Quadratic convergence of an SQP method for some optimization problems with applications to control theory

TL;DR

This paper proves quadratic convergence of a specific SQP algorithm for certain control theory problems under particular optimality conditions, supported by examples and computational comparisons.

Contribution

It establishes stability and quadratic convergence of an SQP method for a class of control problems with no-gap second-order conditions and strict complementarity.

Findings

Quadratic convergence in L^q for all q in [p, ∞], p ≥ 2.

Applicability to many optimal control problems of PDEs.

Computational comparison with other SQP methods shows competitive performance.

Abstract

We analyze a sequential quadratic programming algorithm for solving a class of abstract optimization problems. Assuming that the initial point is in an $L^2$ neighborhood of a local solution that satisfies no-gap second-order sufficient optimality conditions and a strict complementarity condition, we obtain stability and quadratic convergence in $L^q$ for all $q\in[p,\infty]$ where $p\geq 2$ depends on the problem. Many of the usual optimal control problems of partial differential equations fit into this abstract formulation. Some examples are given in the paper. Finally, a computational comparison with other versions of the SQP method is presented.