Economic lot scheduling problem - Revision history

en>HillSquirrel at 19:23, 14 August 2014

2014-08-14T19:23:36Z

← Older revision		Revision as of 21:23, 14 August 2014
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	~~In mathematics, '''Fenchel's duality theorem'''~~ is ~~a result in~~ the ~~theory of convex functions named after [[Werner Fenchel]].~~		Wilber Berryhill is the name his parents gave him and he completely digs that name. To climb is something I truly enjoy doing. For a whilst I've been in Alaska but I will have to move in a year or two. For years she's been working as a journey agent.<br><br>My page - best psychic readings, [http://Koreanyelp.com/index.php?document_srl=1798&mid=SchoolNews koreanyelp.com],

	~~Let ''&fnof;'' be a [[proper convex function]] on '''R'''<sup>''n''</sup>~~ and ~~let ''g'' be a proper concave function on '''R'''<sup>''n''</sup>~~. ~~Then, if regularity conditions are satisfied,~~

	~~:<math>\min_x ( f(x)-g(x) ) = \max_p ( g_\star(p)-f^\star(p))~~.~~\,</math>~~

	~~where~~ '~~'&fnof;''<sup> </sup> is the [[convex conjugate]] of ''&fnof;'' (also referred~~ to ~~as the Fenchel–Legendre transform) and ''g''<sub> </sub> is the [[concave conjugate]] of ''g''. That is,~~

	~~:<math>f^{\star} \left( x^{} \right) := \sup \left \{ \left. \left\langle x^{} , x \right\rangle - f \left( x \right) \right\| x \~~in ~~\mathbb{R}^n \right\}</math>~~

	~~:<math>g_{\star} \left( x^{} \right) := \inf \left \{ \left~~. ~~\left\langle x^{} , x \right\rangle - g \left( x \right) \right\| x \in \mathbb{R}^n \right\}</math>~~

	~~==Mathematical theorem==~~
	~~Let~~ '~~'X'' and ''Y'' be [[Banach spaces]], <math>f: X \to \mathbb{R} \cup \{+\infty\}</math> and <math>g: Y \to \mathbb{R} \cup \{+\infty\}</math> be convex functions and <math>A: X \to Y</math> be~~ a ~~[[bounded operator\|bounded]] [[linear map]]~~. ~~Then the Fenchel problems:~~
	~~:<math>p^* = \inf_{x \in X} \{f(x) + g(Ax)\}~~<~~/math~~>
	:<~~math~~>~~d^* = \sup_{y^* \in Y^} \{-f^(A^y^)~~ - ~~g^(-y^)\}</math>~~
	satisfy [[weak duality]], i.e. <math>p^* \geq d^</math>. Note that <math>f^,g^</math> are the convex conjugates of ''f'',''g'' respectively, and <math>A^</math> is the [[adjoint operator]]. The [[perturbation function]] for this [[dual problem]] is given by <math>F(x,y) = f(x) + g(Ax - y)</math>.

	~~Suppose that ''f'',''g''~~, ~~and ''A'' satisfy either~~
	~~# ''f'' and ''g'' are [[lower semi-continuous]] and <math>0 \in \operatorname{core}(\operatorname{dom}g - A \operatorname{dom}f)</math> where <math>\operatorname{core}</math> is the~~ [~~[algebraic interior]] and <math>\operatorname{dom}h</math> where ''h'' is some function is the set <math>\{z~~: ~~h(z) < +\infty\}</math>, or~~
	~~# <math>A \operatorname{dom}f \cap \operatorname{cont}g \neq \emptyset<~~/~~math> where <math>\operatorname{cont}</math> are the points where the function is [[continuous function\|continuous]].~~
	~~Then [[strong duality]] holds, i.e. <math>p^* = d^<~~/~~math>~~. ~~If <math>d^ \in \mathbb{R}<~~/~~math> then [[supremum]] is attained~~.~~<ref>{{cite book\|title~~=~~Techniques of Variational Analysis\|last1=Borwein\|first1=Jonathan\|last2=Zhu\|first2=Qiji\|year=2005\|publisher=Springer\|isbn=978-1-4419-2026-3\|pages=135–137}}</ref>~~

	=~~=One-dimensional illustration==~~

	~~In the following figure, the minimization problem on the left side of the equation is illustrated~~. One seeks to vary ''x'' such that the vertical distance between the convex and concave curves at ''x'' is as small as possible. The position of the vertical line in the figure is the (approximate) optimum.

	~~[[File:FencheDual02.png~~]]

	The next figure illustrates the maximization problem on the right hand side of the above equation. Tangents are drawn to each of the two curves such that both tangents have the same slope ''p''. The problem is to adjust ''p'' in such a way that the two tangents are as far away from each other as possible (more precisely, such that the point where they intersect the y-axis are as far from each other as possible). Imagine the two tangents as metal bars with vertical springs between them that push them apart and against the two parabolas that are fixed in place.

	~~[[File:FenchelDual01.png]]~~

	~~Fenchel's theorem states that the two problems have the same solution. The points having the minimum vertical separation are also the tangency points for the maximally separated parallel tangents.~~

	~~==See also==~~
	*[[Legendre transformation]]
	*[[Convex conjugate]]
	*[[Moreau's theorem]]
	*[[Wolfe duality]]
	*[[Werner Fenchel]]

	~~==References==~~
	~~{{Reflist}}~~
	* {{cite book \| authorlink=R. Tyrrell Rockafellar\|last=Rockafellar\|first=Ralph Tyrrell \| title=Convex Analysis \| publisher=Princeton University Press \| year=1996 \| isbn=0-691-01586-4}} See page 327.

	~~[[Category:Theorems in analysis]]~~
	~~[[Category:Mathematical optimization]]~~
	~~[[Category:Convex analysis]]~~
	~~[[Category:Convex optimization]]~~

en>Yobot: Reference before punctuation and other fixes using AWB (9510)

2013-10-04T14:01:43Z

Reference before punctuation and other fixes using AWB (9510)

← Older revision		Revision as of 16:01, 4 October 2013
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	~~Hi there~~, ~~I am Alyson Pomerleau~~ and ~~I think it seems fairly great when you say it~~. I'~~ve usually loved residing in Mississippi. One~~ of the ~~issues she enjoys most~~ is ~~canoeing and she~~'~~s been performing it for quite a whilst~~. ~~Office supervising~~ is ~~exactly where her primary income arrives from~~.<br><br>~~my webpage~~ :: ~~clairvoyant psychic,~~ [~~http~~://~~breenq~~.~~com~~/~~index~~.~~php?do~~=/~~profile~~-~~1144~~/~~info~~/ ~~http~~://~~breenq~~.~~com~~/~~index~~.~~php?do~~=/~~profile~~-~~1144/info/~~],		In mathematics, '''Fenchel's duality theorem''' is a result in the theory of convex functions named after [[Werner Fenchel]].

			Let ''&fnof;'' be a [[proper convex function]] on '''R'''<sup>''n''</sup> and let ''g'' be a proper concave function on '''R'''<sup>''n''</sup>. Then, if regularity conditions are satisfied,

			:<math>\min_x ( f(x)-g(x) ) = \max_p ( g_\star(p)-f^\star(p)).\,</math>

			where ''&fnof;''<sup> </sup> is the [[convex conjugate]] of ''&fnof;'' (also referred to as the Fenchel–Legendre transform) and ''g''<sub> </sub> is the [[concave conjugate]] of ''g''. That is,

			:<math>f^{\star} \left( x^{} \right) := \sup \left \{ \left. \left\langle x^{} , x \right\rangle - f \left( x \right) \right\| x \in \mathbb{R}^n \right\}</math>

			:<math>g_{\star} \left( x^{} \right) := \inf \left \{ \left. \left\langle x^{} , x \right\rangle - g \left( x \right) \right\| x \in \mathbb{R}^n \right\}</math>

			==Mathematical theorem==
			Let ''X'' and ''Y'' be [[Banach spaces]], <math>f: X \to \mathbb{R} \cup \{+\infty\}</math> and <math>g: Y \to \mathbb{R} \cup \{+\infty\}</math> be convex functions and <math>A: X \to Y</math> be a [[bounded operator\|bounded]] [[linear map]]. Then the Fenchel problems:
			:<math>p^* = \inf_{x \in X} \{f(x) + g(Ax)\}</math>
			:<math>d^* = \sup_{y^* \in Y^} \{-f^(A^y^) - g^(-y^)\}</math>
			satisfy [[weak duality]], i.e. <math>p^* \geq d^</math>. Note that <math>f^,g^</math> are the convex conjugates of ''f'',''g'' respectively, and <math>A^</math> is the [[adjoint operator]]. The [[perturbation function]] for this [[dual problem]] is given by <math>F(x,y) = f(x) + g(Ax - y)</math>.

			Suppose that ''f'',''g'', and ''A'' satisfy either
			# ''f'' and ''g'' are [[lower semi-continuous]] and <math>0 \in \operatorname{core}(\operatorname{dom}g - A \operatorname{dom}f)</math> where <math>\operatorname{core}</math> is the [[algebraic interior]] and <math>\operatorname{dom}h</math> where ''h'' is some function is the set <math>\{z: h(z) < +\infty\}</math>, or
			# <math>A \operatorname{dom}f \cap \operatorname{cont}g \neq \emptyset</math> where <math>\operatorname{cont}</math> are the points where the function is [[continuous function\|continuous]].
			Then [[strong duality]] holds, i.e. <math>p^* = d^</math>. If <math>d^ \in \mathbb{R}</math> then [[supremum]] is attained.<ref>{{cite book\|title=Techniques of Variational Analysis\|last1=Borwein\|first1=Jonathan\|last2=Zhu\|first2=Qiji\|year=2005\|publisher=Springer\|isbn=978-1-4419-2026-3\|pages=135–137}}</ref>

			==One-dimensional illustration==

			In the following figure, the minimization problem on the left side of the equation is illustrated. One seeks to vary ''x'' such that the vertical distance between the convex and concave curves at ''x'' is as small as possible. The position of the vertical line in the figure is the (approximate) optimum.

			[[File:FencheDual02.png]]

			The next figure illustrates the maximization problem on the right hand side of the above equation. Tangents are drawn to each of the two curves such that both tangents have the same slope ''p''. The problem is to adjust ''p'' in such a way that the two tangents are as far away from each other as possible (more precisely, such that the point where they intersect the y-axis are as far from each other as possible). Imagine the two tangents as metal bars with vertical springs between them that push them apart and against the two parabolas that are fixed in place.

			[[File:FenchelDual01.png]]

			Fenchel's theorem states that the two problems have the same solution. The points having the minimum vertical separation are also the tangency points for the maximally separated parallel tangents.

			==See also==
			*[[Legendre transformation]]
			*[[Convex conjugate]]
			*[[Moreau's theorem]]
			*[[Wolfe duality]]
			*[[Werner Fenchel]]

			==References==
			{{Reflist}}
			* {{cite book \| authorlink=R. Tyrrell Rockafellar\|last=Rockafellar\|first=Ralph Tyrrell \| title=Convex Analysis \| publisher=Princeton University Press \| year=1996 \| isbn=0-691-01586-4}} See page 327.

			[[Category:Theorems in analysis]]
			[[Category:Mathematical optimization]]
			[[Category:Convex analysis]]
			[[Category:Convex optimization]]

en>AnomieBOT: Dating maintenance tags: {{Fact}} {{Refimprove}}

2012-08-16T11:37:20Z

Dating maintenance tags: {{Fact}} {{Refimprove}}

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