417 research outputs found
Optimality conditions for scalar and vector optimization problems with quasiconvex inequality constraints
Let X be a real linear space, X0 X a convex set, Y and Z topological real linear spaces. The constrained optimization problem minCf(x), g(x) 2 -K is considered, where f : X0 ! Y and g : X0 ! Z are given (nonsmooth) functions, and C Y and K Z are closed convex cones. The weakly efficient solutions (w-minimizers) of this problem are investigated. When g obeys quasiconvex properties, first-order necessary and first-order sufficient optimality conditions in terms of Dini directional derivatives are obtained. In the special case of problems with pseudoconvex data it is shown that these conditions characterize the global w-minimizers and generalize known results from convex vector programming. The obtained results are applied to the special case of problems with finite dimensional image spaces and ordering cones the positive orthants, in particular to scalar problems with quasiconvex constraints. It is shown, that the quasiconvexity of the constraints allows to formulate the optimality conditions using the more simple single valued Dini derivatives instead of the set valued ones. Key words: Vector optimization, nonsmooth optimization, quasiconvex vector functions, pseudoconvex vector functions, Dini derivatives, quasiconvex programming, Kuhn-Tucker conditions..
Local maximizers of generalized convex vector-valued functions
Any local maximizer of an explicitly quasiconvex real-valued function is actually a global minimizer, if it belongs to the intrinsic core of the function's domain. In this paper we show that similar properties hold for componentwise explicitly quasiconvex vector-valued functions, with respect to the concepts of ideal, strong and weak optimality. We illustrate these results in the particular framework of linear fractional multicriteria optimization problems.Any local maximizer of an explicitly quasiconvex real-valued
function is actually a global minimizer, if it belongs to the intrinsic core
of the function's domain. In this paper we show that similar properties
hold for componentwise explicitly quasiconvex vector-valued functions,
with respect to the concepts of ideal, strong and weak optimality. We
illustrate these results in the particular framework of linear fractional
multicriteria optimization problems
Effective behavior of nematic elastomer membranes
We derive the effective energy density of thin membranes of liquid crystal
elastomers as the Gamma-limit of a widely used bulk model. These membranes can
display fine-scale features both due to wrinkling that one expects in thin
elastic membranes and due to oscillations in the nematic director that one
expects in liquid crystal elastomers. We provide an explicit characterization
of the effective energy density of membranes and the effective state of stress
as a function of the planar deformation gradient. We also provide a
characterization of the fine-scale features. We show the existence of four
regimes: one where wrinkling and microstructure reduces the effective membrane
energy and stress to zero, a second where wrinkling leads to uniaxial tension,
a third where nematic oscillations lead to equi-biaxial tension and a fourth
with no fine scale features and biaxial tension. Importantly, we find a region
where one has shear strain but no shear stress and all the fine-scale features
are in-plane with no wrinkling
A Primal-Dual Approach of Weak Vector Equilibrium Problems
In this paper we provide some new sufficient conditions that ensure the
existence of the solution of a weak vector equilibrium problem in Hausdorff
topological vector spaces ordered by a cone. Further, we introduce a dual
problem and we provide conditions that assure the solution set of the original
problem and its dual coincide. We show that many known problems from the
literature can be treated in our primal-dual model. We provide several
coercivity conditions in order to obtain solution existence of the primal-dual
problems without compactness assumption. We pay a special attention to the case
when the base space is a reflexive Banach space. We apply the results obtained
to perturbed vector equilibrium problems.Comment: 20 page
Dual Representation of Quasiconvex Conditional Maps
We provide a dual representation of quasiconvex maps between two lattices of
random variables in terms of conditional expectations. This generalizes the
dual representation of quasiconvex real valued functions and the dual
representation of conditional convex maps.Comment: Date changed Added one remark on assumption (c), page
Local maximum points of explicitly quasiconvex functions
This work concerns generalized convex real-valued functions defined on a nonempty convex subset of a real topological linear space. Its aim is twofold: first, to show that any local maximum point of an explicitly quasiconvex function is a global minimum point whenever it belongs to the intrinsic core of the function’s domain and second, to characterize strictly convex normed spaces by applying this property for a particular class of convex functions
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