1,721,010 research outputs found
The origins of quasi-concavity: a development between mathematics and economics
No full textThe origins of the notion of quasi-concave function are considered, with special interest in some work by John von Neumann, Bruno de Finetti, and W. Fenchel. The development of such pioneering studies subsequently led to a whole field of research, known as “generalized convexity.” The different styles of the three authors and the various motivations for introducing quasi-concavity are compared, without losing sight of economic applications characteristic of the whole field of generalized convexity
Stability of properly efficient points and isolated minimizers of constrained vector optimization problems
No full textIn this paper the constrained vector optimization problem min Cf(x), g(x) ∈ - K, is considered, where f:R n → R m g:R n → R p are locally Lipschitz functions and C ⊂ R m and K ⊂ R p are closed convex cones. Several solution concepts are recalled, among them the concept of a properly efficient point (p-minimizer) and an isolated minimizer (i-minimizer). On the base of certain first-order optimalitty conditions it is shown that there is a close relation between the solutions of the constrained problem and some unconstrained problem. This consideration allows to "double" the solution concepts of the given constrained problem, calling sense II optimality concepts for the constrained problem the respective solutions of the related unconstrained problem, retaining the name of sense I concepts for the originally defined optimality solutions. The paper investigates the stability properties of thep-minimizers andi-minimizers. It is shown, that thep-minimizers are stable under perturbations of the cones, while thei-minimizers are stable under perturbations both of the cones and the functions in the data set. Further, it is shown, that sense I concepts are stable under perturbations of the objective data, while sense II concepts are stable under perturbations both of the objective and the constraints. Finally, the so called structural stability is discused. © 2007 Springer
ON THE NOTION OF PROPER EFFICIENCY IN VECTOR OPTIMIZATION
No full textWe consider the main definitions of proper efficiency for a vector optimization problem in topological linear spaces. The implications among these definitions generalize the inclusion structure holding in Euclidean spaces with componentwise ordering
Multiobjective problems with set constraints
No full textThe aim of this paper is to study necessary optimality conditions for vector valued problems having inequality constraints, equality ones and a set constraint. The study is based on the so called image space approach, which allows to state a necessary optimality condition in the image space and, as consequences, some necessary optimality conditions in the decision space more general than the ones known in the literature. The differentiable case and the convex case are also deepened on
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