An efficient spatial branch-and-bound algorithm using an adaptive branching rule for linear multiplicative programming

In this paper, a spatial branch-and-bound algorithm with an adaptive branching rule is proposed for solving linear multiplicative programming (LMP) problem. In the solution algorithm, LMP problem is first transformed into an equivalent problem, and a novel reformulation is then introduced to convert the nonconvex constraints into differences of square terms form, subsequently using a piecewise linear approximation for the concave part. By using the proposed adaptive branching rule for dividing rectangles and iteratively refining the piecewise linear approximations, the process of solving LMP problem can be translated into solving a series of second order cone relaxations (SOCR). Also, we discuss the bound on the optimality gap as a function of the approximation errors at the iterate, and estimate the computational complexity in the order of O(ɛ) to attain an ɛ-optimal solution. Finally, preliminary numerical results demonstrate that the proposed algorithm can efficiently find the global optimal solutions for test instances.