A spatial branch and bound algorithm for solving the sum of linear ratios optimization problem

Abstract

In this paper, we consider the sum of linear ratios problem (SLR) that is known to be NP-hard and often arises in various practical applications such as data envelopment analysis and financial investment. We first introduce an equivalent problem (EP) of SLR that involves differences of square terms in inequality constraints. Subsequently, the concave parts of the non-convex constraints in problem (EP) are replaced with the piecewise linear functions. Using the resulting second-order cone program (SOCP), we design a spatial branch and bound algorithm, which iteratively refines the piecewise linear approximations by dividing rectangles and solving a series of problems (SOCP) to obtain the solution of the original problem. Also, a region compression technique is proposed to accelerate the convergence of the algorithm. Furthermore, we demonstrate that the bound on the optimality gap is a function of approximation errors at the iteration and estimate that the worst-case number of iterations is in the order of $\boldsymbol {O(\sqrt {\varepsilon })}$ to attain an ε-optimal solution. Numerical results illustrate that the proposed algorithm scales better than both the existing LP-based algorithms and the off-the-shelf solvers SCIP to solve the problem (SLR). It is worth mentioning that the proposed algorithm takes significantly less time to reach four-digit accuracy than the time required by the known algorithms on small to medium problem instances.