Superlinear convergence of affine-scaling interior-point Newton methods for infinite-dimensional nonlinear problems with pointwise bounds

We develop and analyze a superlinearly convergent affine-scaling interior-p oint Newton method for infinite-dimensional problems with pointwise bounds in L-p-space. The problem formulation is motivated by optimal control probl ems with L-p-controls and pointwise control constraints. The finite-dimensi onal convergence theory by Coleman and Li [SIAM J. Optim., 6 (1996), pp. 41 8-445] makes essential use of the equivalence of norms and the exact identi fiability of the active constraints close to an optimizer with strict compl ementarity. Since these features are not available in our infinite-dimensio nal framework, algorithmic changes are necessary to ensure fast local conve rgence. The main building block is a Newton-like iteration for an affine-sc aling formulation of the KKT-condition. We demonstrate in an example that a stepsize rule to obtain an interior iterate may require very small stepsiz es even arbitrarily close to a nondegenerate solution. Using a pointwise pr ojection instead we prove superlinear convergence under a weak strict compl ementarity condition and convergence with Q-rate >1 under a slightly strong er 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. Nume rical results for the control of a heating process confirm our theoretical findings.