A polynomial projective algorithm for convex feasibility problems with positive-definite constraints

TL;DR

This paper introduces a polynomial-time algorithm for convex feasibility problems with positive-definite constraints, utilizing projective transformations of spectraplexes to efficiently find solutions or valid inequalities.

Contribution

It proposes a novel polynomial projective algorithm based on spectraplex transformations for convex feasibility problems with positive-definite constraints.

Findings

Algorithm runs in polynomial time under certain conditions.

Provides more precise complexity bounds for specific problem cases.

Either finds a feasible solution or improves the solution set iteratively.

Abstract

We study a class of projective transformations of spectraplexes associated with self-dual cones and, on this basis, propose a polynomial-time algorithm for convex feasibility problems with positive definite constraints. At each iteration of the algorithm, either a feasible solution is found or a suitable valid inequality inducing a projective transformation allowing to bring the solution set closer to the center of an associated spectraplex. The closeness to the center is measured in terms of a potential function. The running time of our algorithm makes the existing complexity bounds more precise for the case when the number of equations linking the positive definite variable matrices is not less than the sum of the ranks of the respective positive-semidefinite cones.