Optimality Conditions and a Method of Centers for Minimax Fractional Programs with Difference of Convex Functions
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Optimality Conditions and a Method of Centers for Minimax Fractional Programs with Difference of Convex Functions Karima Boufi1 · Mostafa El Haffari2 · Ahmed Roubi1 Received: 14 January 2020 / Accepted: 13 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We are concerned in this paper with minimax fractional programs whose objective functions are the maximum of finite ratios of difference of convex functions, with constraints also described by difference of convex functions. Like Dinkelbach-type algorithms, the method of centers for generalized fractional programs fails to work for such problems, since the parametric subproblems may be nonconvex, whereas the latters need a global optimal solution for these subproblems. We first give necessary optimality conditions for these problems, by means of convex analysis tools, and then extend the last method to solve such programs. The method is based on solving a sequence of parametric convex problems. We show that every cluster point of the sequence of optimal solutions of these subproblems satisfies necessary optimality conditions of Karush–Kuhn–Tucker criticality type. Keywords Fractional programming · Difference of convex functions · Optimality conditions · Method of centers Mathematics Subject Classification 90C32 · 90C26 · 90C25 · 90C47 · 90C46
1 Introduction We look for optimality conditions and algorithms for finding a solution of generalized fractional problems (GFP), whose objective functions are the maximum of finite ratios
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Ahmed Roubi [email protected] Karima Boufi [email protected] Mostafa El Haffari [email protected]
1
Laboratoire MISI, Faculté des Sciences et Techniques, Univ. Hassan 1, Settat, Morocco
2
Laboratoire MISI & Ecole Normale Supérieure, Univ. Abdelmalek Essaâdi, Tétouan, Morocco
123
Journal of Optimization Theory and Applications
of difference of convex functions, with constraints also described by difference of convex functions. Problems of this form include ordinary convex programs, GFP with: ratios of convex and concave functions, ratios of convex functions, ratios of concave functions, ratios of concave and convex functions, etc. Another important class of such problems is the GFP for which the functions may be expressed as a difference of convex functions (see e.g., [1] for such functions). For solving a GFP, there have been several primal Dinkelbach-type algorithms in the literature [2–9]; and dual algorithms and results [10–23]. See also [24–29] for more references on fractional programming. These algorithms are based on auxiliary parametric problems having simpler structures than the original GFP. For the primal algorithms, the auxiliary problems furnish sequences of approximate optimal values converging decreasingly to the optimal value of the considered problem, whereas the sequences of values generated by the dual algorithms converge increasingly toward that optimal value. Apart from the situation when the parametric subproblems are convex, the primal approaches fail to wo
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