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en>Nihiltres: Hyphenated ISBN

2014-11-23T23:14:11Z

Hyphenated ISBN

← Older revision		Revision as of 01:14, 24 November 2014
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	~~{{about\|the notion in convex analysis\|the notion in several complex variables\|pseudoconvex domain}}~~		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.<br><br>My weblog [http://conniecolin.com/xe/community/24580 psychic solutions by lynne]
	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]].~~

	~~==Formal definition==~~
	Formally, a real-valued differentiable function ''&fnof;'' 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 ∇''&fnof;'' is the [[gradient]] of ''&fnof;'', defined by
	~~:<math>\nabla f = \left(\frac{\partial f}{\partial x_1},\dots,\frac{\partial f}{\partial x_n}\right).</math>~~

	~~==Properties==~~
	~~Every convex function is pseudoconvex, but the converse~~ is not ~~true. For example,~~ the ~~function {{nowrap\|''&fnof;''(''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\|''&fnof;''(''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 ''&fnof;'' if~~ and ~~only if it is a [[stationary point]] of ''&fnof;'', which is to say that the [[gradient]] of ''&fnof;'' vanishes at ''x''*:~~
	~~:<math>\nabla f(x^*) = 0.</math><ref>{{harvnb\|Mangasarian\|1965}}</ref>~~

	~~==Generalization to nondifferentiable functions==~~
	The notion of pseudoconvexity can be generalized to nondifferentiable functions as follows.<ref>{{harvnb\|Floudas\|Pardalos\|2001}}</ref> Given any function {{nowrap\|''&fnof;'' : ''X'' → '''R'''}} we can define the upper [[Dini derivative]] of ''&fnof;'' by
	~~:<math>f^+(x,u) = \limsup_{h\to 0^+} \frac{f(x+hu) - f(x)}{h}</math>~~
	~~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]] ∂''&fnof;'' as follows:~~
	For all {{nowrap\|''x'', ''y'' ∈ ''X''}}, if there exists an {{nowrap\|''x'' ∈ ∂''&fnof;''(''x'')}} such that <math> \langle x^* , y - x \rangle \ge 0 \,,</math> then {{nowrap\|''&fnof;''(''x'') ≤ ''&fnof;''(''z'')}} for ~~all ''z'' on the line segment adjoining ''x'' and ''y''.~~

	~~==Related notions==~~
	~~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>~~
	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}}~~
	</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}}~~
	</~~ref>~~

	~~==See also==~~
	* [[Pseudoconvexity]]
	* [[Convex function]]
	* [[Quasiconvex function]]

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	* {{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}}.
	* {{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}}.~~
	* {{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}}

	~~[[Category:Convex analysis]]~~
	~~[[Category:Mathematical optimization]]~~
	~~[[Category:Types of functions]]~~
	~~[[Category:Generalized convexity]~~]

en>Monkbot: Fix CS1 deprecated date parameter errors

2014-01-17T13:32:35Z

Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 15:32, 17 January 2014
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	~~Hi there~~, ~~I am Andrew Berryhill~~. ~~I am really fond~~ of ~~handwriting~~ but I can'~~t make~~ it ~~my occupation truly~~. ~~Ohio~~ is ~~exactly where her home~~ is. ~~Credit authorising~~ is ~~how she makes~~ a ~~living~~.<br><br>~~My blog post free online tarot card readings~~ ([~~http~~:~~//breenq~~.~~com/index~~.~~php?do~~=/~~profile~~-~~1144~~/~~info~~/ ~~http:~~//~~breenq~~.~~com~~/~~index~~.~~php?do~~=/~~profile~~-~~1144/info~~])		{{about\|the notion in convex analysis\|the notion in several complex variables\|pseudoconvex domain}}
			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]].

			==Formal definition==
			Formally, a real-valued differentiable function ''&fnof;'' 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 ∇''&fnof;'' is the [[gradient]] of ''&fnof;'', defined by
			:<math>\nabla f = \left(\frac{\partial f}{\partial x_1},\dots,\frac{\partial f}{\partial x_n}\right).</math>

			==Properties==
			Every convex function is pseudoconvex, but the converse is not true. For example, the function {{nowrap\|''&fnof;''(''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\|''&fnof;''(''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 ''&fnof;'' if and only if it is a [[stationary point]] of ''&fnof;'', which is to say that the [[gradient]] of ''&fnof;'' vanishes at ''x''*:
			:<math>\nabla f(x^*) = 0.</math><ref>{{harvnb\|Mangasarian\|1965}}</ref>

			==Generalization to nondifferentiable functions==
			The notion of pseudoconvexity can be generalized to nondifferentiable functions as follows.<ref>{{harvnb\|Floudas\|Pardalos\|2001}}</ref> Given any function {{nowrap\|''&fnof;'' : ''X'' → '''R'''}} we can define the upper [[Dini derivative]] of ''&fnof;'' by
			:<math>f^+(x,u) = \limsup_{h\to 0^+} \frac{f(x+hu) - f(x)}{h}</math>
			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]] ∂''&fnof;'' as follows:
			For all {{nowrap\|''x'', ''y'' ∈ ''X''}}, if there exists an {{nowrap\|''x'' ∈ ∂''&fnof;''(''x'')}} such that <math> \langle x^* , y - x \rangle \ge 0 \,,</math> then {{nowrap\|''&fnof;''(''x'') ≤ ''&fnof;''(''z'')}} for all ''z'' on the line segment adjoining ''x'' and ''y''.

			==Related notions==
			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>
			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}}
			</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}}
			</ref>

			==See also==
			* [[Pseudoconvexity]]
			* [[Convex function]]
			* [[Quasiconvex function]]

			==Notes==
			{{reflist}}

			==References==
			* {{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}}.
			* {{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}}.
			* {{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}}

			[[Category:Convex analysis]]
			[[Category:Mathematical optimization]]
			[[Category:Types of functions]]
			[[Category:Generalized convexity]]

en>Bender235: /* References */Typo fixing, typos fixed: , → , using AWB

2012-06-05T15:41:20Z

References: Typo fixing, typos fixed: , → , using AWB

New page

Hi there, I am Andrew Berryhill. I am really fond of handwriting but I can't make it my occupation truly. Ohio is exactly where her home is. Credit authorising is how she makes a living.<br><br>My blog post free online tarot card readings ([http://breenq.com/index.php?do=/profile-1144/info/ http://breenq.com/index.php?do=/profile-1144/info])