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A Piecewise Linear Dual Procedure in Mixed Integer Programming

Istvan Maros

Book Chapter
New Trends in Mathematical Programming
pp.159–170
1998
Kluwer Academic Publishers
Abstract

Most solution methods for Mixed Integer Programming (MIP) problems require repeated solution of continuous linear programming (LP) problems. It is typical that subsequent LP problems differ just slightly. In most cases it is practically the right-hand-side that changes. For such a situation the dual simplex algorithm (DSA) appears to be the best solution method.

The LP relaxation of a MIP problem contains many bounded variables. This necessitates such an implementation of the DSA where variables of arbitrary type are allowed. The paper presents an algorithm called BSD for the efficient handling of bounded variables. This leads to a ``mini'' nonlinear optimization in each step of the DSA. Interestingly, this technique enables several cheap iterations per selection making the whole algorithm very attractive. The implementational implications and some computational experiences on large scale MIP problems are also reported. BSD is included in FortMP optimization system of Brunel University.

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