27,209 research outputs found
How to project onto extended second order cones
The extended second order cones were introduced by S. Z. N\'emeth and G.
Zhang in [S. Z. N\'emeth and G. Zhang. Extended Lorentz cones and variational
inequalities on cylinders. J. Optim. Theory Appl., 168(3):756-768, 2016] for
solving mixed complementarity problems and variational inequalities on
cylinders. R. Sznajder in [R. Sznajder. The Lyapunov rank of extended second
order cones. Journal of Global Optimization, 66(3):585-593, 2016] determined
the automorphism groups and the Lyapunov or bilinearity ranks of these cones.
S. Z. N\'emeth and G. Zhang in [S.Z. N\'emeth and G. Zhang. Positive operators
of Extended Lorentz cones. arXiv:1608.07455v2, 2016] found both necessary
conditions and sufficient conditions for a linear operator to be a positive
operator of an extended second order cone. This note will give formulas for
projecting onto the extended second order cones. In the most general case the
formula will depend on a piecewise linear equation for one real variable which
will be solved by using numerical methods
NEW APPROXIMATE SCHEMES FOR GENERALIZED GENERAL SET-VALUED MIXED QUASI VARIATIONAL INEQUALITIES
Abstract. In this paper, we suggest and consider a class of new three-step approximation schemes for generalized general set-valued mixed quasi variational inequalities. We also consider and analyze a new class of extragradienttype methods for solving generalized set-valued variational inequalities. The proposed methods include several new and known methods as special cases. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Introduction In recent years, variational inequalities have been generalized and extended in many different directions using novel and innovative techniques to study wider classes of unrelated problems in mechanics, physics, optimization and control, nonlinear programming, economics, regional, structural, transportation, elasticity, and applied sciences, etc, see [1]-[9] and the reference therein. An important and useful generalization of variational inequalities is called generalized general set-valued mixed quasi variational inequality involving the nonlinear bifunction, which is introduced and studied by Chao Feng Shi, San Yang Liu and Jun Li Lia
Suboptimality Conditions for Mathematical Programs with Equilibrium Constraints
In this paper we study mathematical programs with equilibrium constraints (MPECs) described by generalized equations in the extended form 0 is an element of the set G(x,y) + Q(x,y), where both mappings G and Q are set-valued. Such models arise, in particular, from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish new weak and strong suboptimality conditions for the general MPEC problems under consideration in finite-dimensional and infinite-dimensional spaces that do not assume the existence of optimal solutions. This issue is particularly important for infinite-dimensional optimization problems, where the existence of optimal solutions requires quite restrictive assumptions. Our techriiques are mainly based on modern tools of variational analysis and generalized differentiation revolving around the fundamental extremal principle in variational analysis and its analytic counterpart known as the subdifferential variational principle
Weak Minimizers, Minimizers and Variational Inequalities for set valued Functions. A blooming wreath?
In the literature, necessary and sufficient conditions in terms of
variational inequalities are introduced to characterize minimizers of convex
set valued functions with values in a conlinear space. Similar results are
proved for a weaker concept of minimizers and weaker variational inequalities.
The implications are proved using scalarization techniques that eventually
provide original problems, not fully equivalent to the set-valued counterparts.
Therefore, we try, in the course of this note, to close the network among the
various notions proposed. More specifically, we prove that a minimizer is
always a weak minimizer, and a solution to the stronger variational inequality
always also a solution to the weak variational inequality of the same type. As
a special case we obtain a complete characterization of efficiency and weak
efficiency in vector optimization by set-valued variational inequalities and
their scalarizations. Indeed this might eventually prove the usefulness of the
set-optimization approach to renew the study of vector optimization
Existence and solution methods for equilibria
Equilibrium problems provide a mathematical framework which includes optimization, variational inequalities, fixed-point and saddle point problems, and noncooperative games as particular cases. This general format received an increasing interest in the last decade mainly because many theoretical and algorithmic results developed for one of these models can be often extended to the others through the unifying language provided by this common format. This survey paper aims at covering the main results concerning the existence of equilibria and the solution methods for finding them
Variational Principle of Bogoliubov and Generalized Mean Fields in Many-Particle Interacting Systems
The approach to the theory of many-particle interacting systems from a
unified standpoint, based on the variational principle for free energy is
reviewed. A systematic discussion is given of the approximate free energies of
complex statistical systems. The analysis is centered around the variational
principle of N. N. Bogoliubov for free energy in the context of its
applications to various problems of statistical mechanics and condensed matter
physics. The review presents a terse discussion of selected works carried out
over the past few decades on the theory of many-particle interacting systems in
terms of the variational inequalities. It is the purpose of this paper to
discuss some of the general principles which form the mathematical background
to this approach, and to establish a connection of the variational technique
with other methods, such as the method of the mean (or self-consistent) field
in the many-body problem, in which the effect of all the other particles on any
given particle is approximated by a single averaged effect, thus reducing a
many-body problem to a single-body problem. The method is illustrated by
applying it to various systems of many-particle interacting systems, such as
Ising and Heisenberg models, superconducting and superfluid systems, strongly
correlated systems, etc. It seems likely that these technical advances in the
many-body problem will be useful in suggesting new methods for treating and
understanding many-particle interacting systems. This work proposes a new,
general and pedagogical presentation, intended both for those who are
interested in basic aspects, and for those who are interested in concrete
applications.Comment: 60 pages, Refs.25
Variational Inequalities in Critical-State Problems
Similar evolutionary variational inequalities appear as convenient
formulations for continuous quasistationary models for sandpile growth,
formation of a network of lakes and rivers, magnetization of type-II
superconductors, and elastoplastic deformations. We outline the main steps of
such models derivation and try to clarify the origin of this similarity. New
dual variational formulations, analogous to mixed variational inequalities in
plasticity, are derived for sandpiles and superconductors.Comment: Submitted for publicatio
- …