Local Convergence Results for Sequential Quadratic Programming with Complementarity Constraints

TL;DR

This paper presents new local convergence results for the SQPCC method applied to MPCCs, showing convergence to S-stationary points under weaker conditions than previous analyses.

Contribution

The paper introduces a novel convergence analysis for SQPCC, requiring no strict complementarity and building on classical SQP results, with implications for active-set identification.

Findings

Existence of converging sequences of S-stationary points

Conditions for local convergence and uniqueness

Active-set stabilization after finitely many iterations

Abstract

Mathematical programs with complementarity constraints (MPCCs) are a challenging class of nonlinear optimization problems, because their nonlinear programming reformulations violate standard constraint qualifications at every feasible point. This paper analyzes sequential quadratic programming with complementarity constraints (SQPCC). In this method, the complementarity constraints are retained in the subproblems, yielding quadratic programs with complementarity constraints (QPCCs). The main contribution of the paper is a new local convergence result for the SQPCC method to S-stationary points. We show that there exists at least one sequence of QPCC S-stationary points converging to a reference S-stationary point of the MPCC, and we characterize conditions under which each such sequence converges and under which such a sequence is locally unique. In contrast to previous results, the analysis is established under weaker second-order sufficient conditions and requires no upper-level strict complementarity. Our result builds upon local convergence results for the classical SQP method. We present local convergence results for SQP within the kappa-omega setting, which also cover linearly convergent variants. These assumptions, which are widely used in the analysis of Newton-type methods, provide a natural framework for quantifying subproblem approximation errors, their effect on the convergence speed, and the effect of nonlinearity on the size of the local convergence region. Furthermore, we establish an active-set stabilization result for SQPCC, identifying conditions under which the optimal complementarity and inequality active sets are identified after finitely many iterations, and conditions under which such identification occurs only asymptotically. Numerical examples illustrate the theoretical findings and highlight some advantages of SQPCC over classical SQP applied to MPCCs.