A variable metric method for nonsmooth convex constrained optimization

We propose an algorithm for solving nonsmooth convex constrained problems, which combines the ideas of the variable metric methods with the filter strategy for evaluating candidate points. The resulted algorithm inherits some attractive features from both approaches. On the one hand, the time-consuming quadratic programming sub-problems do not need to be solved. On the other hand, the dominance concept of multiobjective optimization was used and the new iterate must satisfy so-called filter criterion. Some encouraging preliminary computational results are reported.

[1]  L. Luksan,et al.  Globally Convergent Variable Metric Method for Nonconvex Nondifferentiable Unconstrained Minimization , 2001 .

[2]  Sven Leyffer,et al.  A bundle filter method for nonsmooth nonlinear optimization , 1999 .

[3]  K. Kiwiel.  A constraint linearization method for nondifferentiable convex minimization , 1987 .

[4]  Sven Leyffer,et al.  On the Global Convergence of a Filter--SQP Algorithm , 2002, SIAM J. Optim..

[5]  Klaus Schittkowski,et al.  More test examples for nonlinear programming codes , 1981 .

[6]  Claudia A. Sagastizábal,et al.  An Infeasible Bundle Method for Nonsmooth Convex Constrained Optimization without a Penalty Function or a Filter , 2005, SIAM J. Optim..

[7]  Jacek Gondzio,et al.  ACCPM — A library for convex optimization based on an analytic center cutting plane method☆ , 1996 .

[8]  K. Kiwiel.  An Exact Penalty Function Algorithm for Non-smooth Convex Constrained Minimization Problems , 1985 .

[9]  Robert Mifflin,et al.  An Algorithm for Constrained Optimization with Semismooth Functions , 1977, Math. Oper. Res..

[10]  Jochem Zowe,et al.  A Version of the Bundle Idea for Minimizing a Nonsmooth Function: Conceptual Idea, Convergence Analysis, Numerical Results , 1992, SIAM J. Optim..

[11]  L. Luksan,et al.  Globally Convergent Variable Metric Method for Convex Nonsmooth Unconstrained Minimization1 , 1999 .

[12]  Pravin M. Vaidya,et al.  A cutting plane algorithm for convex programming that uses analytic centers , 1995, Math. Program..

[13]  K. Kiwiel.  An ellipsoid trust region bundle method for nonsmooth convex minimization , 1989 .

[14]  Sven Leyffer,et al.  Nonlinear programming without a penalty function , 2002, Math. Program..

[15]  Krzysztof C. Kiwiel,et al.  Proximity control in bundle methods for convex nondifferentiable minimization , 1990, Math. Program..

[16]  P. Neittaanmäki,et al.  Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control , 1992 .

[17]  Jean-Louis Goffin,et al.  A Nonlinear Analytic Center Cutting Plane Method for a Class of Convex Programming Problems , 1996, SIAM J. Optim..

[18]  H. Kunzi,et al.  Lectu re Notes in Economics and Mathematical Systems , 1975 .

[19]  Krzysztof C. Kiwiel,et al.  Efficiency of the Analytic Center Cutting Plane Method for Convex Minimization , 1997, SIAM J. Optim..

[20]  Krzysztof C. Kiwiel,et al.  Exact penalty functions in proximal bundle methods for constrained convex nondifferentiable minimization , 1991, Math. Program..