Linear Reformulation of Polynomial Discrete Programming for Fast Computation
提出一组方程高效线性化离散交叉乘积项，将立方多项式离散规划的线性约束从O(h³n³)降至O(hn)，计算时间减少两个数量级，适用于求解难解的多项式离散规划问题。
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(h 3 n 3 ) 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 (10 2 -times) reduction in computational time for some randomly generated cubic polynomial discrete programming problems. There is a Video Overview associated with this article, available as supplemental material.