A DC approach for minimax fractional optimization programs with ratios of convex functions

Abstract

This paper deals with minimax fractional programs whose objective functions are the maximum of finite ratios of convex functions, with arbitrary convex constraints set. For such problems, Dinkelbach-type algorithms fail to work since the parametric subproblems may be nonconvex, whereas the latter need a global optimal solution of these subproblems. We give necessary optimality conditions for such problems, by means of convex analysis tools. We then propose a method, based on solving approximately a sequence of parametric convex problems, which acts as dc (difference of convex functions) algorithm, if the parameter is positive and as Dinkelbach algorithm if not. We show that every cluster point of the sequence of optimal solutions of these subproblems satisfies necessary optimality conditions of KKT criticality type, that are also of Clarke stationarity type. Finally we end with some numerical tests to illustrate the behaviour of the algorithm.

Keywords:

• Fractional programming
• quotient of convex functions
• difference of convex functions
• optimality conditions
• Dinkelbach algorithms

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

A. Ghazi

A. Ghazi is a PhD student at Hassan 1 University, Settat, Morocco. His research interest is numerical optimization.

A. Roubi

A. Roubi obtained his PhD degree from the University of Burgundy, Dijon, French, in 1994. He is actually professor at Hassan 1 University, Settat, Morocco. His research focuses on numerical optimization.