An Infinite-Dimensional Convergence Theory for Reduced SQP Methods in Hilbert Space

Abstract

We develop a general convergence theory for a class of reduced successive quadratic programming (SQP) methods for infinite-dimensional equality constrained optimization problems in Hilbert space. The methods under consideration approximate second order information of the Lagrangian restricted to the null space of $h'$, the derivative of the constraints. In particular, a sufficient condition for two-step superlinear convergence will be given, and a general result for secant methods will be established which can serve as a tool to prove superlinear convergence in an infinite-dimensional framework. This result is applied to prove local two-step superlinear convergence of reduced SQP methods under the assumption that exact reduced second order information is known up to a compact perturbation. To test our convergence theory we consider optimal control problems, and we formulate a reduced quasi-Newton algorithm which presents a new approach to an efficient solution of these problems. Our main motivation for studying reduced SQP is that these methods allow us to exploit special structure of the null space of $h'$ which usually leads to more reasonable storage requirements. Such a structure is often the result of a natural separability of variables which appears frequently in infinite-dimensional applications, as for example in optimal control and in parameter identification problems. This special structure suggests the use of a representation of the null space of $h'$ which is nonstandard in the reduced SQP context.