Local cuts for mixed-integer programming (original) (raw)

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.

Mixed integer non-linear programming using cutting plane techniques

Computer Aided Chemical Engineering, 2000

In the present paper a modification of the extended cutting plane (ECP) method is described and illustrated. It is shown how it is possible to solve general MINLP (Mixed Integer Non-Linear Programming) problems with pseudo-convex objective as well as constraints to global optimality by a sophisticated cutting plane approach. The method relies on the ability to construct valid cutting planes for the entire feasible region of the problem. The method is illustrated on a simple test example and on some demanding practical scheduling problems. A comparison with a recently developped branch-and-bound approach is also given.

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.

An optimality cut for mixed integer linear programs

European Journal of Operational Research, 1999

We derive the penalty cut, a simple optimality cut of general applicability in pure or mixed linear programs. This cut is tested on a number of examples and comparisons with the classical Gomory cut are provided.

Cutting planes for branch‐and‐price algorithms

2011

This article presents a general framework for formulating cutting planes in the context of column generation for integer programs. Valid inequalities can be derived using the variables of an equivalent compact formulation (i.e., the subproblem variables) or the master problem variables. In the first case, cuts are added to the compact formulation, either at the master level or the subproblem level, and the decomposition process is reapplied.

Primal cutting plane algorithms revisited

Mathematical Methods of Operations Research (ZOR), 2002

Dual fractional cutting plane algorithms, in which cutting planes are used to iteratively tighten a linear relaxation of an integer program, are wellknown and form the basis of the highly successful branch-and-cut method. It is rather less well-known that various primal cutting plane algorithms were developed in the 1960s, for example by Young. In a primal algorithm, the main role of the cutting planes is to enable a feasible solution to the original problem to be improved. Research on these algorithms has been almost nonexistent. In this paper we argue for a re-examination of these primal methods. We describe a new primal algorithm for pure 0-1 problems based on strong valid inequalities and give some encouraging computational results. Possible extensions to the case of general mixed-integer programs are also discussed.

Solving pseudo-convex mixed integer optimization problems by cutting plane techniques

Optimization and Engineering, 2002

In the present paper a cutting plane approach to solve mixed-integer non-linear programming (MINLP) problems, containing pseudo-convex functions, is given. It is shown how valid cutting planes for pseudo convex functions can be obtained and, furthermore, it is shown how a class of non-convex MINLP problems with a pseudoconvex objective function and pseudo-convex constraints, can be solved to global optimality with the considered cutting plane technique. Finally the numerical efficiency of the procedure, when solving some example problems, is illustrated.