Linear Reformulation of Polynomial Discrete Programming for Fast Computation

Polynomial discrete programming problems are commonly faced but hard to solve. Treating the nonconvex cross-product terms is the key. State-of-the-art methods usually convert such a problem into a 0-1 mixed-integer linear programming problem and then adopt the branch-and-bound scheme to find an optimal solution. Much effort has been spent on reducing the required numbers of variables and linear constraints as well as on avoiding unbalanced branch-and-bound trees. This study presents a set of equations that linearize the discrete cross-product terms in an extremely effective manner. It is shown that embedding the proposed “equations for linearizing discrete products” into those state-of-the-art methods in the literature not only significantly reduces the required number of linear constraints from O(h³n³) to O(hn) for a cubic polynomial discrete program with n variables in h possible values but also tighten these methods with much more balanced branch-and-bound trees. Numerical experiments confirm a two-order (10²-times) reduction in computational time for some randomly generated cubic polynomial discrete programming problems.

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