Interior-Point Methods for Large-Scale Cone Programming

• Published in: Optimization for Machine Learning p. 55
• Main Authors: Martin Andersen, Joachim Dahl, Zhang Liu, Lieven Vandenberghe
• Format: Book Chapter
• Language: English
• Published: United States The MIT Press 30.09.2011; MIT Press
• Subjects: Algebra; Algorithms; Applied sciences; Artificial Intelligence; Business; Category theory; Cholesky factorization; Computer industry; Computer programming; Computer Science; Conic sections; Direct products; Factorization; Geometric shapes; Geometry; High technology industries; Industrial sectors; Industry; Knowledge industries; Linear algebra; Linear equations; Linear programming; Machine learning; Machine Learning & Neural Networks; Mathematical procedures; Mathematical programming; Mathematics; Matrices; Matrix theory; Programming languages; Pure mathematics; Software industry; Software packages
• ISBN: 026201646X; 9780262016469
• DOI: 10.7551/mitpress/8996.003.0005

The cone programming formulation has been popular in the recent literature on convex optimization. In this chapter we define acone linear program(cone LP or conic LP) as an optimization problem of the form minimize${c^T}x$ subject to$Gx{\underline \prec _C}h$(3.1) $Ax = b$ with optimization variablex. The inequality$Gx{\underline \prec _C}h$is ageneralized inequality, which means that$h - Gx \in C$, whereCis a closed, pointed, convex cone with nonempty interior. We will also encountercone quadratic programs(cone QPs), minimize$(1/2){x^T}Px + {c^T}x$(3.2) subject to$Gx{\underline \prec _C}h$ $Ax = b$, withPpositive semidefinite. If$C = R_ + ^p$(the nonnegative orthant in${R^p}$), the generalized inequality is a componentwise