Comparison of bundle and classical column generation

When a column generation approach is applied to decomposable mixed integer programming problems, it is standard to formulate and solve the master problem as a linear program. Seen in the dual space, this results in the algorithm known in the nonlinear programming community as the cutting-plane algorithm of Kelley and Cheney-Goldstein. However, more stable methods with better theoretical convergence rates are known and have been used as alternatives to this standard. One of them is the bundle method; our aim is to illustrate its differences with Kelley’s method. In the process we review alternative stabilization techniques used in column generation, comparing them from both primal and dual points of view. Numerical comparisons are presented for five applications: cutting stock (which includes bin packing), vertex coloring, capacitated vehicle routing, multi-item lot sizing, and traveling salesman. We also give a sketchy comparison with the volume algorithm.

[1]  Michael Patriksson,et al.  Ergodic, primal convergence in dual subgradient schemes for convex programming , 1999, Mathematical programming.

[2]  Francisco Barahona,et al.  The volume algorithm: producing primal solutions with a subgradient method , 2000, Math. Program..

[3]  Yurii Nesterov,et al.  Complexity estimates of some cutting plane methods based on the analytic barrier , 1995, Math. Program..

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

[5]  Claude Lemaréchal,et al.  Lagrangian Relaxation , 2000, Computational Combinatorial Optimization.

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

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

[8]  Anass Nagih,et al.  An Adapted Step Size Algorithm for a 0-1 Biknapsack Lagrangean Dual , 2005, Ann. Oper. Res..

[9]  Arthur M. Geoffrion,et al.  Lagrangian Relaxation for Integer Programming , 2010, 50 Years of Integer Programming.

[10]  Richard M. Karp,et al.  The traveling-salesman problem and minimum spanning trees: Part II , 1971, Math. Program..

[11]  Pierre Hansen,et al.  Stabilized column generation , 1998, Discret. Math..

[12]  Yu. M. Ermol’ev.  Methods of solution of nonlinear extremal problems , 1966 .

[13]  Krzysztof C. Kiwiel.  An Inexact Bundle Approach to Cutting-Stock Problems , 2010, INFORMS J. Comput..

[14]  Naum Zuselevich Shor,et al.  Minimization Methods for Non-Differentiable Functions , 1985, Springer Series in Computational Mathematics.

[15]  Krzysztof C. Kiwiel,et al.  A Proximal Bundle Method with Approximate Subgradient Linearizations , 2006, SIAM J. Optim..

[16]  Krzysztof C. Kiwiel,et al.  An aggregate subgradient method for nonsmooth convex minimization , 1983, Math. Program..

[17]  C. Lemaréchal,et al.  Bundle methods applied to the unit-commitment problem , 1996 .

[18]  John E. Beasley,et al.  OR-Library: Distributing Test Problems by Electronic Mail , 1990 .

[19]  J. Hiriart-Urruty,et al.  Fundamentals of Convex Analysis , 2004 .

[20]  Richard M. Karp,et al.  The Traveling-Salesman Problem and Minimum Spanning Trees , 1970, Oper. Res..

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

[22]  Michael A. Trick,et al.  A Column Generation Approach for Graph Coloring , 1996, INFORMS J. Comput..

[23]  Nelson Maculan,et al.  The volume algorithm revisited: relation with bundle methods , 2002, Math. Program..

[24]  R. E. Marsten,et al.  The Boxstep Method for Large-Scale Optimization , 2011, Oper. Res..

[25]  Claude Lemaréchal,et al.  An Algorithm for Minimizing Convex Functions , 1974, IFIP Congress.

[26]  K. Kiwiel.  A Cholesky dual method for proximal piecewise linear programming , 1994 .

[27]  C. Lemaréchal.  Nonsmooth Optimization and Descent Methods , 1978 .

[28]  Claude Lemaréchal,et al.  Variable metric bundle methods: From conceptual to implementable forms , 1997, Math. Program..

[29]  Jun-Yeon Lee,et al.  A descent method with linear programming subproblems for nondifferentiable convex optimization , 1995, Math. Program..

[30]  C. Lemaréchal,et al.  An inexact conic bundle variant suited to column generation , 2006 .

[31]  Andreas Griewank,et al.  On constrained optimization by adjoint based quasi-Newton methods , 2002, Optim. Methods Softw..

[32]  François Vanderbeck.  Extending Dantzig's bound to the bounded multiple-class binary Knapsack problem , 2002, Math. Program..

[33]  Yinyu Ye,et al.  Complexity Analysis of an Interior Cutting Plane Method for Convex Feasibility Problems , 1996, SIAM J. Optim..

[34]  Gerhard Wäscher,et al.  CUTGEN1: A problem generator for the standard one-dimensional cutting stock problem , 1995 .

[35]  Zeger Degraeve,et al.  Optimal Integer Solutions to Industrial Cutting-Stock Problems: Part 2, Benchmark Results , 2003, INFORMS J. Comput..

[36]  Michel Gendreau,et al.  An exact algorithm for the elementary shortest path problem with resource constraints: Application to some vehicle routing problems , 2004, Networks.

[37]  H. Uzawa,et al.  Preference, production, and capital: Iterative methods for concave programming , 1989 .

[38]  N. Z. Shor.  Utilization of the operation of space dilatation in the minimization of convex functions , 1972 .

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

[40]  John Darzentas,et al.  Problem Complexity and Method Efficiency in Optimization , 1983 .

[41]  Michel Gendreau,et al.  Interior point stabilization for column generation , 2007, Oper. Res. Lett..

[42]  Yurii Nesterov,et al.  Homogeneous Analytic Center Cutting Plane Methods for Convex Problems and Variational Inequalities , 1999, SIAM J. Optim..

[43]  Jean-Philippe Vial,et al.  Convex nondifferentiable optimization: A survey focused on the analytic center cutting plane method , 2002, Optim. Methods Softw..

[44]  W. Ben‐Ameur,et al.  Acceleration of cutting-plane and column generation algorithms: Applications to network design , 2007 .

[45]  K. Kiwiel.  Methods of Descent for Nondifferentiable Optimization , 1985 .

[46]  R. Tyrrell Rockafellar,et al.  A dual approach to solving nonlinear programming problems by unconstrained optimization , 1973, Math. Program..

[47]  Claude Lemaréchal,et al.  The omnipresence of Lagrange , 2003, 4OR.

[48]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[49]  R. Prim.  Shortest connection networks and some generalizations , 1957 .

[50]  Yurii Nesterov,et al.  New variants of bundle methods , 1995, Math. Program..

[51]  K. Kiwiel.  A Dual Method for Certain Positive Semidefinite Quadratic Programming Problems , 1989 .