A Balasian-Based Algorithm for Zero-One Polynomial Programming

Abstract

A new algorithm is developed for solving zero-one polynomial problems. The algorithm is a direct extension of the well-known Balasian algorithm for zero-one linear programs. The basic idea underlying the development is that every zero-one polynomial programming problem can be converted into an equivalent zero-one linear system with nonlinear secondary constraints representing the polynomial terms. Moreover, the optimal solution to the original polynomial problem must be a feasible solution to the converted linear problem. Thus, the optimal solution is determined by searching for the best solution among the feasible solutions to the linear problem such that the secondary constraints are satisfied. The important point is that the secondary constraints are considered implicitly so that the size of the problem is determined by its equivalent linear form. Computational experience indicates that the new algorithm compares favorably with the existing ones due to Lawler and Bell [Lawler, E., M. Bell. 1967. A method for solving discrete optimization problems. Oper. Res. 15 1098–1112.] and to Watters [Watters, L. J. 1967. Reduction of integer polynomial problems to zero-one linear programming problems. Oper. Res. 15 1171–1174.]. The application of the algorithm to a special class of nonlinear binary problems is also investigated.