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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] |
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| A '''second-order cone program''' ('''SOCP''') is a [[convex optimization]] problem of the form
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| :minimize <math>\ f^T x \ </math>
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| :subject to
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| ::<math>\lVert A_i x + b_i \rVert_2 \leq c_i^T x + d_i,\quad i = 1,\dots,m</math>
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| ::<math>Fx = g \ </math>
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| 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>
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| 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]].
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| ==Example: Quadratic constraint==
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| Consider a [[quadratically constrained quadratic program|quadratic constraint]] of the form
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| :<math> x^T A^T A x + b^T x + c \leq 0. </math>
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| This is equivalent to the SOC constraint
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| :<math> \left\|
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| \begin{matrix}
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| (1 + b^T x +c)/2\\
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| Ax
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| \end{matrix} \right\|_2
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| \leq (1 - b^T x -c)/2.</math>
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| ==Example: Stochastic programming==
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| Consider a [[stochastic linear program]] in inequality form
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| :minimize <math>\ c^T x \ </math>
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| :subject to
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| :: <math>P(a_i^T(x) \geq b_i) \geq p, \quad i = 1,\dots,m </math>
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| 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
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| :minimize <math>\ c^T x \ </math>
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| :subject to
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| :: <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>
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| where <math>\Phi^{-1} \ </math> is the inverse [[error function]].<ref name="boyd"/>
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| ==Solvers and scripting (programming) languages==
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| {| class="wikitable"
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| |-
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| !Name
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| !License
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| !Brief info
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| |Xpress||commercial|| from 7.6 release
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| |[[CPLEX]]||commercial||
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| |[[Gurobi]]||commercial||parallel SOCP barrier algorithm
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| |JOptimizer||[[Apache License]]|| Java library for convex optimization (open source)
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| |[[MOSEK]]||commercial||
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| |[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>
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| |[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>
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| |[[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.
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| |}
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| ==References==
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| {{reflist}}
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| [[Category:Mathematical optimization]]
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| [[Category:Convex optimization]]
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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.
Also visit my webpage - online psychic chat