A Globally Convergent LP and SOCP-based algorithm for Semidefinite Programming

TL;DR

This paper introduces a new algorithm that guarantees global convergence for solving general semidefinite programs efficiently by reducing them to simpler LP and SOCP problems, improving scalability.

Contribution

It presents a polynomial-time, globally convergent algorithm for SDPs that leverages LP and SOCP relaxations, with bounds on iteration complexity.

Findings

Algorithm successfully solves large SDPs with reduced computational resources

Demonstrates effectiveness on random and benchmark SDPs

Provides theoretical bounds on convergence and iteration count

Abstract

Semidefinite programs (SDP) are one of the most versatile frameworks in numerical optimization, serving as generalizations of many conic programs and as relaxations of NP-hard combinatorial problems. Their main drawback is their computational and memory complexity, which sets a practical limit to the size of problems solvable by off-the-shelf SDP solvers. To circumvent this fact, many algorithms have been proposed to exploit the structure of particular problems and increase the scalability of SDPs for those problem instances. Progress has been less steep, however, for general-case SDPs. In this paper, motivated by earlier results by Ahmadi and Hall, we show that a general SDP can be solved to $\epsilon$-optimality, in polynomial time, by performing a sequence of less computationally demanding Linear or Second Order Cone programs. In addition, we provide a bound on the number of iterations required to achieve $\epsilon$-optimality. These results are illustrated using random SDPs and well-known problems from the SDPLib dataset.