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-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.

[1]  Ekkehard W. Sachs,et al.  A Parabolic Control Problem with a Boundary Condition of the Stefan‐Boltzmann Type , 1978 .

[2]  Matthias Heinkenschloss.  Projected Sequential Quadratic Programming Methods , 1996, SIAM J. Optim..

[3]  Ekkehard W. Sachs,et al.  A Trust Region Method for Parabolic Boundary Control Problems , 1999, SIAM J. Optim..

[4]  M. Heinkenschloss.  A Trust Region Method for Norm Constrained Problems , 1998 .

[5]  D. Bertsekas.  Projected Newton methods for optimization problems with simple constraints , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[6]  J. C. Dunn,et al.  Variants of the Kuhn-Tucker sufficient conditions in cones of nonnegative functions , 1992 .

[7]  J. C. Dunn,et al.  On the Gradient Projection Method for Optimal Control Problems with Nonnegative ${\cal L}^2$ Inputs , 1994 .

[8]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[9]  Ekkehard W. Sachs,et al.  Solution of Optimal Control Problems by a Pointwise Projected Newton Method , 1995 .

[10]  M. Heinkenschloss,et al.  Global Convergence of Trust-Region Interior-Point Algorithms for Infinite-Dimensional Nonconvex Mini , 1999 .

[11]  L. N. Vicente,et al.  Trust-Region Interior-Point SQP Algorithms for a Class of Nonlinear Programming Problems , 1998 .

[12]  Thomas F. Coleman,et al.  On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds , 1994, Math. Program..

[13]  Thomas F. Coleman,et al.  A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems , 1999, SIAM J. Sci. Comput..

[14]  W. Alt.  The lagrange-newton method for infinite-dimensional optimization problems , 1990 .

[15]  Ekkehard W. Sachs,et al.  Multilevel Algorithms for Constrained Compact Fixed Point Problems , 1994, SIAM J. Sci. Comput..

[16]  A. Griewank.  The local convergence of Broyden-like methods on Lipschitzian problems in Hilbert spaces , 1987 .

[17]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[18]  Ekkehard W. Sachs,et al.  Numerical solution of a nonlinear parabolic control problem by a reduced SQP method , 1992, Comput. Optim. Appl..

[19]  J. Burger,et al.  Functional and numerical solution of a control problem originating from heat transfer , 1991 .

[20]  Luis S. Vicente.  On interior-point Newton algorithms for discretized optimal control problems with state constraints ∗ , 1998 .

[21]  J. C. Dunn,et al.  On L 2 Sufficient Conditions and the Gradient Projection Method for Optimal Control Problems , 1996 .

[22]  H. Maurer.  First and second order sufficient optimality conditions in mathematical programming and optimal control , 1981 .

[23]  Joseph C. Dunn.  Second-Order Optimality Conditions in Sets of ${\sf L}^\infty$ Functions with Range in a Polyhedron , 1995 .

[24]  M. Heinkenschloss,et al.  ANALYSIS OF INEXACT TRUST – REGION INTERIOR – POINT SQP ALGORITHMS , 1995 .

[25]  C. Kelley,et al.  Quasi Newton methods and unconstrained optimal control problems , 1986, 1986 25th IEEE Conference on Decision and Control.

[26]  Thomas F. Coleman,et al.  An Interior Trust Region Approach for Nonlinear Minimization Subject to Bounds , 1993, SIAM J. Optim..