Column-Generation in Integer Linear Programming (original) (raw)
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Branch-and-Price: Column Generation for Solving Huge Integer Programs
Operations Research, 1998
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 branchand-bound tree. We present classes of models for which t h i s a p p r o a c h decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. W e t h e n discuss computational issues and implementation of column generation, branch-andbound algorithms, including special branching rules and e cient w ays to solve t h e LP relaxation.
On compact formulations for integer programs solved by column generation
2005
Column generation has become a powerful tool in solving large scale integer programs. It is well known that most of the often reported compatibility issues between pricing subproblem and branching rule disappear when branching decisions are based on imposing constraints on the subproblem's variables. This can be generalized to branching on variables of a socalled compact formulation. We constructively show that such a formulation always exists under mild assumptions. It has a block diagonal structure with identical subproblems, each of which contributes only one column in an integer solution. This construction has an interpretation as reversing a Dantzig-Wolfe decomposition. Our proposal opens the way for the development of branching rules adapted to the subproblem's structure and to the linking constraints.
Progress in Linear Programming-Based Algorithms for Integer Programming: An Exposition
INFORMS Journal on Computing, 2000
This paper is about modeling and solving mixed integer programming (MIP) problems. In the last decade, the use of mixed integer programming models has increased dramatically. Fifteen years ago, mainframe computers were required to solve problems with a hundred integer variables. Now it is possible to solve problems with thousands of integer variables on a personal computer and obtain provably good approximate solutions to problems such as set partitioning with millions of binary variables. These advances have been made possible by developments in modeling, algorithms, software, and hardware. This paper focuses on effective modeling, preprocessing, and the methodologies of branch-and-cut and branch-and-price, which are the techniques that make it possible to treat problems with either a very large number of constraints or a very large number of variables. We show how these techniques are useful in important application areas such as network design and crew scheduling. Finally, we discuss the relatively new research areas of parallel integer programming and stochastic integer programming.
An integer optimality condition for column generation on zero–one linear programs
Discrete Optimization, 2019
Column generation is a linear programming method that, when combined with appropriate integer programming techniques, has been successfully used for solving huge integer programs. The method alternates between a restricted master problem and a column generation subproblem. The latter step is founded on dual information from the former one; often an optimal dual solution to the linear programming relaxation of the restricted master problem is used. We consider a zero-one linear programming problem that is approached by column generation and present a generic sufficient optimality condition for the restricted master problem to contain the columns required to find an integer optimal solution to the complete problem. The condition is based on dual information, but not necessarily on an optimal dual solution. It is however most natural to apply the condition in a situation when an optimal or near-optimal dual solution is at hand. We relate our result to a few special cases from the literature, and make some suggestions regarding possible exploitation of the optimality condition in the construction of column generation methods for integer programs.
AN ALGORITHM FOR SOLVING INTEGER LINEAR PROGRAMMING PROBLEMS
The paper describes a method to solve an ILP by describing whether an approximated integer solution to the RLP is an optimal solution to the ILP. If the approximated solution fails to satisfy the optimality condition, then a search will be conducted on the optimal hyperplane to obtain an optimal integer solution using a modified form of Branch and Bound Algorithm.
A Decomposition Technique For Solving Integer Programming Problems
GANIT: Journal of Bangladesh Mathematical Society, 2014
Dantzig-Wolfe decomposition as applied to an integer program is a specific form of problem reformulation that aims at providing a tighter linear programming relaxation bound due to the non-convexity of an integer problem. In this paper, we develop an algorithm for solving large scale integer program relying on column generation method. We implemented our algorithm for solving Capital budgeting and scheduling type problems. Moreover, we used the Computer Aided System (CAS) AMPL to convert our algorithm into programming codes and illustrated the same problem in our program. We demonstrate our method by illustrating some numerical examples. DOI: http://dx.doi.org/10.3329/ganit.v33i0.17649 GANIT J. Bangladesh Math. Soc.Vol. 33 (2013) 1-11
Computers & Operations Research, 2013
The well-known column generation scheme is often an efficient approach for solving the linear relaxation of large-size Covering Integer Programs (CIP). In this paper, this technique is hybridized with an extension of the best-known CIP approximation heuristic, taking advantage of distinct criteria of columns selection. This extension uses fractional optimization for solving pricing subproblems. Numerical results on a real-case transportation planning problem show that the hybrid scheme accelerates the convergence of column generation both in terms of number of iterations and computational time. The integer solutions generated at the end of the process can also be improved for a significant proportion of instances, highlighting the potential of diversification of the approximation heuristic. & 2013 Elsevier Ltd. All rights reserved. 1. Accelerate the column generation process that solves the linear relaxation of the CIP master problem, 2. output an integer solution that is strictly better than both the CG þMIP solution and the Gr þ solution.
Cutting planes in integer and mixed integer programming
Discrete Applied Mathematics, 2002
This survey presents cutting planes that are useful or potentially useful in solving mixed integer programs. Valid inequalities for (i) general integer programs, (ii) problems with local structure such as knapsack constraints, and (iii) problems with 0-1 coe cient matrices, such as set packing, are examined in turn. Finally, the use of valid inequalities for classes of problems with structure, such as network design, is explored.
Decomposition and Dynamic Cut Generation in Integer Programming: Theory and Algorithms
Abstract Decomposition algorithms such as Lagrangian relaxation and Dantzig-Wolfe decomposition are well-known methods that can be used to compute bounds for integer programming problems. We discuss a framework for integrating dynamic cut generation with traditional decomposition methods in order to obtain improved bounds and present a new paradigm for separation called decompose and cut.