Superlinear Convergence of Affine-Scaling Interior-Point Newton Methods for Infinite-Dimensional Nonlinear Problems with Pointwise Bounds

Abstract

We develop and analyze a superlinearly convergent affine-scaling interior-point Newton method for infinite-dimensional problems with pointwise bounds in Lp-space. The problem formulation is motivated by optimal control problems with Lp-controls and pointwise control constraints. The finite-dimensional convergence theory by Coleman and Li [SIAM J. Optim., 6 (1996), pp. 418--445] makes essential use of the equivalence of norms and the exact identifiability of the active constraints close to an optimizer with strict complementarity. Since these features are not available in our infinite-dimensional framework, algorithmic changes are necessary to ensure fast local convergence. The main building block is a Newton-like iteration for an affine-scaling formulation of the KKT-condition. We demonstrate in an example that a stepsize rule to obtain an interior iterate may require very small stepsizes even arbitrarily close to a nondegenerate solution. Using a pointwise projection instead we prove superlinear convergence under a weak strict complementarity condition and convergence with \mbox{Q-rate $>$1} under a slightly stronger condition if a smoothing step is available. We discuss how the algorithm can be embedded in the class of globally convergent trust-region interior-point methods recently developed by M. Heinkenschloss and the authors. Numerical results for the control of a heating process confirm our theoretical findings.