A Limiting Analysis on Regularization of Singular SDP and its Implication to Infeasible Interior-point Algorithms

TL;DR

This paper analyzes the asymptotic behavior of the optimal value in ill-posed semidefinite programs under small perturbations, revealing convergence properties and implications for infeasible interior-point algorithms.

Contribution

It provides a novel asymptotic analysis of perturbed SDPs without assumptions like compactness, showing convergence of optimal values and insights into infeasible interior-point methods.

Findings

Optimal values of perturbed problems converge between primal and dual values.

Infeasible interior-point algorithms converge to a value between primal and dual solutions.

Analysis applies without requiring constraint qualifications or compactness.

Abstract

We consider primal-dual pairs of semidefinite programs and assume that they are ill-posed, i.e., both primal and dual are either weakly feasible or weakly infeasible. Under such circumstances, strong duality may break down and the primal and dual might have a nonzero duality gap. Nevertheless, there are arbitrary small perturbations to the problem data which makes the perturbed primal-dual pair strongly feasible thus zeroing the duality gap. In this paper, we conduct an asymptotic analysis of the optimal value as the perturbation is driven to zero. Specifically, we fix two positive definite matrices (typically the identity matrices), and shift the associated affine spaces of the primal and dual slightly in the direction of the two positive definite matrices possibly in a different proportion so that the perturbed problems have interior feasible solutions, and analyze the behavior of the optimal value of the perturbed problem when the perturbation is reduced to zero keeping the proportion. A key feature of our analysis is that no further assumptions such as compactness or constraint qualifications are ever made. It will be shown that the optimal value of the perturbed problem converges to a value between the primal and dual optimal values of the original problem. Finally, the analysis leads us to the relatively surprising consequence that the infeasible interior-point algorithms for SDP generates a sequence converging to a number between the primal and dual optimal values, even in the presence of a nonzero duality gap. We expect that this property might be particularly useful in solving mixed integer SDPs with infeasible interior-point methods.