Generalized convexity and concavity of the optimal value function in nonlinear programming.(English)Zbl 0643.90071
Let f \(*(\epsilon)=\inf \{f(x,\epsilon)|\) \(x\in R(\epsilon)\}\), where f: E \(n\times E\) \(k\to E\) 1 and R: E \(k\to {\mathfrak P}(E\) n). In a previous paper, the authors [J. Optimization Theory Appl. 48, 95-126 (1986;Zbl 0556.90085)] studied the convexity and concavity properties of f *. In this new article, several sufficient conditions for the G-convexity and G-concavity of f *, under different assumptions on f and R, are presented. The notions of G-convexity and G-concavity, introduced here, depend on a function G of four real arguments. It is shown that, for suitable choices of G, G-convexity becomes convexity, quasi-convexity or paraconvexity, among other classes of generalized convex functions (and analogously for G-concavity). The particular case when R corresponds to the feasible region of a parametric inequality-equality constrained nonlinear programming problem is examined. Moreover, a diagram showing all the relationships between the classes of point-to-set maps considered in the paper is provided.
Reviewer: J.-E.Martinez-Legaz
MSC:
| 90C30 | Nonlinear programming |
| 90C31 | Sensitivity, stability, parametric optimization |
| 26B25 | Convexity of real functions of several variables, generalizations |
Keywords:
optimal value function;sufficient conditions;G-convexity;G-concavity;quasi-convexity;paraconvexity;parametric inequality-equality constrained nonlinear programming;point-to-set mapsCitations:
Zbl 0556.90085Cite
Full Text:DOI
References:
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