Bundle-based relaxation methods for multicommodity capacitated fixed charge network design

Abstract To efficiently derive bounds for large-scale instances of the capacitated fixed-charge network design problem, Lagrangian relaxations appear promising. This paper presents the results of comprehensive experiments aimed at calibrating and comparing bundle and subgradient methods applied to the optimization of Lagrangian duals arising from two Lagrangian relaxations. This study substantiates the fact that bundle methods appear superior to subgradient approches because they converge faster and are more robust relative to different relaxations, problem characteristics, and selection of the initial parameter values. It also demonstrates that effective lower bounds may be computed efficiently for large-scale instances of the capacitated fixed-charge network design problem. Indeed, in a fraction of the time required by a standard simplex approach to solve the linear programming relaxation, the methods we present attain very high-quality solutions.

[1]  J. Goffin,et al.  A Lagrangian Relaxation of the Capacitated Multi-Item Lot Sizing Problem Solved with an Interior Poi , 1997 .

[2]  Michel Gendreau,et al.  A Simplex-Based Tabu Search Method for Capacitated Network Design , 2000, INFORMS J. Comput..

[3]  Di Yuan,et al.  A Lagrangian Heuristic Based Branch-and-Bound Approach for the Capacitated Network Design Problem , 2000, Oper. Res..

[4]  P. Camerini,et al.  On improving relaxation methods by modified gradient techniques , 1975 .

[5]  Prakash Mirchandani,et al.  Modeling and Solving the Capacitated Network Loading Problem , 1991 .

[6]  Antonio Frangioni,et al.  Solving semidefinite quadratic problems within nonsmooth optimization algorithms , 1996, Comput. Oper. Res..

[7]  Thomas L. Magnanti,et al.  Modeling and Solving the Two-Facility Capacitated Network Loading Problem , 1995, Oper. Res..

[8]  J. Goffin,et al.  Decomposition and nondifferentiable optimization with the projective algorithm , 1992 .

[9]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[10]  Jacek Gondzio,et al.  Solving nonlinear multicommodity flow problems by the analytic center cutting plane method , 1997, Math. Program..

[11]  Teodor Gabriel Crainic,et al.  RELAXATIONS FOR MULTICOMMODITY CAPACITATED NETWORK DESIGN PROBLEMS. , 1994 .

[12]  Jean-Philippe Vial,et al.  On Improvements to the Analytic Center Cutting Plane Method , 1998, Comput. Optim. Appl..

[13]  Giorgio Gallo,et al.  A Bundle Type Dual-Ascent Approach to Linear Multicommodity Min-Cost Flow Problems , 1999, INFORMS J. Comput..

[14]  Teodor Gabriel Crainic,et al.  Multicommodity Capacitated Network Design , 1999 .

[15]  Brunilde Sansò,et al.  Telecommunications Network Planning , 1998 .

[16]  Bezalel Gavish,et al.  Lower Bounding Procedures for Multiperiod Telecommunications Network Expansion Problems , 1995, Oper. Res..

[17]  Thomas L. Magnanti,et al.  Working Paper Alfred P. Sloan School of Management "models for Planning Capacity Expansion in Local Access Telecommunication Networks" "models for Planning Capacity Expansion in Local Access Telecommunication Networks" Models for Planning Capacity Expansion in Local Access Telecommunication Networks , 2008 .

[18]  M. Patriksson,et al.  Conditional subgradient optimization -- Theory and applications , 1996 .

[19]  Jeffery L. Kennington,et al.  A generalization of Polyak's convergence result for subgradient optimization , 1987, Math. Program..

[20]  Peter L. Hammer,et al.  Discrete Applied Mathematics , 1993 .

[21]  Oktay Günlük,et al.  Capacitated Network Design - Polyhedral Structure and Computation , 1996, INFORMS J. Comput..

[22]  Antonio Frangioni,et al.  Generalized Bundle Methods , 2002, SIAM J. Optim..

[23]  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..

[24]  Bezalel Gavish,et al.  Topological design of telecommunication networks-local access design methods , 1991, Ann. Oper. Res..

[25]  Michel Minoux,et al.  Graphes et algorithmes , 1995 .

[26]  Teodor Gabriel Crainic,et al.  BOUNDING PROCEDURES FOR MULTICOMMODITY CAPACITATED FIXED CHARGE NETWORK DESIGN PROBLEMS. , 1995 .

[27]  Boris Polyak.  Minimization of unsmooth functionals , 1969 .

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

[29]  Thomas L. Magnanti,et al.  Network Design and Transportation Planning: Models and Algorithms , 1984, Transp. Sci..

[30]  Philip Wolfe,et al.  Validation of subgradient optimization , 1974, Math. Program..

[31]  Michel Minoux,et al.  Networks synthesis and optimum network design problems: Models, solution methods and applications , 1989, Networks.