A Branch-and-Bound Based Method for Solving Monotone Optimization Problems

Monotone optimization problems are an important class of global optimization problems with various applications. In this paper, we propose a new exact method for monotone optimization problems. The method is of branch-and-bound framework that combines three basic strategies: partition, convexification and local search. The partition scheme is used to construct a union of subboxes that covers the boundary of the feasible region. The convexification outer approximation is then applied to each subbox to obtain an upper bound of the objective function on the subbox. The performance of the method can be further improved by incorporating the method with local search procedure. Illustrative examples describe how the method works. Computational results for small randomly generated problems are reported. Dedicated to Professor Alex Rubinov on the occasion of his 65th birthday. The authors appreciate very much the discussions with Professor Alex Rubinov and his suggestion of using local search. Research supported by the National Natural Science Foundation of China under Grants 10571116 and 10261001, and Guangxi University Scientific Research Foundation (No. X051022).