Solving fractional problems with dynamic multistart improving hit-and-run

Abstract

Fractional programming has numerous applications in economy and engineering. While some fractional problems are easy in the sense that they are equivalent to an ordinary linear program, other problems like maximizing a sum or product of several ratios are known to be hard, as these functions are highly nonconvex and multimodal. In contrast to the standard Branch-and-Bound type algorithms proposed for specific types of fractional problems, we treat general fractional problems with stochastic algorithms developed for multimodal global optimization. Specifically, we propose Improving Hit-and-Run with restarts, based on a theoretical analysis of Multistart Pure Adaptive Search (cf. the dissertation of Khompatraporn (2004)) which prescribes a way to utilize problem specific information to sample until a certain level α of confidence is achieved. For this purpose, we analyze the Lipschitz properties of fractional functions, and then utilize a unified method to solve general fractional problems. The paper ends with a report on numerical experiments.