1,720,989 research outputs found
Quasiconcavity of sets and connectedness of the efficient frontier in ordered vector spaces
No full text
Scalarizations and its stability in vector optimization
No full textA class of scalarizations of vector optimization problems is
studied in order to characterize weakly efficient, efficient, and properly efficient points of a nonconvex vector problem. A parallelism is established between the different solutions of the scalarized problem and the various efficient frontiers. In particular, properly efficient points correspond to stable solutions with respect to suitable perturbations of the feasible set
Convergence of minimal sets in convex vector optimization
No full textWe study the behavior of the minimal sets of a sequence of convex sets converging to a given set The main feature of the present work is the use of convexity properties of the sets and to obtain upper and lower convergence of the minimal frontiers. We emphasize that we study both Kuratowski--Painlevé convergence and Attouch--Wets convergence of minimal sets. Moreover, we prove stability results that hold in a normed linear space ordered by a general cone, in order to deal with the most common spaces ordered by their natural nonnegative orthants (e.g., , and for ). We also make a comparison with the existing results related to the topics considered in our work
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
Well-posedness and convexity in vector optimization
No full textWe study a notion of well-posedness in vector optimization
through the behaviour of minimizing sequences of sets, defined in terms of Hausdorff set-convergence. We show that the notion of strict efficiency is related to the notion of well-posedness. Using the obtained results we identify a class of well-posed vector optimization problems: the convex problems with compact efficient frontiers
Well-posedness and stability for abstract spline problems
No full textIn this work well-posedness and stability properties of the abstract spline problem are studied in the
framework of reflexive spaces. Tykhonov well-posedness is proved without restrictive assumptions. In the
context of Hilbert spaces, also the stronger notion of Levitin–Polyak well-posedness is established. A sequence
of parametric problems converging to the given abstract spline problem is considered in order to
study stability. Under natural assumptions, convergence results for sequences of solutions of the perturbed
problems are obtained
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