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

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 st...

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