5 research outputs found
Formulae for the Conjugate and the Subdifferential of the Supremum Function
The aim of this work is to provide formulae for the subdifferential and the
conjungate function of the supremun function over an arbitrary family of
functions. The work is principally motivated by the case when data functions
are lower semicontinuous proper and convex. Nevertheless, we explore the case
when the family of functions is arbitrary, but satisfying that the biconjugate
of the supremum functions is equal to the supremum of the biconjugate of the
data functions. The study focuses its attention on functions defined in
finite-dimensional spaces, in this case the formulae can be simplified under
certain qualification conditions. However, we show how to extend these results
to arbitrary locally convex spaces without any qualification condition.Comment: submitte
New extremal principles with applications to stochastic and semi-infinite programming
This paper develops new extremal principles of variational analysis that are
motivated by applications to constrained problems of stochastic programming and
semi-infinite programming without smoothness and/or convexity assumptions.
These extremal principles concern measurable set-valued mappings/multifunctions
with values in finite-dimensional spaces and are established in both
approximate and exact forms. The obtained principles are instrumental to derive
via variational approaches integral representations and upper estimates of
regular and limiting normals cones to essential intersections of sets defined
by measurable multifunctions, which are in turn crucial for novel applications
to stochastic and semi-infinite programming.Comment: 26 page
Error Bounds for Parametric Polynomial Systems with Applications to Higher-Order Stability Analysis and Convergence Rates
The paper addresses parametric inequality systems described by polynomial
functions in finite dimensions, where state-dependent infinite parameter sets
are given by finitely many polynomial inequalities and equalities. Such systems
can be viewed, in particular, as solution sets to problems of generalized
semi-infinite programming with polynomial data. Exploiting the imposed
polynomial structure together with powerful tools of variational analysis and
semialgebraic geometry, we establish a far-going extension of the \L ojasiewicz
gradient inequality to the general nonsmooth class of supremum marginal
functions as well as higher-order (H\"older type) local error bounds results
with explicitly calculated exponents. The obtained results are applied to
higher-order quantitative stability analysis for various classes of
optimization problems including generalized semi-infinite programming with
polynomial data, optimization of real polynomials under polynomial matrix
inequality constraints, and polynomial second-order cone programming. Other
applications provide explicit convergence rate estimates for the cyclic
projection algorithm to find common points of convex sets described by matrix
polynomial inequalities and for the asymptotic convergence of trajectories of
subgradient dynamical systems in semialgebraic settings
Subdifferential formulae for the supremum of an arbitrary family of functions
This work provides calculus for the Fr\'echet and limiting subdifferential of
the pointwise supremum given by an arbitrary family of lower semicontinuous
functions. We start our study showing fuzzy results about the Fr\'echet
subdifferential of the supremum function. Posteriorly, we study in finite- and
infinite-dimensional settings the limiting subdifferential of the supremum
function. Finally, we apply our results to the study of the convex
subdifferential; here we recover general formulae for the subdifferential of an
arbitrary family of convex functions.Comment: 27 pages, Submitte
SUBDIFFERENTIALS OF NONCONVEX SUPREMUM FUNCTIONS AND THEIR APPLICATIONS TO SEMI-INFINITE AND INFINITE PROGRAMS WITH LIPSCHITZIAN DATA
The paper is devoted to the subdifferential study and applications of the supremum of uniformly Lipschitzian functions over arbitrary index sets with no topology. Based on advanced techniques of variational analysis, we evaluate major subdifferentials of the supremum functions in the general framework of Asplund (in particular, reflexive) spaces with no convexity or relaxation assumptions. The results obtained are applied to deriving new necessary optimality conditions for nonsmooth and nonconvex problems of semi-infinite and infinite programming