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The sum of ratios problem has a variety of important applications ineconomics and management science, but it is difficult to globally solve thisproblem. In this paper, we consider the minimization problem of a sum of anumber of nondifferentiable quasi-convex component functions over a closed andconvex set, which includes the sum of ratios problem as a special case. The sumof quasi-convex component functions is not necessarily to be quasi-convex, andso, this study goes beyond quasi-convex optimization. Exploiting the structureof the sum-minimization problem, we propose a new incremental subgradientmethod for this problem and investigate its convergence properties to a globaloptimal solution when using the constant, diminishing or dynamic stepsize rulesand under a homogeneous assumption and the H\"{o}lder condition of order $p$.To economize on the computation cost of subgradients of a large number ofcomponent functions, we further propose a randomized incremental subgradientmethod, in which only one component function is randomly selected to constructthe subgradient direction at each iteration. The convergence properties areobtained in terms of function values and distances of iterates from the optimalsolution set with probability 1. The proposed incremental subgradient methodsare applied to solve the sum of ratios problem, as well as the multipleCobb-Douglas productions efficiency problem, and the numerical results showthat the proposed methods are efficient for solving the large sum of ratiosproblem.

Incremental Subgradient Methods for Minimizing The Sum of Quasi-convex Functions