Linearly Convergent First-Order Algorithms for Semi-definite Programming

TL;DR

This paper introduces new linearly convergent first-order algorithms for solving semi-definite programming problems, including smooth, non-smooth, and special cases, with a bundle-level method that does not require smoothness information.

Contribution

It presents novel algorithms with linear convergence for both smooth and non-smooth LMIs, and introduces a bundle-level method applicable without smoothness assumptions.

Findings

Algorithms achieve linear convergence rates.

Bundle-level method works without smoothness information.

Applicable to special cases with weaker assumptions.

Abstract

In this paper, we consider two formulations for Linear Matrix Inequalities (LMIs) under Slater type constraint qualification assumption, namely, SDP smooth and non-smooth formulations. We also propose two first-order linearly convergent algorithms for solving these formulations. Moreover, we introduce a bundle-level method which converges linearly uniformly for both smooth and non-smooth problems and does not require any smoothness information. The convergence properties of these algorithms are also discussed. Finally, we consider a special case of LMIs, linear system of inequalities, and show that a linearly convergent algorithm can be obtained under a weaker assumption.