 Maximization of Generalized Convex Functionals in Locally Convex SpacesJ. Haberl
Journal of Optimization Theory and Applications, 2004, vol. 121, issue 2, No 5, 327-359 Abstract: Abstract The major part of the investigation is related to the problem of maximizing an upper semicontinuous quasiconvex functional f over a compact (possibly nonconvex) subset K of a real Hausdorff locally convex space E. A theorem by Bereanu (Ref. 1) says that the condition “f is quasiconvex (quasiconcave) on K” is sufficient for the existence of maximum (minimum) point of f over K among the extreme points of K. But, as we prove by a counterexample, this is not true in general. On the further condition that the convex hull of the set of extreme points of K is closed, we show that it is sufficient to claim that f is “induced-quasiconvex” on K to achieve an equivalent conclusion. This new concept of quasiconvexity, which we define by requiring that each lower-level set of f can be represented as the intersection of K with some convex set, is suitable for functionals with a nonconvex domain. Under essentially the same conditions, we prove that an induced-quasiconvex functional f is directionally monotone in the sense that, for each y ∈ K, the functional f is increasing along a line segment starting at y and running to some extreme point of K. In order to guarantee the existence of maximum points on the relative boundary r ∂ K of K, it suffices to make weaker demands on the function f and the space E. By introducing a weaker kind of directional monotonicity, we are able to obtain the following result: If f is i.s.d.-increasing i.e., for each y y ∈ K, there is a half-line emanating from y such that f is increasing along this half-line, then f attains its maximum at r∂K , even if E is a topological linear Hausdorff space (infinite-dimensional and not necessarily locally convex). We state further a practical method of proving i.s.d.-monotonicity for functions in finite-dimensional spaces and we discuss also some aspects of classification. Keywords: Generalized convexity; quasiconvex maximization; quasiconvex functionals with nonconvex domain; directional monotonicity; global optimization; topological vector spaces (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:121:y:2004:i:2:d:10.1023_b:jota.0000037408.31141.e4 Ordering information: This journal article can be ordered from DOI: 10.1023/B:JOTA.0000037408.31141.e4 Access Statistics for this article Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull More articles in Journal of Optimization Theory and Applications from Springer |
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