Range of the displacement operator of PDHG with applications to quadratic and conic programming

TL;DR

This paper develops a new theoretical framework to analyze the displacement operator of PDHG, enabling it to diagnose infeasibility or unboundedness in quadratic and conic programming problems.

Contribution

It introduces a novel formula for the range of the displacement operator using monotone operator theory, specialized to quadratic and conic programming.

Findings

PDHG can diagnose infeasible or unbounded quadratic programming instances.

The analysis applies to the ellipsoid-separation problem within SOCP.

New theoretical insights into the behavior of PDHG in convex optimization.

Abstract

Primal-dual hybrid gradient (PDHG) is a first-order method for saddle-point problems and convex programming introduced by Chambolle and Pock. Recently, Applegate et al.\ analyzed the behavior of PDHG when applied to an infeasible or unbounded instance of linear programming, and in particular, showed that PDHG is able to diagnose these conditions. Their analysis hinges on the notion of the infimal displacement vector in the closure of the range of the displacement mapping of the splitting operator that encodes the PDHG algorithm. In this paper, we develop a novel formula for this range using monotone operator theory. The analysis is then specialized to conic programming and further to quadratic programming (QP) and second-order cone programming (SOCP). A consequence of our analysis is that PDHG is able to diagnose infeasible or unbounded instances of QP and of the ellipsoid-separation problem, a subclass of SOCP.