Second order cone programming

From Wikipedia, the free encyclopedia

A Second order cone program (SOCP) is a convex optimization problem of the form

minimize $\ f^T x \$ subject to

$\lVert A_i x + b_i \rVert_2 \leq c_i^T x + d_i,\quad i = 1,\dots,m$

$Fx = g \$

where the problem parameters are $f \in \mathbb{R}^n, \ A_i \in \mathbb{R}^{{n_i}\times n}, \ b_i \in \mathbb{R}^{n_I}, \ c_i \in \mathbb{R}^n, \ d_i \in \mathbb{R}, \ F \in \mathbb{R}^{p\times n}$ , and $g \in \mathbb{R}^p$ . Here $x\in\mathbb{R}^n$ is the optimization variable. When $A i = 0$ for $i = 1,\dots,m$ , the SOCP reduces to a linear program. When $c i = 0$ for $i = 1,\dots,m$ , the SOCP is equivalent to a convex Quadratically constrained quadratic program. SOCPs can be solved with great efficiency by interior point methods.

Consider a stochastic linear program in inequality form

minimize $\ c^T x \$ subject to

$P(a_i^T(x) \leq b_i) \leq \eta, \quad i = 1,\dots,m$

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

minimize $\ c^T x \$ subject to

$\bar{a}_i^T (x) + \Phi^{-1}(\eta) \lVert \Sigma_i^{1/2} x \rVert_2 \leq b_i , \quad i = 1,\dots,m$

where $\Phi^{-1} \$ is the inverse error function.

Stephen Boyd and Lieven Vandenberghe, Convex Optimization (book in pdf)

Software

MOSEK — The first commercially available software package for solution SOCP.

This mathematics-related article is a stub. You can help Wikipedia by expanding it.

Retrieved from "http://en.wikipedia.org../../../s/e/c/Second_order_cone_programming.html"

Categories: Optimization | Mathematics stubs

Advanced Search include all of these words: include this exact phrase: include at least one of these words: exclude these words: language: file type: last updated: limit domain to: that link to: related to:

close

Top Matching Results

Occasionally Search.com will highlight specialized results that are based on the context of your query. Examples of specialized results include specific links to news, images, or video.

Top Matching Results may highlight information from other Search.com pages, content from the CNET Network of sites, or third party content. The listings are based purely on relevance. Search.com does not receive payment for listings in this section but our partners that provide this data may get paid for listing these products.

Sponsored Links
This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Search.com sends your search query to several search engines at one time and integrates the results into one list which has been sorted by relevance using Search.com's proprietary algorithm. You can customize the list of search engines included in your metasearch from the preferences.

The search engines that are used in your metasearch may allow companies to pay to have their Web sites included within the results. To view the Paid Inclusion policy for a specific search engine, please visit their Web site. Search.com does not accept payment or share revenue with any search engine partner for listings in this section.