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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Ulbrich, Michael Ulbrich, Stefan
Copyright Year	2000
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.
Starting Page	1938
Ending Page	1984
Page Count	47
File Format	PDF
ISSN	03630129
DOI	10.1137/S0363012997325915
e-ISSN	10957138
Issue Number	6
Volume Number	38
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-26
Access Restriction	Subscribed
Subject Keyword	nonlinear programming optimal control Programming in abstract spaces affine scaling infinite-dimensional optimization bound constraints interior-point algorithms Problems in abstract spaces optimality conditions trust-region methods Mathematical programming methods Newton-type methods superlinear convergence Nonlinear programming Methods of nonlinear programming type
Content Type	Text
Resource Type	Article
Subject	Control and Optimization Applied Mathematics

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