A numerical feasible interior point method for linear semidefinite programs

Abstract

This paper presents a feasible primal algorithm for linear semidefinite programming. The algorithm starts with a strictly feasible solution, but in case where no such a solution is known, an application of the algorithm to an associate problem allows to obtain one. Finally, we present some numerical experiments which show that the algorithm works properly.

A new efficient primal-dual projective interior point method for semidefinite programming

“…In this section, we recall some results given in [24] and derive the prototype of the algorithm which gives an optimal primal-dual solution of (SDP) and (DSDP).…”

“…The Cholesky factorization of the positive definite matrix X k gives a lower triangular matrix L k with positive diagonal entries such that L k L t k = X k . We use the projective transformation given by Benterki, Crouzeix and Merikhi [24] to bring the current iterate to the centre which is the identity matrix. Define T k :…”

“…To facilities the study of (TSDP3), Benterki, Crouzeix and Merikhi [24] set V = X − I and v = x − 1. (TSDP3) is equivalent to getting the optimal solution of the convex optimization problem…”

“…Consequently, various primal-dual interior-point algorithms were proposed by different authors; see, e.g., [2,11,16,17,18,19,20,21,22,23] and the references therein. Inspired by Alizadeh [2] and Benterki, Crouzeix and Merikhi [24], the main aim of this paper is to construct a polynomial feasible primal-dual projective interior point algorithm with two phases. The first phase gives an initial strictly feasible solution of (SDP), and the second one gives a primal-dual optimal solution of (SDP) and (DSDP).…”

A logarithmic barrier approach for linear programming

“…In semidefinite programming problems, effective and less expensive procedures are proposed by several researchers to avoid line search methods on one hand and to accelerate the convergence of algorithm on the other hand [2,4].…”