Dynamic Bundle Methods: Application to Combinatorial Optimization

Lagrangian relaxation is a popular technique to solve difficult optimization problems. However, the applicability of this technique depends on having a relatively low number of hard constraints to dualize. When there are exponentially many hard constraints, it is preferable to relax them dynamically, according to some rule depending on which multipliers are active. For instance, only the most violated constraints at a given iteration could be dualized. From the dual point of view, this approach yields multipliers with varying dimensions and a dual objective function that changes along iterations. We discuss how to apply a bundle methodology to solve this kind of dual problems. We analyze the resulting dynamic bundle method giving a positive answer for its convergence properties, including finite termination and a primal result for polyhedral problems. We also report preliminary numerical experience on Linear Ordering and Traveling Salesman Problems.

[1]  Christoph Helmberg.  A Cutting Plane Algorithm for Large Scale Semidefinite Relaxations , 2004, The Sharpest Cut.

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

[3]  Nicos Christofides,et al.  Extensions to a Lagrangean relaxation approach for the capacitated warehouse location problem , 1983 .

[4]  C. C. Gonzaga,et al.  On Constraint Dropping Schemes and Optimality Functions for a Class of Outer Approximations Algorithms , 1979 .

[5]  Jean Charles Gilbert,et al.  Numerical Optimization: Theoretical and Practical Aspects , 2003 .

[6]  C. C. Gonzaga,et al.  An improved algorithm for optimization problems with functional inequality constraints , 1980 .

[7]  Franz Rendl,et al.  Computational experience with a bundle approach for semidefinite cutting plane relaxations of Max-Cut and Equipartition , 2006, Math. Program..

[8]  Jorge J. Moré,et al.  Digital Object Identifier (DOI) 10.1007/s101070100263 , 2001 .

[9]  Ulrich Faigle,et al.  A Lagrangian relaxation approach to the edge-weighted clique problem , 2001, Eur. J. Oper. Res..

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

[11]  Jean-Philippe Vial,et al.  Solving Large-Scale Linear Multicommodity Flow Problems with an Active Set Strategy and Proximal-ACCPM , 2006, Oper. Res..

[13]  Giovanni Rinaldi,et al.  New approaches for optimizing over the semimetric polytope , 2005, Math. Program..

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

[15]  Gerhard Reinelt,et al.  A Cutting Plane Algorithm for the Linear Ordering Problem , 1984, Oper. Res..

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

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

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

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

[20]  Alexandre Belloni.  Lagrangian Heuristics to Linear Ordering , 2001 .

[21]  Bezalel Gavish,et al.  Augmented Lagrangean Based Algorithms for Centralized Network Design , 1985, IEEE Trans. Commun..

[22]  Cid C. de Souza,et al.  Optimal rectangular partitions , 2003, Networks.

[23]  Kavindra Malik,et al.  A Lagrangian relax-and-cut approach for the sequential ordering problem with precedence relationships , 1994, Ann. Oper. Res..

[24]  J. Frédéric Bonnans,et al.  Numerical Optimization: Theoretical and Practical Aspects (Universitext) , 2006 .

[25]  J. Beasley.  A lagrangian heuristic for set‐covering problems , 1990 .

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

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

[28]  G. Reinelt,et al.  Combinatorial optimization and small polytopes , 1996 .

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