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.

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) |