1,756 research outputs found
Quantitative Stability and Optimality Conditions in Convex Semi-Infinite and Infinite Programming
This paper concerns parameterized convex infinite (or semi-infinite)
inequality systems whose decision variables run over general
infinite-dimensional Banach (resp. finite-dimensional) spaces and that are
indexed by an arbitrary fixed set T . Parameter perturbations on the right-hand
side of the inequalities are measurable and bounded, and thus the natural
parameter space is . Based on advanced variational analysis, we
derive a precise formula for computing the exact Lipschitzian bound of the
feasible solution map, which involves only the system data, and then show that
this exact bound agrees with the coderivative norm of the aforementioned
mapping. On one hand, in this way we extend to the convex setting the results
of [4] developed in the linear framework under the boundedness assumption on
the system coefficients. On the other hand, in the case when the decision space
is reflexive, we succeed to remove this boundedness assumption in the general
convex case, establishing therefore results new even for linear infinite and
semi-infinite systems. The last part of the paper provides verifiable necessary
optimality conditions for infinite and semi-infinite programs with convex
inequality constraints and general nonsmooth and nonconvex objectives. In this
way we extend the corresponding results of [5] obtained for programs with
linear infinite inequality constraints
Optimal boundary control of a simplified Ericksen--Leslie system for nematic liquid crystal flows in
In this paper, we investigate an optimal boundary control problem for a two
dimensional simplified Ericksen--Leslie system modelling the incompressible
nematic liquid crystal flows. The hydrodynamic system consists of the
Navier--Stokes equations for the fluid velocity coupled with a convective
Ginzburg--Landau type equation for the averaged molecular orientation. The
fluid velocity is assumed to satisfy a no-slip boundary condition, while the
molecular orientation is subject to a time-dependent Dirichlet boundary
condition that corresponds to the strong anchoring condition for liquid
crystals. We first establish the existence of optimal boundary controls. Then
we show that the control-to-state operator is Fr\'echet differentiable between
appropriate Banach spaces and derive first-order necessary optimality
conditions in terms of a variational inequality involving the adjoint state
variables
Relative Pareto Minimizers to Multiobjective Problems: Existence and Optimality Conditions
In this paper we introduce and study enhanced notions of relative Pareto minimizers to constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency/minimality. Using advanced tools of variational analysis and generalized differentiation, we establish the existence of relative Pareto minimizers to general multiobjective problems under a refined version of the subdifferential Palais-Smale condition for set-valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers (as well as for conventional efficient and weak efficient counterparts) that are new in both finite-dimensional and infinite-dimensional settings. Our proofs are based on variational and extremal principles of variational analysis; in particular, on new versions of the Ekeland variational principle and the subdifferential variational principle for set-valued and single-valued mappings in infinite-dimensional spaces
Variational Principles for Set-Valued Mappings with Applications to Multiobjective Optimization
This paper primarily concerns the study of general classes of constrained multiobjective optimization problems (including those described via set-valued and vector-valued cost mappings) from the viewpoint of modern variational analysis and generalized differentiation. To proceed, we first establish two variational principles for set-valued mappings, which~being certainly of independent interest are mainly motivated by applications to multiobjective optimization problems considered in this paper. The first variational principle is a set-valued counterpart of the seminal derivative-free Ekeland variational principle, while the second one is a set-valued extension of the subdifferential principle by Mordukhovich and Wang formulated via an appropriate subdifferential notion for set-valued mappings with values in partially ordered spaces. Based on these variational principles and corresponding tools of generalized differentiation, we derive new conditions of the coercivity and Palais-Smale types ensuring the existence of optimal solutions to set-valued optimization problems with noncompact feasible sets in infinite dimensions and then obtain necessary optimality and suboptimality conditions for nonsmooth multiobjective optmization problems with general constraints, which are new in both finite-dimensional and infinite-dimensional settings
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