An Infeasible-Start Framework for Convex Quadratic Optimization, with Application to Constraint-Reduced Interior-Point Methods

TL;DR

This paper introduces a new framework for solving convex quadratic programs from infeasible starting points using an existing feasible-start algorithm, with applications to interior-point methods and demonstrated improvements on imbalanced problems.

Contribution

The paper develops an infeasible-start framework for convex quadratic programming that guarantees convergence and provides infeasibility certificates, extending interior-point methods to infeasible initial points.

Findings

Framework converges under mild assumptions

Produces infeasibility certificates at minimal additional cost

Outperforms standard solvers on imbalanced problems

Abstract

A framework is proposed for solving general convex quadratic programs (CQPs) from an infeasible starting point by invoking an existing feasible-start algorithm tailored for inequality-constrained CQPs. The central tool is an exact penalty function scheme equipped with a penalty-parameter updating rule. The feasible-start algorithm merely has to satisfy certain general requirements, and so is the updating rule. Under mild assumptions, the framework is proved to converge on CQPs with both inequality and equality constraints and, at a negligible additional cost per iteration, produces an infeasibility certificate, together with a feasible point for an (approximately) $\ell_1$-least relaxed feasible problem when the given problem does not have a feasible solution. The framework is applied to a feasible-start constraint-reduced interior-point algorithm previously proved to be highly performant on problems with many more constraints than variables ("imbalanced"). Numerical comparison with popular codes (SDPT3, SeDuMi, MOSEK) is reported on both randomly generated problems and support-vector machine classifier training problems. The results show that the former typically outperforms the latter on imbalanced problems.