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Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. (English) Zbl 0591.90073
Convexity and duality in optimization, Proc. Symp., Groningen/Neth. 1984, Lect. Notes Econ. Math. Syst. 256, 37-70 (1985).
[For the entire collection see Zbl 0569.00010.]
A large number of optimization problems of practical interest actually involve d.c. functions, i.e. functions that can be expressed as a difference of two convex functions. From a theoretical point of view, the importance of such functions stems from their relationship to convex functions and the fact that they constitute a linear space which is a dense subset of the space of continuous functions over a compact set. The reviewed article gives an excellent survey of the main known results on the analysis and optimization of d.c. functions. The following questions are discussed: differential properties, characterization of d.c. functions among locally Lipschitz functions, finding the ”best” d.c. representation of a given function, duality results, in particular the basic Toland’s duality relation. Also a preview on procedures for globally minimizing a d.c. function is presented.
Reviewer: Hoang Tuy

MSC:
90C30 Nonlinear programming
49M37 Numerical methods based on nonlinear programming
26B05 Continuity and differentiation questions
26B25 Convexity of real functions of several variables, generalizations
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
49N15 Duality theory (optimization)