4,880 research outputs found
Nonsmooth multiobjective optimization using limiting subdifferentials
AbstractIn this study, using the properties of limiting subdifferentials in nonsmooth analysis and regarding a separation theorem, some weak Pareto-optimality (necessary and sufficient) conditions for nonsmooth multiobjective optimization problems are proved
Codifferentials and Quasidifferentials of the Expectation of Nonsmooth Random Integrands and Two-Stage Stochastic Programming
This work is devoted to an analysis of exact penalty functions and optimality
conditions for nonsmooth two-stage stochastic programming problems. To this
end, we first study the co-/quasi-differentiability of the expectation of
nonsmooth random integrands and obtain explicit formulae for its co- and
quasidifferential under some natural assumptions on the integrand. Then we
analyse exact penalty functions for a variational reformulation of two-stage
stochastic programming problems and obtain sufficient conditions for the global
exactness of these functions with two different penalty terms. In the end of
the paper, we combine our results on the co-/quasi-differentiability of the
expectation of nonsmooth random integrands and exact penalty functions to
derive optimality conditions for nonsmooth two-stage stochastic programming
problems in terms of codifferentials
Optimality and duality for a class of nonsmooth fractional multiobjective optimization problems (Nonlinear Analysis and Convex Analysis)
In this paper, we establish necessary optimality conditions for (weakly) efficient solutions of a nonsmooth fractional multiobjective optimization problem with inequality and equality constraints by employing some advanced tools of variational analysis and generalized differentiation. Sufficient optimality conditions for such solutions to the considered problem are also provided by means of introducing (strictly) convex-affine functions. Along with optimality conditions, we formulate a dual problem to the primal one and explore weak, strong and converse duality relations between them under assumptions of (strictly) convex-affine functions
Constrained Nonsmooth Problems of the Calculus of Variations
The paper is devoted to an analysis of optimality conditions for nonsmooth
multidimensional problems of the calculus of variations with various types of
constraints, such as additional constraints at the boundary and isoperimetric
constraints. To derive optimality conditions, we study generalised concepts of
differentiability of nonsmooth functions called codifferentiability and
quasidifferentiability. Under some natural and easily verifiable assumptions we
prove that a nonsmooth integral functional defined on the Sobolev space is
continuously codifferentiable and compute its codifferential and
quasidifferential. Then we apply general optimality conditions for nonsmooth
optimisation problems in Banach spaces to obtain optimality conditions for
nonsmooth problems of the calculus of variations. Through a series of simple
examples we demonstrate that our optimality conditions are sometimes better
than existing ones in terms of various subdifferentials, in the sense that our
optimality conditions can detect the non-optimality of a given point, when
subdifferential-based optimality conditions fail to disqualify this point as
non-optimal.Comment: A number of small mistakes and typos was corrected in the second
version of the paper. Moreover, the paper was significantly shortened.
Extended and improved versions of the deleted sections on nonsmooth Noether
equations and nonsmooth variational problems with nonholonomic constraints
will be published in separate submission
KKT reformulation and necessary conditions for optimality in nonsmooth bilevel optimization
For a long time, the bilevel programming problem has essentially been considered as a special case of mathematical programs with equilibrium constraints (MPECs), in particular when the so-called KKT reformulation is in question. Recently though, this widespread believe was shown to be false in general. In this paper, other aspects of the difference between both problems are revealed as we consider the KKT approach for the nonsmooth bilevel program. It turns out that the new inclusion (constraint) which appears as a consequence of the partial subdifferential of the lower-level Lagrangian (PSLLL) places the KKT reformulation of the nonsmooth bilevel program in a new class of mathematical program with both set-valued and complementarity constraints. While highlighting some new features of this problem, we attempt here to establish close links with the standard optimistic bilevel program. Moreover, we discuss possible natural extensions for C-, M-, and S-stationarity concepts. Most of the results rely on a coderivative estimate for the PSLLL that we also provide in this paper
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