Interior Point Methods in Function Space

A primal-dual interior point method for optimal control problems is considered. The algorithm is directly applied to the infinite-dimensional problem. Existence and convergence of the central path are analyzed, and linear convergence of a short-step path-following method is established.

[1]  Peter Deuflhard,et al.  The Central Path towards the Numerical Solution of Optimal Control Problems , 2001 .

[2]  Michael Ulbrich,et al.  Semismooth Newton Methods for Operator Equations in Function Spaces , 2002, SIAM J. Optim..

[3]  Karl Kunisch,et al.  A Comparison of a Moreau-Yosida-Based Active Set Strategy and Interior Point Methods for Constrained Optimal Control Problems , 2000, SIAM J. Optim..

[4]  F. Potra,et al.  Asymptotic mesh independence of Newton-Galerkin methods via a refined Mysovskii theorem , 1992 .

[5]  Matthias Heinkenschloss,et al.  Analysis of the Lagrange-SQP-Newton Method for the Control of a Phase Field Equation , 1998 .

[6]  P. Deuflhard A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting , 1974 .

[7]  H. Bock,et al.  A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems , 1984 .

[8]  S. Ito,et al.  Inexact primal-dual interior point iteration for linear programs in function spaces , 1995, Comput. Optim. Appl..

[9]  Stefan Ulbrich,et al.  Superlinear Convergence of Affine-Scaling Interior-Point Newton Methods for Infinite-Dimensional Nonlinear Problems with Pointwise Bounds , 2000, SIAM J. Control. Optim..

[10]  Dietrich Braess,et al.  A multigrid method for a parameter dependent problem in solid mechanics , 1990 .

[11]  S. Kurcyusz On the existence and nonexistence of Lagrange multipliers in Banach spaces , 1976 .

[12]  William W. Hager,et al.  Runge-Kutta methods in optimal control and the transformed adjoint system , 2000, Numerische Mathematik.

[13]  Gary R. Consolazio,et al.  Finite Elements , 2007, Handbook of Dynamic System Modeling.

[14]  E. Zeidler Nonlinear functional analysis and its applications , 1988 .

[15]  U. Ascher,et al.  A New Basis Implementation for a Mixed Order Boundary Value ODE Solver , 1987 .

[16]  O. V. Stryk,et al.  Numerical Solution of Optimal Control Problems by Direct Collocation , 1993 .

[17]  G. Tallini,et al.  ON THE EXISTENCE OF , 1996 .

[18]  William W. Hager,et al.  Uniform Convergence and Mesh Independence of Newton's Method for Discretized Variational Problems , 2000, SIAM J. Control. Optim..

[19]  Robert D. Russell,et al.  Collocation Software for Boundary-Value ODEs , 1981, TOMS.

[20]  Fredi Tröltzsch,et al.  On the Lagrange--Newton--SQP Method for the Optimal Control of Semilinear Parabolic Equations , 1999, SIAM J. Control. Optim..

[21]  Bintong Chen,et al.  A Global Linear and Local Quadratic Noninterior Continuation Method for Nonlinear Complementarity Problems Based on Chen-Mangasarian Smoothing Functions , 1999, SIAM J. Optim..

[22]  Kazimierz Malanowski,et al.  The Lagrange-Newton method for state constrained optimal control problems , 1995, Comput. Optim. Appl..

[23]  Stephen J. Wright,et al.  Warm-Start Strategies in Interior-Point Methods for Linear Programming , 2002, SIAM J. Optim..

[24]  V. Schulz Solving discretized optimization problems by partially reduced SQP methods , 1998 .

[25]  Fredi Troltzsch,et al.  An SQP method for the optimal control of a nonlinear heat equation , 1994 .

[26]  A. Fischer A special newton-type optimization method , 1992 .

[27]  Stephen M. Robinson,et al.  Generalized Equations , 1982, ISMP.

[28]  Anders Forsgren,et al.  Interior Methods for Nonlinear Optimization , 2002, SIAM Rev..

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

[30]  Olvi L. Mangasarian,et al.  A class of smoothing functions for nonlinear and mixed complementarity problems , 1996, Comput. Optim. Appl..

[31]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[32]  F. Brezzi On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[33]  P. Toint,et al.  Numerical Methods for Large-Scale Non-Convex Quadratic Programming , 2002 .

[34]  M. Weiser Function Space Complementarity Methods for Optimal Control Problems , 2001 .

[35]  James V. Burke,et al.  The Global Linear Convergence of a Noninterior Path-Following Algorithm for Linear Complementarity Problems , 1998, Math. Oper. Res..

[36]  K. C. P. Machielsen Numerical solution of state constrained optimal control problems , 1988 .

[37]  P. Deuflhard A stepsize control for continuation methods and its special application to multiple shooting techniques , 1979 .

[38]  Michael Hintermüller,et al.  A SQP-Semismooth Newton-type Algorithm applied to Control of the instationary Navier--Stokes System Subject to Control Constraints , 2006, SIAM J. Optim..

[39]  Christian Kanzow,et al.  Some Noninterior Continuation Methods for Linear Complementarity Problems , 1996, SIAM J. Matrix Anal. Appl..

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

[41]  Robert D. Russell,et al.  Numerical solution of boundary value problems for ordinary differential equations , 1995, Classics in applied mathematics.