Linear programming with positive semi-definite matrices
TLDR
In this article, the authors considered the general linear programming problem over the cone of positive semi-definite matrices and provided a simple sufficient condition for existence of optimal solutions and absence of a duality gap without requiring existence of a strictly feasible solution.Abstract:
We consider the general linear programming problem over the cone of positive semi-definite matrices. We first provide a simple sufficient condition for existence of optimal solutions and absence of a duality gap without requiring existence of a strictly feasible solution. We then simply characterize the analogues of the standard concepts of linear programming, i.e., extreme points, basis, reduced cost, degeneracy, pivoting step as well as a Simplex-like algorithm.read more
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Semidefinite and Cone Programming Bibliography/Comments
TL;DR: This online technical report presents abstracts (short outlines) of papers related to semidefinite programming, grouped by subject.
Journal ArticleDOI
A pivoting procedure for a class of second-order cone programming
TL;DR: Under a nondegeneracy assumption, it is proved that the objective function value is strictly decreasing by a pivot unless the current basic solution is optimal, and an algorithm is proposed which has a global convergence property.
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Two-phase simplex method for linear semidefinite optimization
TL;DR: The variant of the primal simplex method, that generalizes the corresponding method for linear programming problems, is proposed and the local convergence of the method is proven.
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A feasible dual affine scaling steepest descent method for the linear semidefinite programming problem
TL;DR: In this article, the dual affine scaling method was proposed for linear semidefinite programming problems, in which all current iterations belong to the feasible set, and the boundaries of the finite set may be reached.
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A Variant of the Dual Simplex Method for a Linear Semidefinite Programming Problem
TL;DR: A linear semidefinite programming problem in the standard statement is considered, and a variant of the dual simplex method is proposed for its solution.