113 research outputs found
Local strong maximal monotonicity and full stability for parametric variational systems
The paper introduces and characterizes new notions of Lipschitzian and
H\"olderian full stability of solutions to general parametric variational
systems described via partial subdifferential and normal cone mappings acting
in Hilbert spaces. These notions, postulated certain quantitative properties of
single-valued localizations of solution maps, are closely related to local
strong maximal monotonicity of associated set-valued mappings. Based on
advanced tools of variational analysis and generalized differentiation, we
derive verifiable characterizations of the local strong maximal monotonicity
and full stability notions under consideration via some positive-definiteness
conditions involving second-order constructions of variational analysis. The
general results obtained are specified for important classes of variational
inequalities and variational conditions in both finite and infinite dimensions
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
Applying Metric Regularity to Compute a Condition Measure of a Smoothing Algorithm for Matrix Games
We develop an approach of variational analysis and generalized
differentiation to conditioning issues for two-person zero-sum matrix games.
Our major results establish precise relationships between a certain condition
measure of the smoothing first-order algorithm proposed by Gilpin et al.
[Proceedings of the 23rd AAAI Conference (2008) pp. 75-82] and the exact bound
of metric regularity for an associated set-valued mapping. In this way we
compute the aforementioned condition measure in terms of the initial matrix
game data
Charactarizations of Linear Suboptimality for Mathematical Programs with Equilibrium Constraints
The paper is devoted to the study of a new notion of linear suboptimality in constrained mathematical programming. This concept is different from conventional notions of solutions to optimization-related problems, while seems to be natural and significant from the viewpoint of modern variational analysis and applications. In contrast to standard notions, it admits complete characterizations via appropriate constructions of generalized differentiation in nonconvex settings. In this paper we mainly focus on various classes of mathematical programs with equilibrium constraints (MPECs), whose principal role has been well recognized in optimization theory and its applications. Based on robust generalized differential calculus, we derive new results giving pointwise necessary and sufficient conditions for linear suboptimality in general MPECs and its important specifications involving variational and quasi variational inequalities, implicit complementarity problems, etc
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