Generalized affine functions and generalized differentials. (English) Zbl 1267.90141
The article of N. T. H. Linh and J.-P. Penot is a mathematically deep and valuable contribution to the fields of analysis, or (generalized) calculus, and continuous optimization, with a great promise for a better understanding of “what goes beyond” of linearity, convexity and differentiability and, then, of optimization problems and their optimality conditions which include, employ and benefit from these new concepts of functional analysis. Here, “what goes beyond” becomes expressed in generalized terms of differentiation, calculus, characterization and solution.
A generalized affine linear function is both generalized convex and generalized concave. Among them are quasi-affine functions and pseudo-affine functions. In particular, fractional functions are pseudoaffine functions and quadratic pseudoaffine functions can be characterized.
In fact, the authors study some classes of generalized affine functions, using a generalized differential. They investigate some properties and characterizations of these classes and devise several characterizations of solution sets of optimization problems involving such functions or functions of related classes.
The five sections of the article are as follows: 1. Introduction, 2. Notations and definitions, 3. Characterizations of generalized affine functions, 4. Characterization of solution sets, and 5. Conclusions.
Further deep insights and also methods can be expected in the future, initiated and fostered by this research paper. Then, that progress could foster and stimulate activity in science, especially, in mathematical modeling, data mining and statistics, in engineering, economics and social-political decision making, in finance and OR, and, herewith, in the improvements of living conditions of the peoples on earth.
A generalized affine linear function is both generalized convex and generalized concave. Among them are quasi-affine functions and pseudo-affine functions. In particular, fractional functions are pseudoaffine functions and quadratic pseudoaffine functions can be characterized.
In fact, the authors study some classes of generalized affine functions, using a generalized differential. They investigate some properties and characterizations of these classes and devise several characterizations of solution sets of optimization problems involving such functions or functions of related classes.
The five sections of the article are as follows: 1. Introduction, 2. Notations and definitions, 3. Characterizations of generalized affine functions, 4. Characterization of solution sets, and 5. Conclusions.
Further deep insights and also methods can be expected in the future, initiated and fostered by this research paper. Then, that progress could foster and stimulate activity in science, especially, in mathematical modeling, data mining and statistics, in engineering, economics and social-political decision making, in finance and OR, and, herewith, in the improvements of living conditions of the peoples on earth.
Reviewer: Gerhard-Wilhelm Weber (Ankara)
Keywords:
colinvex; colinfine; generalized differential; optimization problem; protoconvex function; pseudoconvex function; pseudolinear function; quasiconvex functionReferences:
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