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| '''Convex analysis''' is the branch of [[mathematics]] devoted to the study of properties of [[convex function]]s and [[convex set]]s, often with applications in [[convex optimization|convex minimization]], a subdomain of [[optimization (mathematics)|optimization theory]].
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| == Convex sets ==
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| {{main|Convex set}}
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| A '''convex set''' is a set ''C'' ⊆ ''X'', for some [[vector space]] ''X'', such that for any ''x'', ''y'' ∈ ''C'' and λ ∈ [0, 1] then<ref name="Rockafellar">{{cite book |author=[[Rockafellar, R. Tyrrell]]|title=Convex Analysis|publisher=Princeton University Press|location=Princeton, NJ|year=1997|origyear=1970|isbn=978-0-691-01586-6}}</ref>
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| :<math>\lambda x + (1 - \lambda)y \in C</math>.
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| == Convex functions ==
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| {{main|Convex function}}
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| A '''convex function''' is any [[extended reals|extended real-valued]] function ''f'' : ''X'' → '''R''' ∪ {±∞} which satisfies [[Jensen's inequality]], i.e. for any ''x'', ''y'' ∈ ''X'' and any λ ∈ [0, 1] then
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| :<math>f(\lambda x + (1 - \lambda)y) \leq \lambda f(x) + (1-\lambda) f(y)</math>.<ref name="Rockafellar" />
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| Equivalently, a convex function is any (extended) real valued function such that its [[epigraph (mathematics)|epigraph]]
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| :<math>\left\{(x,r) \in X \times \mathbf{R}: f(x) \leq r \right\}</math>
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| is a convex set.<ref name="Rockafellar" />
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| == Convex conjugate ==
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| {{main|Convex conjugate}}
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| The '''convex conjugate''' of an extended real-valued (not necessarily convex) function ''f'' : ''X'' → '''R''' ∪ {±∞} is ''f*'' : ''X*'' → '''R''' ∪ {±∞} where ''X*'' is the [[dual space]] of ''X'', and<ref name="Zalinescu" />{{rp|pp.75–79}}
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| : <math>f^*(x^*) = \sup_{x \in X} \left \{\langle x^*,x \rangle - f(x) \right \}.</math>
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| === Biconjugate ===
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| The ''biconjugate'' of a function ''f'' : ''X'' → '''R''' ∪ {±∞} is the conjugate of the conjugate, typically written as ''f**'' : ''X'' → '''R''' ∪ {±∞}. The biconjugate is useful for showing when [[strong duality|strong]] or [[weak duality]] hold (via the [[perturbation function]]).
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| For any ''x'' ∈ ''X'' the inequality ''f**''(''x'') ≤ ''f''(''x'') follows from the ''Fenchel–Young inequality''. For [[Proper convex function|proper functions]], ''f'' = ''f**'' [[if and only if]] ''f'' is convex and [[lower semi-continuous]] by [[Fenchel–Moreau theorem]].<ref name="Zalinescu" />{{rp|pp.75–79}}<ref name="BorweinLewis">{{cite book |last1=Borwein |first1=Jonathan |last2=Lewis |first2=Adrian |title=Convex Analysis and Nonlinear Optimization: Theory and Examples| edition=2 |year=2006 |publisher=Springer |isbn=978-0-387-29570-1|pages=76–77}}</ref> | |
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| == Convex minimization ==
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| {{main|Convex optimization}}
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| A '''convex minimization''' (primal) problem is one of the form
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| :<math>\inf_{x \in M} f(x)</math>
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| such that ''f'' : ''X'' → '''R''' ∪ {±∞} is a convex function and ''M'' ⊆ ''X'' is a convex set.
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| === Dual problem ===
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| {{main|Duality (optimization)}}
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| <!-- Copied from [[Duality (optimization)#Duality principle]] -->
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| In optimization theory, the ''duality principle'' states that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem.
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| In general given two [[dual pair]]s [[separated space|separated]] [[locally convex space]]s (''X'', ''X*'') and (''Y'', ''Y*''). Then given the function ''f'' : ''X'' → '''R''' ∪ {+∞}, we can define the primal problem as finding ''x'' such that
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| :<math>\inf_{x \in X} f(x).</math>
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| If there are constraint conditions, these can be built into the function ''f'' by letting <math>f = f + I_{\mathrm{constraints}}</math> where ''I'' is the [[Characteristic function (convex analysis)|indicator function]]. Then let ''F'' : ''X'' × ''Y'' → '''R''' ∪ {±∞} be a [[perturbation function]] such that ''F''(''x'', 0) = ''f''(''x'').<ref name="BWG">{{cite book |title=Duality in Vector Optimization |author1=Boţ, Radu Ioan |author2=Wanka, Gert|author3=Grad, Sorin-Mihai |year=2009 | publisher=Springer |isbn=978-3-642-02885-4 }}</ref>
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| The ''dual problem'' with respect to the chosen perturbation function is given by
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| :<math>\sup_{y^* \in Y^*} -F^*(0,y^*)</math> | |
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| where ''F*'' is the convex conjugate in both variables of ''F''.
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| The [[duality gap]] is the difference of the right and left hand sides of the inequality<ref name="Zalinescu">{{cite book |last=Zălinescu |first=Constantin |title=Convex analysis in general vector spaces |publisher=World Scientific Publishing Co., Inc. |isbn=981-238-067-1 |mr=1921556 |issue=J |year=2002 |location=River Edge, NJ }}</ref>{{rp|pp. 106–113}}<ref name="BWG" /><ref>{{cite book |title=Overcoming the failure of the classical generalized interior-point regularity conditions in convex optimization. Applications of the duality theory to enlargements of maximal monotone operators |author=Csetnek, Ernö Robert |year=2010 |publisher=Logos Verlag Berlin GmbH |isbn=978-3-8325-2503-3 }}</ref>
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| :<math>\sup_{y^* \in Y^*} -F^*(0,y^*) \le \inf_{x \in X} F(x,0).</math> | |
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| This principle is the same as [[weak duality]]. If the two sides are equal to each other then the problem is said to satisfy [[strong duality]].
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| There are many conditions for strong duality to hold such as:
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| *''F'' = ''F**'' where ''F'' is the [[perturbation function]] relating the primal and dual problems and ''F**'' is the [[convex conjugate|biconjugate]] of ''F'';{{Citation needed|date=January 2012}}
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| * the primal problem is a [[linear optimization|linear optimization problem]];
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| * [[Slater's condition]] for a [[convex optimization|convex optimization problem]].<ref name="borwein">{{cite book |last1=Borwein |first1=Jonathan |last2=Lewis |first2=Adrian |title=Convex Analysis and Nonlinear Optimization: Theory and Examples| edition=2 |year=2006 |publisher=Springer |isbn=978-0-387-29570-1}}</ref><ref name="boyd">{{cite book |last1=Boyd |first1=Stephen |last2=Vandenberghe |first2=Lieven |title=Convex Optimization |publisher=Cambridge University Press |year=2004 |isbn=978-0-521-83378-3 |url=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf |format=pdf |accessdate=October 3, 2011}}</ref>
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| ==== Lagrange duality ====
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| For a convex minimization problem with inequality constraints,
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| ::: min<sub>''x''</sub> ''f''(''x'') subject to ''g<sub>i</sub>''(''x'') ≤ 0 for ''i'' = 1, ..., ''m''.
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| the Lagrangian dual problem is
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| ::: sup<sub>''u''</sub> inf<sub>''x''</sub> ''L''(''x'', ''u'') subject to ''u<sub>i</sub>''(''x'') ≥ 0 for ''i'' = 1, ..., ''m''.
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| where the objective function ''L''(''x'', ''u'') is the Lagrange dual function defined as follows:
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| :<math>L(x,u) = f(x) + \sum_{j=1}^m u_j g_j(x)</math>
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| == See also ==
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| * [[List of convexity topics]]
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| * [[Werner Fenchel]]
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| == References ==
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| {{Reflist}}
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| * {{cite book|author1=J.-B. Hiriart-Urruty|author2=[[C. Lemaréchal]]|title=Fundamentals of convex analysis|publisher=Springer-Verlag |location=Berlin |year=2001 |isbn=978-3-540-42205-1}}
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| * {{cite book|last=Singer|first=Ivan|title=Abstract convex analysis|series=Canadian Mathematical Society series of monographs and advanced texts|publisher=John Wiley & Sons, Inc.|location=New York|year= 1997|pages=xxii+491|isbn=0-471-16015-6|mr=1461544}}
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| * {{cite book|first1=J.|last1=Stoer|first2=C.|last2=Witzgall|title=Convexity and optimization in finite dimensions |volume=1 |publisher=Springer |location=Berlin | year=1970 |isbn=978-0-387-04835-2}}
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| [[Category:Convex analysis|*]]
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| [[Category:Mathematical optimization]]
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| [[Category:Variational analysis]]
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