UBC Theses and Dissertations
Covering relaxation methods for solving the zero-one positive polynomial programming problem Vaessen, Willem
Covering relaxation algorithms were first developed by Granot et al for solving positive 0-1 polynomial programming (PP) problems which maximize a linear objective function in 0-1 variables subject to a set of polynomial inequalities containing only positive coefficients ["Covering Relaxation for Positive 0-1 Polynomial Programs", Management Science, Vol. 25, (1979)]. The covering relaxation approach appears to cope successfully with the non-linearity of the PP problem and is able to solve modest size (40 variables and 40 constraints) sparse PP problems. This thesis develops a more sophisticated covering relaxation method which accelerates the performance of this approach, especially when solving PP problems with many terms in a constraint. Both the original covering relaxation algorithm and the newly introduced accelerated algorithm are cutting plane algorithms in which the relaxed problem is the set covering problem and the cutting planes are linear covering constraints. In contrast with other cutting plane methods in integer programming, the accelerated covering relaxation algorithm developed in this thesis does not solve the relaxed problem to optimality after the introduction of the cutting plane constraints. Rather, the augmented relaxed problem is first solved approximately and only if the approximate solution is feasible to the PP problem is the relaxed problem solved to optimality. The promise of this approach stems from the excellent performance of approximate procedures for solving integer programming problems. Indeed, the extensive computational experiments that were performed show that the accelerated algorithm has reduced both the number of set covering problems to be solved and the overall time required to solve a PP problem. The improvements are particularly significant for PP problems with many terms in a constraint.
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