Overall pressure ratio - Revision history

en>Bhny: /* top */ WP:REFERS

2014-06-19T18:50:52Z

← Older revision		Revision as of 20:50, 19 June 2014
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	~~{{Technical\|date=October 2011}}~~		The author is known as Wilber Pegues. He is an information officer. To perform lacross is the thing I love most of all. Her family life in Ohio.<br><br>Also visit my webpage - [http://ltreme.com/index.php?do=/profile-127790/info/ online psychic chat]

	~~A '''second-order cone program''' ('''SOCP''')~~ is ~~a [[convex optimization]] problem of the form~~

	~~:minimize <math>\ f^T x \ </math>~~
	~~:subject to~~
	~~::<math>\lVert A_i x + b_i \rVert_2 \leq c_i^T x + d_i,\quad i = 1,\dots,m</math>~~
	~~::<math>Fx = g \ </math>~~

	where the problem parameters are <math>f \in \mathbb{R}^n, \ A_i \in \mathbb{R}^{{n_i}\times n}, \ b_i \in \mathbb{R}^{n_i}, \ c_i \in \mathbb{R}^n, \ d_i \in \mathbb{R}, \ F \in \mathbb{R}^{p\times n}</math>, and <math>g \in \mathbb{R}^p</math>. ~~Here <math>x\in\mathbb{R}^n</math>~~ is the optimization variable.<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>

	~~When <math>A_i = 0</math> for <math>i = 1,\dots,m</math>, the SOCP reduces to a [[linear program]]~~. ~~When <math>c_i = 0 </math> for <math>i = 1,\dots,m</math>, the SOCP~~ is ~~equivalent to a convex [[quadratically constrained quadratic program]]. [[Semidefinite programming]] subsumes SOCPs as~~ the SOCP constraints can be written as [[linear matrix inequality\|linear matrix inequalities]] (LMI) and can be reformulated as an instance of semi definite program. SOCPs can be solved with great efficiency by [[interior point methods]].

	~~==Example: Quadratic constraint==~~
	~~Consider a [[quadratically constrained quadratic program\|quadratic constraint]]~~ of ~~the form~~

	~~:<math> x^T A^T A x + b^T x + c \leq 0~~. ~~</math>~~

	~~This is equivalent to the SOC constraint~~

	~~:<math> \left\\|~~
	~~\begin{matrix}~~
	~~(1 + b^T x +c)/2\\~~
	Ax
	~~\end{matrix} \right\\|_2~~

	~~\leq (1 - b^T x -c)/2.</math>~~

	~~==Example: Stochastic programming==~~
	~~Consider a [[stochastic linear program]]~~ in ~~inequality form~~

	~~:minimize <math>\ c^T x \ </math>~~
	~~:subject to~~
	~~:: <math>P(a_i^T(x) \geq b_i) \geq p, \quad i = 1,\dots,m </math>~~

	where the parameters <math>a_i \ </math> are independent Gaussian random vectors with mean <math>\bar{a}_i</math> and covariance <math>\Sigma_i \ </math> and <math>p\geq0.5</math>. This problem can be expressed as the SOCP

	~~:minimize <math>\ c^T x \ </math>~~
	~~:subject to~~
	~~:: <math>\bar{a}_i^T (x) + \Phi^{-1}(1-p) \lVert \Sigma_i^{1/2} x \rVert_2 \geq b_i , \quad i = 1,\dots,m </math>~~

	~~where <math>\Phi^{-1} \ </math> is the inverse [[error function]]~~.<~~ref name="boyd"/~~>

	~~==Solvers and scripting (programming) languages==~~

	~~{\| class="wikitable"~~
	\|-
	~~!Name~~
	~~!License~~
	~~!Brief info~~
	\|-
	~~\|Xpress\|\|commercial\|\| from 7.6 release~~
	\|-
	~~\|[[CPLEX]]\|\|commercial\|\|~~
	\|-
	~~\|[[Gurobi]]\|\|commercial\|\|parallel SOCP barrier algorithm~~
	\|-
	~~\|JOptimizer\|\|[[Apache License]]\|\| Java library for convex optimization (open source)~~
	\|-
	~~\|[[MOSEK]]\|\|commercial\|\|~~
	\|-
	\|[http://sedumi.ie.lehigh.edu/ SeDuMi]\|\|GPL v3\|\|Matlab package with primal–dual interior point methods<ref>{{cite journal\|last=Sturm\|first=Jos F.\|title=Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones\|journal=Optimization Methods and Software\|year=1999\|volume=11-12\|pages=625-653}}<~~/ref~~>
	\|-
	\|[http://~~www~~.~~math.cmu.edu~~/~reha/sdpt3.html SDPT3]\|\|GPL v2\|\|Matlab package with primal–dual interior point methods<ref>{{cite journal\|last=Toh\|first=K.C.\|coauthors=M.J. Todd, and R.H. Tutuncu\|title=SDPT3 - a Matlab software package for semidefinite programming\|journal=Optimization Methods and Software\|year=1999\|volume=11\|pages=545-581}}</ref> <ref>{{cite journal\|last=Tutuncu\|first=R.H.\|coauthors=K.C. Toh, and M.J. Todd\|title=Solving semidefinite-quadratic-linear programs using SDPT3\|journal=Mathematical Programming\|year=2003\|volume=Ser. ~~B, 95\|pages~~=~~189-217}}<~~/~~ref>~~
	\|-
	~~\|[[OpenOpt]]\|\|[[BSD]]\|\|universal cross-platform numerical optimization framework, see its [http:~~//~~openopt.org/SOCP SOCP] page and [http://openopt.org/Problems other problems] involved. Uses [[NumPy]] arrays and [[SciPy]] sparse matrices.~~
	\|}

	~~==References==~~
	~~{{reflist}}~~

	~~[[Category:Mathematical optimization]]~~
	~~[[Category:Convex optimization]~~]

en>Pilot850: /* Examples */

2013-07-07T11:43:58Z

Examples

← Older revision		Revision as of 13:43, 7 July 2013
Line 1:		Line 1:
	~~The author~~'s title ~~is Christy Brookins~~. ~~The preferred pastime~~ for ~~him and his kids~~ is ~~fashion~~ and ~~he'll~~ be ~~starting something else along~~ with it. ~~Credit authorising~~ is ~~how he tends~~ to ~~make money~~. ~~Ohio is~~ where ~~his home is~~ and ~~his family loves it~~.<br><br>~~Here~~ is ~~my webpage~~: ~~online psychic readings (~~[http://www.~~herandkingscounty~~.~~com~~/~~content~~/~~information~~-and-~~facts~~-~~you~~-~~must~~-~~know~~-~~about~~-~~hobbies www~~.~~herandkingscounty~~.~~com~~])		{{Technical\|date=October 2011}}

			A '''second-order cone program''' ('''SOCP''') is a [[convex optimization]] problem of the form

			:minimize <math>\ f^T x \ </math>
			:subject to
			::<math>\lVert A_i x + b_i \rVert_2 \leq c_i^T x + d_i,\quad i = 1,\dots,m</math>
			::<math>Fx = g \ </math>

			where the problem parameters are <math>f \in \mathbb{R}^n, \ A_i \in \mathbb{R}^{{n_i}\times n}, \ b_i \in \mathbb{R}^{n_i}, \ c_i \in \mathbb{R}^n, \ d_i \in \mathbb{R}, \ F \in \mathbb{R}^{p\times n}</math>, and <math>g \in \mathbb{R}^p</math>. Here <math>x\in\mathbb{R}^n</math> is the optimization variable.<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>

			When <math>A_i = 0</math> for <math>i = 1,\dots,m</math>, the SOCP reduces to a [[linear program]]. When <math>c_i = 0 </math> for <math>i = 1,\dots,m</math>, the SOCP is equivalent to a convex [[quadratically constrained quadratic program]]. [[Semidefinite programming]] subsumes SOCPs as the SOCP constraints can be written as [[linear matrix inequality\|linear matrix inequalities]] (LMI) and can be reformulated as an instance of semi definite program. SOCPs can be solved with great efficiency by [[interior point methods]].

			==Example: Quadratic constraint==
			Consider a [[quadratically constrained quadratic program\|quadratic constraint]] of the form

			:<math> x^T A^T A x + b^T x + c \leq 0. </math>

			This is equivalent to the SOC constraint

			:<math> \left\\|
			\begin{matrix}
			(1 + b^T x +c)/2\\
			Ax
			\end{matrix} \right\\|_2

			\leq (1 - b^T x -c)/2.</math>

			==Example: Stochastic programming==
			Consider a [[stochastic linear program]] in inequality form

			:minimize <math>\ c^T x \ </math>
			:subject to
			:: <math>P(a_i^T(x) \geq b_i) \geq p, \quad i = 1,\dots,m </math>

			where the parameters <math>a_i \ </math> are independent Gaussian random vectors with mean <math>\bar{a}_i</math> and covariance <math>\Sigma_i \ </math> and <math>p\geq0.5</math>. This problem can be expressed as the SOCP

			:minimize <math>\ c^T x \ </math>
			:subject to
			:: <math>\bar{a}_i^T (x) + \Phi^{-1}(1-p) \lVert \Sigma_i^{1/2} x \rVert_2 \geq b_i , \quad i = 1,\dots,m </math>

			where <math>\Phi^{-1} \ </math> is the inverse [[error function]].<ref name="boyd"/>

			==Solvers and scripting (programming) languages==

			{\| class="wikitable"
			\|-
			!Name
			!License
			!Brief info
			\|-
			\|Xpress\|\|commercial\|\| from 7.6 release
			\|-
			\|[[CPLEX]]\|\|commercial\|\|
			\|-
			\|[[Gurobi]]\|\|commercial\|\|parallel SOCP barrier algorithm
			\|-
			\|JOptimizer\|\|[[Apache License]]\|\| Java library for convex optimization (open source)
			\|-
			\|[[MOSEK]]\|\|commercial\|\|
			\|-
			\|[http://sedumi.ie.lehigh.edu/ SeDuMi]\|\|GPL v3\|\|Matlab package with primal–dual interior point methods<ref>{{cite journal\|last=Sturm\|first=Jos F.\|title=Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones\|journal=Optimization Methods and Software\|year=1999\|volume=11-12\|pages=625-653}}</ref>
			\|-
			\|[http://www.math.cmu.edu/~reha/sdpt3.html SDPT3]\|\|GPL v2\|\|Matlab package with primal–dual interior point methods<ref>{{cite journal\|last=Toh\|first=K.C.\|coauthors=M.J. Todd, and R.H. Tutuncu\|title=SDPT3 - a Matlab software package for semidefinite programming\|journal=Optimization Methods and Software\|year=1999\|volume=11\|pages=545-581}}</ref> <ref>{{cite journal\|last=Tutuncu\|first=R.H.\|coauthors=K.C. Toh, and M.J. Todd\|title=Solving semidefinite-quadratic-linear programs using SDPT3\|journal=Mathematical Programming\|year=2003\|volume=Ser. B, 95\|pages=189-217}}</ref>
			\|-
			\|[[OpenOpt]]\|\|[[BSD]]\|\|universal cross-platform numerical optimization framework, see its [http://openopt.org/SOCP SOCP] page and [http://openopt.org/Problems other problems] involved. Uses [[NumPy]] arrays and [[SciPy]] sparse matrices.
			\|}

			==References==
			{{reflist}}

			[[Category:Mathematical optimization]]
			[[Category:Convex optimization]]

en>ChrisGualtieri: /* History of overall pressure ratios */Typo fixing, typos fixed: innaccuracies → inaccuracies using AWB

2012-07-22T03:46:28Z

History of overall pressure ratios: Typo fixing, typos fixed: innaccuracies → inaccuracies using AWB

New page

The author's title is Christy Brookins. The preferred pastime for him and his kids is fashion and he'll be starting something else along with it. Credit authorising is how he tends to make money. Ohio is where his home is and his family loves it.<br><br>Here is my webpage: online psychic readings ([http://www.herandkingscounty.com/content/information-and-facts-you-must-know-about-hobbies www.herandkingscounty.com])