| ==References==
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| * {{cite book|last1=Arrow|first1=Kenneth J.|authorlink1=Kenneth Arrow|last2=Hahn|first2=Frank H.|authorlink2=Frank Hahn|year=1980<!-- |chapter=Appendix B: Convex and related sets -->|title=General competitive analysis|publisher=North-Holland|<!-- pages=375–401 -->|series=Advanced Textbooks in Economics|volume=12|edition=reprint of San Francisco, CA: Holden-Day, Inc. Mathematical Economics Texts '''6'''|origyear=1971|location=Amsterdam|isbn=0-444-85497-5|mr=439057|ref=harv}}
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| * {{cite journal|last=Artstein|first=Zvi|title=Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points|journal=SIAM Review|volume=22|year=1980|issue=2|pages=172–185|doi=10.1137/1022026|mr=564562|jstor=2029960|ref=harv}}
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| * {{cite book|last=Carter|first=Michael|title=Foundations of mathematical economics|url=http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=8630|publisher=MIT Press|location=Cambridge, MA|year=2001|pages=xx+649|isbn=0-262-53192-5|mr=1865841|id=([http://michaelcarteronline.com/FOME/ Author's website] with [http://michaelcarteronline.com/FOME/answers.html answers to exercises])|ref=harv}}
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| * {{cite book|first=W. E.|last=Diewert|chapter=12 Duality approaches to microeconomic theory
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| |pages=535–599
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| |url=http://www.sciencedirect.com/science/article/B7P5Y-4FDF0FN-R/2/dcc0f8c9352eb054c96b3ff481976ce7
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| |doi=10.1016/S1573-4382(82)02007-4
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| |title=Handbook of mathematical economics, Volume '''II'''|editor1-link=Kenneth Arrow |editor1-first=Kenneth Joseph|editor1-last=Arrow|editor2-first=Michael D<!-- . -->|editor2-last=Intriligator|series=Handbooks in Economics|volume=1|publisher=North-Holland Publishing Co|location=Amsterdam|year=1982|isbn=978-0-444-86127-6|mr=648778|ref=harv}}
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| * {{cite book|last=Ekeland|first=Ivar|authorlink=Ivar Ekeland|chapter=Appendix I: An ''a priori'' estimate in convex programming|editor1-last=Ekeland|editor1-first=Ivar|editor2-last=Temam|editor2-first=Roger|editor2-link=Roger Temam|title=Convex analysis and variational problems|edition=Corrected reprinting of the North-Holland|origyear=1976|series=Classics in Applied Mathematics|volume=28 |publisher=Society for Industrial and Applied Mathematics (SIAM)|location=Philadelphia, PA|year=1999|pages=357–373|isbn=0-89871-450-8|mr=1727362|ref=harv}}
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| * {{cite book|first1=Jerry|last1=Green|first2=Walter P.|last2=Heller|chapter=1 Mathematical analysis and convexity with applications to economics|pages=15–52|url=http://www.sciencedirect.com/science/article/B7P5Y-4FDF0FN-5/2/613440787037f7f62d65a05172503737|doi=10.1016/S1573-4382(81)01005-9|title=Handbook of mathematical economics, Volume '''I'''|editor1-link=Kenneth Arrow |editor1-first=Kenneth Joseph|editor1-last=Arrow|editor2-first=Michael D<!-- . -->|editor2-last=Intriligator|series=Handbooks in Economics|volume=1|publisher=North-Holland Publishing Co|location=Amsterdam|year=1981|isbn=0-444-86126-2|mr=634800|ref=harv}}
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| * {{cite book|last=Guesnerie|first=Roger|authorlink=Roger Guesnerie|year=1989|chapter=First-best allocation of resources with nonconvexities <!-- original, NOT "non–convexities" --> in production|pages=99–143|editor-first=Bernard|editor-last=Cornet|editor2-first=Henry|editor2-last=Tulkens|title=Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987)|publisher=MIT Press|location=Cambridge, MA|isbn=0-262-03149-3|mr=1104662|ref=harv}}
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| * {{cite book|last=Mas-Colell|first=A.|authorlink=Andreu Mas-Colell|chapter=Non-convexity|title=[[The New Palgrave Dictionary of Economics|The new Palgrave: A dictionary of economics]]|editor1-first=John|editor1-last=Eatwell|editor1-link=John Eatwell, Baron Eatwell|editor2-first=Murray|editor2-last=Milgate|editor2-link=Murray Milgate|editor3-first=Peter|editor3-last=Newman|editor3-link=Peter Kenneth Newman|publisher=Palgrave Macmillan|year=1987|edition=first|doi=10.1057/9780230226203.3173<!-- SNAFU at NP? 30 Jan 2011-->|pages=653–661|url=http://www.dictionaryofeconomics.com/article?id=pde1987_X001573|id=([http://www.econ.upf.edu/~mcolell/research/art_083b.pdf PDF file at Mas-Colell's homepage])|ref=harv}}
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| * {{cite book|last=Rockafellar|first=R. Tyrrell|authorlink=R. Tyrrell Rockafellar|title=Convex analysis|edition=Reprint of the 1970 ({{MR|274683}}) Princeton Mathematical Series '''28'''|series=Princeton Landmarks in Mathematics|publisher=Princeton University Press|location=Princeton, NJ|year=1997|pages=xviii+451|isbn=0-691-01586-4|mr=1451876|ref=harv}}
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| * {{cite book|last=Schneider|first=Rolf|title=Convex bodies: The Brunn–Minkowski theory|series=Encyclopedia of Mathematics and its Applications|volume=44|publisher=Cambridge University Press|location=Cambridge|year=1993|pages=xiv+490|ref=harv|isbn=0-521-35220-7|mr=1216521}}
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| * {{citation|last=Starr|first=Ross M.|authorlink=Ross Starr|issue=1|journal=Econometrica|pages=25–38|title=Quasi-equilibria in markets with non-convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37)|volume=37|year=1969|jstor=1909201|ref=harv}}
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| * {{cite book|last=Starr|first=Ross M.|<!-- |authorlink=Ross Starr -->|chapter=Shapley–Folkman theorem|title=[[The New Palgrave Dictionary of Economics|The new Palgrave dictionary of economics]]|editor-first=Steven N.|editor-last=Durlauf|editor2-first=Lawrence E<!-- . -->|editor2-last=Blume|editor1-link=Steven N. Durlauf|editor2-link=Lawrence E. Blume|publisher=Palgrave Macmillan|year=2008|edition=Second|pages=317–318 (1st ed.)|url=http://www.dictionaryofeconomics.com/article?id=pde2008_S000107|doi=10.1057/9780230226203.1518|ref=harv}}
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| ==External links==
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| * {{citation|title=Economics 201B: Nonconvex preferences and approximate equilibria|chapter=1 The Shapley–Folkman theorem|pages=1–5|<!-- date=2005–03–14 -->|year=2005|month=<!-- 3 -->March|first=Robert M.|last=Anderson|authorlink=<!-- NOT WP's Robert M. Anderson -->|location=Berkeley, CA|publisher=Economics Department, University of California, Berkeley|url=http://elsa.berkeley.edu/users/anderson/Econ201B/NonconvexHandout.pdf|accessdate=15 January 2011}}
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| * {{citation|title=On the tendency toward convexity of the vector sum of sets|authorlink=Roger Evans Howe|last=Howe|first=Roger|year=1979|month=<!-- 3 -->November|publisher=[[Cowles Foundation|Cowles Foundation for Research in Economics]], Yale University|series=Cowles Foundation discussion papers|location=Box 2125 Yales Station, New Haven, CT 06520|volume=538 |url=http://cowles.econ.yale.edu/P/cd/d05a/d0538.pdf|<!-- url-2=http://econpapers.repec.org/RePEc:cwl:cwldpp:538 -->|accessdate=15 January 2011}}
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| * {{citation|last=Starr|first=Ross M.|authorlink=Ross Starr|chapter=8 Convex sets, separation theorems, and non-convex sets in '''R'''<sup>''N''</sup> (Section 8.2.3 Measuring non-convexity, the Shapley–Folkman theorem)|title=General equilibrium theory: An introduction|edition=|publisher=|year=2009|month=<!-- 21 -->September|pages=3–6|url=http://www.econ.ucsd.edu/~rstarr/113Winter2010/Webpage/PDFonlyCUPsubmission/Chap8-2009/2009CHAP-08092109.pdf|mr=1462618|id=(Draft of second edition, from Starr's course at the Economics Department of the University of California, San Diego)|accessdate=15 January 2011}}
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| * {{citation|last=Starr|first=Ross M.|authorlink=Ross Starr|title=Shapley–Folkman theorem|month=<!-- 19 -->May|year=2007|pages=1–3|url=http://www.econ.ucsd.edu/~rstarr/SFarticle.pdf|accessdate=15 January 2011|id=(Draft of article for the second edition of ''New Palgrave Dictionary of Economics'')}}
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