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| {{about|the notion in convex analysis|the notion in several complex variables|pseudoconvex domain}}
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| In [[convex analysis]] and the [[calculus of variations]], branches of [[mathematics]], a '''pseudoconvex function''' is a [[function (mathematics)|function]] that behaves like a [[convex function]] with respect to finding its [[local extrema|local minima]], but need not actually be convex. Informally, a differentiable function is pseudoconvex if it is increasing in any direction where it has a positive [[directional derivative]].
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| ==Formal definition==
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| Formally, a real-valued differentiable function ''ƒ'' defined on a (nonempty) [[convex set|convex]] [[open set]] ''X'' in the finite-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup> is said to be '''pseudoconvex''' if, for all {{nowrap|''x'', ''y'' ∈ ''X''}} such that <math>\nabla f(x)\cdot(y-x) \ge 0</math>, we have <math>f(y)\ge f(x)</math>.<ref>{{harvnb|Mangasarian|1965}}</ref> Here ∇''ƒ'' is the [[gradient]] of ''ƒ'', defined by
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| :<math>\nabla f = \left(\frac{\partial f}{\partial x_1},\dots,\frac{\partial f}{\partial x_n}\right).</math>
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| ==Properties==
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| Every convex function is pseudoconvex, but the converse is not true. For example, the function {{nowrap|''ƒ''(''x'') {{=}} ''x'' + ''x''<sup>3</sup>}} is pseudoconvex but not convex. Any pseudoconvex function is [[quasiconvex function|quasiconvex]], but the converse is not true since the function {{nowrap|''ƒ''(''x'') {{=}} ''x''<sup>3</sup>}} is quasiconvex but not pseudoconvex. Pseudoconvexity is primarily of interest because a point ''x''* is a local minimum of a pseudoconvex function ''ƒ'' if and only if it is a [[stationary point]] of ''ƒ'', which is to say that the [[gradient]] of ''ƒ'' vanishes at ''x''*:
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| :<math>\nabla f(x^*) = 0.</math><ref>{{harvnb|Mangasarian|1965}}</ref>
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| ==Generalization to nondifferentiable functions==
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| The notion of pseudoconvexity can be generalized to nondifferentiable functions as follows.<ref>{{harvnb|Floudas|Pardalos|2001}}</ref> Given any function {{nowrap|''ƒ'' : ''X'' → '''R'''}} we can define the upper [[Dini derivative]] of ''ƒ'' by
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| :<math>f^+(x,u) = \limsup_{h\to 0^+} \frac{f(x+hu) - f(x)}{h}</math>
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| where ''u'' is any [[unit vector]]. The function is said to be pseudoconvex if it is increasing in any direction where the upper Dini derivative is positive. More precisely, this is characterized in terms of the [[subdifferential]] ∂''ƒ'' as follows:
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| *For all {{nowrap|''x'', ''y'' ∈ ''X''}}, if there exists an {{nowrap|''x''* ∈ ∂''ƒ''(''x'')}} such that <math> \langle x^* , y - x \rangle \ge 0 \,,</math> then {{nowrap|''ƒ''(''x'') ≤ ''ƒ''(''z'')}} for all ''z'' on the line segment adjoining ''x'' and ''y''.
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| ==Related notions==
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| A '''{{visible anchor|pseudoconcave function}}''' is a function whose negative is pseudoconvex. A '''{{visible anchor|pseudolinear function}}''' is a function that is both pseudoconvex and pseudoconcave.<ref>{{harvnb|Rapcsak|1991}}</ref> For example, [[linear-fractional programming|linear–fractional program]]s have pseudolinear [[objective function]]s and [[linear programming|linear–inequality constraints]]: These properties allow fractional–linear problems to be solved by a variant of the [[simplex algorithm]] (of [[George B. Dantzig]]).<ref>
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| Chapter five: {{cite book| last=Craven|first=B. D.|title=Fractional programming|series=Sigma Series in Applied Mathematics|volume=4|publisher=Heldermann Verlag|location=Berlin|year=1988|pages=145|isbn=3-88538-404-3 |mr=949209}}</ref><ref>{{cite article | last1=Kruk | first1=Serge|last2=Wolkowicz|first2=Henry|title=Pseudolinear programming | jstor=2653207 |journal=[[SIAM Review]]|volume=41 |year=1999 |number=4 |pages=795-805 |mr=1723002 | jstor = 2653207 |doi=10.1137/S0036144598335259}}
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| </ref><ref>{{cite article | last1=Mathis|first1=Frank H.|last2=Mathis|first2=Lenora Jane|title=A nonlinear programming algorithm for hospital management |jstor=2132826|journal=[[SIAM Review]]|volume=37 |year=1995 |number=2 |pages=230-234|mr=1343214 | jstor = 2132826 |doi=10.1137/1037046}}
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| </ref>
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| ==See also==
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| * [[Pseudoconvexity]]
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| * [[Convex function]]
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| * [[Quasiconvex function]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{citation|first1=Christodoulos A.|last1=Floudas|first2=Panos M.|last2=Pardalos|title=Encyclopedia of Optimization|chapter=Generalized monotone multivalued maps|publisher=Springer|year=2001|isbn=978-0-7923-6932-5|page=227}}.
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| * {{cite journal|ref=harv|title=Pseudo-Convex Functions|journal=Journal of the Society for Industrial and Applied Mathematics Series A|volume=3|issue=2|pages=281–290 |date=January 1965|doi=10.1137/0303020|first=O. L.|last=Mangasarian|issn=0363-0129}}.
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| * {{cite journal|ref=harv|first=T.|last=Rapcsak|title=On pseudolinear functions|journal=European Journal of Operational Research|volume=50|issue=3|date=1991-02-15|pages=353–360|issn=0377-2217|doi=10.1016/0377-2217(91)90267-Y}}
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| [[Category:Convex analysis]]
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| [[Category:Mathematical optimization]]
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| [[Category:Types of functions]]
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| [[Category:Generalized convexity]]
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Hi there, I am Alyson Boon although it is not the title on my birth certificate. To climb is something I really enjoy doing. My wife and I live in Mississippi and I love every working day residing right here. Invoicing is what I do for a living but I've always needed my personal business.
My weblog psychic solutions by lynne