Minimizing the difference of two quasiconvex functions
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Minimizing the difference of two quasiconvex functions S. Dempe1 · N. Gadhi2 · K. Hamdaoui2 Received: 20 September 2018 / Accepted: 19 August 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this note, we are concerned with an optimization problem (P) where the objective function is the difference of two quasiconvex functions. Using a suitable subdifferential introduced by Suzuki and Kuroiwa (Nonlinear Anal 74:1279–1285, 2011), we give necessary optimality conditions. An example is given to illustrate the result. Keywords Quasiconvex function · Difference of quasiconvex functions · Q-subdifferential · Optimality conditions
1 Introduction Generalized convexity has been played a very important role in optimization by enlarging a lot of results such as optimality conditions and duality to recover some classes of functions larger than the class of convex ones and containing it. The concept of a quasiconvex function has been introduced in the first half of the last century as an important extension of convex ones. Penot [5] gives a comprehensive overview of quasiconvex analysis, Penot and Z˘alinescu [7] investigate different subdifferentials for quasiconvex functions. Suzuki and Kuroiwa [8,9] as well as Enkhbat and Ibaraki [4] and Z˘a linescu [11] describe optimality conditions for quasiconvex optimization problems. Quasidifferential calculus [3] is one of the most interesting tools in nonsmooth optimization. Differences of convex functions are quasidifferentiable. Differences of quasidifferentiable functions, as well as generalizations of the quasidif-
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K. Hamdaoui [email protected] S. Dempe [email protected] N. Gadhi [email protected]
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Department of Mathematics and Computers Sciences, Technical University Bergakademie Freiberg, Freiberg, Germany
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Department of Mathematics, LSO, Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, Fès, Morocco
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S. Dempe et al.
ferential (i.e. exhausters and convexificators [10]), are often used to formulate models of applied problems and investigate their properties. In this note, we consider the following DQC (difference of quasiconvex) optimization problem inf f (x) − g (x) (P) subject to: x ∈ X , where X is a Banach space and f , g : X → R = [−∞, +∞] are quasiconvex functions. Using the Q-subdifferential introduced by Suzuki and Kuroiwa [8], we give necessary optimality conditions for (P) . In convex programming, the subdifferential, which is a generalized notion of the differential, plays very important roles to discuss optimality conditions. For example, it is well known that x0 is a global minimizer of a convex function f if and only if 0 ∈ ∂ f (x0 ) and this result is used extensively in various studies. During the recent decades, the problems with functions representable as a difference of two convex functions (i.e. d.c. functions) can be considered as some of the most attractive in nonconvex optimization; similar to what preceded, if x0 is a local minimizer of a d.c. function h = f − g, one has ∂ g (x0 ) ⊆ ∂ f (x0 ) .
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