6,905 research outputs found
Second-order subdifferential calculus with applications to tilt stability in optimization
The paper concerns the second-order generalized differentiation theory of
variational analysis and new applications of this theory to some problems of
constrained optimization in finitedimensional spaces. The main attention is
paid to the so-called (full and partial) second-order subdifferentials of
extended-real-valued functions, which are dual-type constructions generated by
coderivatives of frst-order subdifferential mappings. We develop an extended
second-order subdifferential calculus and analyze the basic second-order
qualification condition ensuring the fulfillment of the principal secondorder
chain rule for strongly and fully amenable compositions. The calculus results
obtained in this way and computing the second-order subdifferentials for
piecewise linear-quadratic functions and their major specifications are applied
then to the study of tilt stability of local minimizers for important classes
of problems in constrained optimization that include, in particular, problems
of nonlinear programming and certain classes of extended nonlinear programs
described in composite terms
Fuzzy games with a countable space of actions and applications to systems of generalized quasi-variational inequalities
In this paper, we introduce an abstract fuzzy economy (generalized fuzzy
game) model with a countable space of actions and we study the existence of the
fuzzy equilibrium. As applications, two types of results are obtained. The
first ones concern the existence of the solutions for systems of generalized
quasi-variational inequalities with random fuzzy mappings which we define. The
last ones are new random fixed point theorems for correspondences with values
in complete countable metric spaces.Comment: 18 page
A Douglas-Rachford splitting for semi-decentralized equilibrium seeking in generalized aggregative games
We address the generalized aggregative equilibrium seeking problem for
noncooperative agents playing average aggregative games with affine coupling
constraints. First, we use operator theory to characterize the generalized
aggregative equilibria of the game as the zeros of a monotone set-valued
operator. Then, we massage the Douglas-Rachford splitting to solve the monotone
inclusion problem and derive a single layer, semi-decentralized algorithm whose
global convergence is guaranteed under mild assumptions. The potential of the
proposed Douglas-Rachford algorithm is shown on a simplified resource
allocation game, where we observe faster convergence with respect to
forward-backward algorithms.Comment: arXiv admin note: text overlap with arXiv:1803.1044
Generalized Newton's Method based on Graphical Derivatives
This paper concerns developing a numerical method of the Newton type to solve
systems of nonlinear equations described by nonsmooth continuous functions. We
propose and justify a new generalized Newton algorithm based on graphical
derivatives, which have never been used to derive a Newton-type method for
solving nonsmooth equations. Based on advanced techniques of variational
analysis and generalized differentiation, we establish the well-posedness of
the algorithm, its local superlinear convergence, and its global convergence of
the Kantorovich type. Our convergence results hold with no semismoothness
assumption, which is illustrated by examples. The algorithm and main results
obtained in the paper are compared with well-recognized semismooth and
-differentiable versions of Newton's method for nonsmooth Lipschitzian
equations
New results on systems of generalized vector quasi-equilibrium problems
In this paper, we firstly prove the existence of the equilibrium for the
generalized abstract economy. We apply these results to show the existence of
solutions for systems of vector quasi-equilibrium problems with multivalued
trifunctions. Secondly, we consider the generalized strong vector
quasi-equilibrium problems and study the existence of their solutions in the
case when the correspondences are weakly naturally quasi-concave or weakly
biconvex and also in the case of weak-continuity assumptions. In all
situations, fixed-point theorems are used.Comment: 24 page
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