Penalty and Smoothing Methods for Convex Semi-Infinite Programming

In this paper we consider min-max convex semi-infinite programming. To solve these problems we introduce a unified framework concerning Remez-type algorithms and integral methods coupled with penalty and smoothing methods. This framework subsumes well-known classical algorithms, but also provides some new methods with interesting properties. Convergence of the primal and dual sequences are proved under minimal assumptions.

[1]  R. Sheu,et al.  Solving Continuous Min-Max Problems by an Iterative Entropic Regularization Method , 2004 .

[2]  Marc Teboulle.  Nonlinear perturbations for linear semi-infinite optimization problems , 1990, 29th IEEE Conference on Decision and Control.

[3]  Johannes O. Royset,et al.  Algorithms for Finite and Semi-Infinite Min–Max–Min Problems Using Adaptive Smoothing Techniques , 2003 .

[4]  David Q. Mayne,et al.  A barrier function method for minimax problems , 1992, Math. Program..

[5]  A. A. Goldstein,et al.  Newton's method for convex programming and Tchebycheff approximation , 1959, Numerische Mathematik.

[6]  M. Teboulle,et al.  A smoothing technique for nondifferentiable optimization problems , 1988 .

[7]  C. Goh,et al.  A simple computational procedure for optimization problems with functional inequality constraints , 1987 .

[8]  Rembert Reemtsen,et al.  Numerical Methods for Semi-Infinite Programming: A Survey , 1998 .

[9]  J. Frédéric Bonnans,et al.  Perturbation Analysis of Optimization Problems , 2000, Springer Series in Operations Research.

[10]  Ulrich Schättler,et al.  An interior-point method for semi-infinite programming problems , 1996, Ann. Oper. Res..

[11]  Susana Gómez,et al.  A regularization method for solving the finite convex min-max problem , 1990 .

[12]  Yurii Nesterov,et al.  Smooth minimization of non-smooth functions , 2005, Math. Program..

[13]  D. Bertsekas.  Approximation procedures based on the method of multipliers , 1977 .

[14]  Kok Lay Teo,et al.  A new computational algorithm for functional inequality constrained optimization problems , 1993, Autom..

[15]  Alfred Auslender.  Penalty and Barrier Methods: A Unified Framework , 1999, SIAM J. Optim..

[16]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[17]  Chih-Jen Lin,et al.  An Unconstrained Convex Programming Approach to Linear Semi-Infinite Programming , 1998, SIAM J. Optim..

[18]  Johannes O. Royset,et al.  Algorithms with Adaptive Smoothing for Finite Minimax Problems , 2003 .

[19]  Bernard Martinet,et al.  Algorithmes pour la résolution de problèmes d'optimisation et de minimax , 1972 .

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

[21]  M. Teboulle,et al.  Asymptotic cones and functions in optimization and variational inequalities , 2002 .

[22]  Shu-Cherng Fang,et al.  Solving min-max problems and linear semi-infinite programs , 1996 .

[23]  J. E. Kelley,et al.  The Cutting-Plane Method for Solving Convex Programs , 1960 .

[24]  Soon-Yi Wu,et al.  Combined Entropic Regularization and Path-Following Method for Solving Finite Convex Min-max Problems Subject to Infinitely Many Linear Constraints , 1999 .

[25]  Roberto Cominetti,et al.  Asymptotic Analysis for Penalty and Barrier Methods in Convex and Linear Programming , 1997, Math. Oper. Res..

[26]  Jack Elzinga,et al.  A central cutting plane algorithm for the convex programming problem , 1975, Math. Program..

[27]  A. Auslender.  Méthodes et théorèmes de dualité , 1970 .

[28]  Arthur F. Veinott,et al.  The Supporting Hyperplane Method for Unimodal Programming , 1967, Oper. Res..

[29]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.