Iterative Inner/outer Approximations for Scalable Semidefinite Programs using Block Factor-width-two Matrices

TL;DR

This paper introduces iterative inner and outer approximation algorithms for semidefinite programs using block factor-width-two matrices, providing scalable solutions with guaranteed bounds and improved efficiency.

Contribution

The paper develops a novel iterative approximation framework for SDPs based on block factor-width-two matrices, enhancing scalability and solution quality.

Findings

Algorithms generate monotonically decreasing upper bounds and increasing lower bounds.

Numerical results confirm the effectiveness and efficiency of the proposed methods.

The approach generalizes scaled diagonally dominance approximations.

Abstract

In this paper, we propose iterative inner/outer approximations based on a recent notion of block factor-width-two matrices for solving semidefinite programs (SDPs). Our inner/outer approximating algorithms generate a sequence of upper/lower bounds of increasing accuracy for the optimal SDP cost. The block partition in our algorithms offers flexibility in terms of both numerical efficiency and solution quality, which includes the approach of scaled diagonally dominance (SDD) approximation as a special case. We discuss both the theoretical results and numerical implementation in detail. Our main theorems guarantee that the proposed iterative algorithms generate monotonically decreasing upper (increasing lower) bounds. Extensive numerical results confirm our findings.