Primal superlinear convergence of SQP methods in piecewise linear-quadratic composite optimization

TL;DR

This paper establishes the primal superlinear convergence of quasi-Newton SQP methods for piecewise linear-quadratic composite optimization, under conditions related to Lagrange multipliers and second-order sufficiency.

Contribution

It extends convergence analysis for SQP methods in composite problems by linking noncriticality, second-order conditions, and Dennis-More criteria.

Findings

Primal superlinear convergence is justified under noncriticality and Dennis-More conditions.

Replacing noncriticality with second-order sufficiency makes convergence equivalent to Dennis-More.

Recovery of Bonnans' primal-dual superlinear convergence result under specific conditions.

Abstract

This paper mainly concerns with the primal superlinear convergence of the quasi-Newton sequential quadratic programming (SQP) method for piecewise linear-quadratic composite optimization problems. We show that the latter primal superlinear convergence can be justified under the noncriticality of Lagrange multipliers and a version of the Dennis-More condition. Furthermore, we show that if we replace the noncriticality condition with the second-order sufficient condition, this primal superlinear convergence is equivalent with an appropriate version of the Dennis-More condition. We also recover Bonnans' result in [1] for the primal-dual superlinear of the basic SQP method for this class of composite problems under the second-order sufficient condition and the uniqueness of Lagrange multipliers. To achieve these goals, we first obtain an extension of the reduction lemma for convex Piecewise linear-quadratic functions and then provide a comprehensive analysis of the noncriticality of Lagrange multipliers for composite problems. We also establish certain primal estimates for KKT systems of composite problems, which play a significant role in our local convergence analysis of the quasi-Newton SQP method.