A reduced space branch and bound algorithm for a class of sum of ratios problems

Sum of ratios problem occurs frequently in various areas of engineering practice and management science, but most solution methods for this kind of problem are often designed for determining local solutions . In this paper, we develop a reduced space branch and bound algorithm for globally solving sum of convex-concave ratios problem. By introducing some auxiliary variables, the initial problem is converted into an equivalent problem where the objective function is linear. Then the convex relaxation problem of the equivalent problem is established by relaxing auxiliary variables only in the outcome space. By integrating some acceleration and reduction techniques into branch and bound scheme, the presented global optimization algorithm is developed for solving these kind of problems. Convergence and optimality of the algorithm are presented and numerical examples taken from some recent literature and MINLPLib are carried out to validate the performance of the proposed algorithm.

Global Optimization for a Class of Nonlinear Sum of Ratios Problem

We present a branch and bound algorithm for globally solving the sum of ratios problem. In this problem, each term in the objective function is a ratio of two functions which are the sums of the absolute values of affine functions with coefficients. This problem has an important application in financial optimization, but the global optimization algorithm for this problem is still rare in the literature so far. In the algorithm we presented, the branch and bound search undertaken by the algorithm uses rectangular partitioning and takes place in a space which typically has a much smaller dimension than the space to which the decision variables of this problem belong. Convergence of the algorithm is shown. At last, some numerical examples are given to vindicate our conclusions.