Topic
Column generation
About: Column generation is a research topic. Over the lifetime, 2870 publications have been published within this topic receiving 85480 citations.
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TL;DR: A technique is presented for the decomposition of a linear program that permits the problem to be solved by alternate solutions of linear sub-programs representing its several parts and a coordinating program that is obtained from the parts by linear transformations.
Abstract: A technique is presented for the decomposition of a linear program that permits the problem to be solved by alternate solutions of linear sub-programs representing its several parts and a coordinating program that is obtained from the parts by linear transformations. The coordinating program generates at each cycle new objective forms for each part, and each part generates in turn from its optimal basic feasible solutions new activities columns for the interconnecting program. Viewed as an instance of a “generalized programming problem” whose columns are drawn freely from given convex sets, such a problem can be studied by an appropriate generalization of the duality theorem for linear programming, which permits a sharp distinction to be made between those constraints that pertain only to a part of the problem and those that connect its parts. This leads to a generalization of the Simplex Algorithm, for which the decomposition procedure becomes a special case. Besides holding promise for the efficient computation of large-scale systems, the principle yields a certain rationale for the “decentralized decision process” in the theory of the firm. Formally the prices generated by the coordinating program cause the manager of each part to look for a “pure” sub-program analogue of pure strategy in game theory, which he proposes to the coordinator as best he can do. The coordinator finds the optimum “mix” of pure sub-programs using new proposals and earlier ones consistent with over-all demands and supply, and thereby generates new prices that again generates new proposals by each of the parts, etc. The iterative process is finite.
2,281 citations
TL;DR: In this paper, column generation methods for integer programs with a huge number of variables are discussed, including implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branch-and-bound tree.
Abstract: We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branch-and-bound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. We then discuss computational issues and implementation of column generation, branch-and-bound algorithms, including special branching rules and efficient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality.
2,248 citations
IBM1
TL;DR: In this paper, a technique is described for overcoming the difficulty in the linear programming formulation of the cutting-stock problem, which enables one to compute always with a matrix which has no more columns than it has rows.
Abstract: The cutting-stock problem is the problem of filling an order at minimum cost for specified numbers of lengths of material to be cut from given stock lengths of given cost. When expressed as an integer programming problem the large number of variables involved generally makes computation infeasible. This same difficulty persists when only an approximate solution is being sought by linear programming. In this paper, a technique is described for overcoming the difficulty in the linear programming formulation of the problem. The technique enables one to compute always with a matrix which has no more columns than it has rows.
1,933 citations
TL;DR: In this article, a simple and effective metaheuristic for combinatorial and global optimization, called variable neighborhood search (VNS), is presented, which can easily be implemented using any local search algorithm as a subroutine.
1,824 citations
TL;DR: This paper presents a new optimization algorithm capable of optimally solving 100-customer problems of the vehicle routing problem with time windows VRPTW and indicates that this algorithm proved to be successful on a variety of practical sized benchmark VRPTw test problems.
Abstract: The vehicle routing problem with time windows VRPTW is a generalization of the vehicle routing problem where the service of a customer can begin within the time window defined by the earliest and the latest times when the customer will permit the start of service. In this paper, we present the development of a new optimization algorithm for its solution. The LP relaxation of the set partitioning formulation of the VRPTW is solved by column generation. Feasible columns are added as needed by solving a shortest path problem with time windows and capacity constraints using dynamic programming. The LP solution obtained generally provides an excellent lower bound that is used in a branch-and-bound algorithm to solve the integer set partitioning formulation. Our results indicate that this algorithm proved to be successful on a variety of practical sized benchmark VRPTW test problems. The algorithm was capable of optimally solving 100-customer problems. This problem size is six times larger than any reported to date by other published research.
1,085 citations