134 research outputs found
Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods
The convex feasibility problem (CFP) is at the core of the modeling of many
problems in various areas of science. Subgradient projection methods are
important tools for solving the CFP because they enable the use of subgradient
calculations instead of orthogonal projections onto the individual sets of the
problem. Working in a real Hilbert space, we show that the sequential
subgradient projection method is perturbation resilient. By this we mean that
under appropriate conditions the sequence generated by the method converges
weakly, and sometimes also strongly, to a point in the intersection of the
given subsets of the feasibility problem, despite certain perturbations which
are allowed in each iterative step. Unlike previous works on solving the convex
feasibility problem, the involved functions, which induce the feasibility
problem's subsets, need not be convex. Instead, we allow them to belong to a
wider and richer class of functions satisfying a weaker condition, that we call
"zero-convexity". This class, which is introduced and discussed here, holds a
promise to solve optimization problems in various areas, especially in
non-smooth and non-convex optimization. The relevance of this study to
approximate minimization and to the recent superiorization methodology for
constrained optimization is explained.Comment: Mathematical Programming Series A, accepted for publicatio
Some remarks on the class of continuous (Semi-) strictcly quasiconvex functions
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Well-Behavior, Well-Posedness and Nonsmooth Analysis
AMS subject classification: 90C30, 90C33.We survey the relationships between well-posedness and well-behavior. The latter
notion means that any critical sequence (xn) of a lower semicontinuous function
f on a Banach space is minimizing. Here “critical” means that the remoteness of
the subdifferential ∂f(xn) of f at xn (i.e. the distance of 0 to ∂f(xn)) converges
to 0. The objective function f is not supposed to be convex or smooth and the
subdifferential ∂ is not necessarily the usual Fenchel subdifferential. We are thus
led to deal with conditions ensuring that a growth property of the subdifferential
(or the derivative) of a function implies a growth property of the function itself.
Both qualitative questions and quantitative results are considered
(Convex) level sets integration
The paper addresses the problem of recovering a pseudoconvex function from the normal cones to its level sets that we call the convex level sets integration problem. An important application is the revealed preference problem. Our main result can be described as integrating a maximally cyclically pseudoconvex multivalued map that sends vectors or “bundles” of a Euclidean space to convex sets in that space. That is, we are seeking a pseudoconvex (real) function such that the normal cone at each boundary point of each of its lower level sets contains the set value of the multivalued map at the same point. This raises the question of uniqueness of that function up to rescaling. Even after normalizing the function long an orienting direction, we give a counterexample to its uniqueness. We are, however, able to show uniqueness under a condition motivated by the classical theory of ordinary differential equations
Evenly convex sets, and evenly quasiconvex functions, revisited
Since its appearance, even convexity has become a remarkable notion in convex analysis. In the fifties, W. Fenchel introduced the evenly convex sets as those sets solving linear systems containing strict inequalities. Later on, in the eighties, evenly quasiconvex functions were introduced as those whose sublevel sets are evenly convex. The significance of even convexity relies on the different areas where it enjoys applications, ranging from convex optimization to microeconomics. In this paper, we review some of the main properties of evenly convex sets and evenly quasiconvex functions, provide further characterizations of evenly convex sets, and present some new results for evenly quasiconvex functions.This research has been partially supported by MINECO of Spain and ERDF of EU, Grants PGC2018-097960-B-C22 and ECO2016-77200-P
A Minty variational principle for set optimization
Extremal problems are studied involving an objective function with values in
(order) complete lattices of sets generated by so called set relations.
Contrary to the popular paradigm in vector optimization, the solution concept
for such problems, introduced by F. Heyde and A. L\"ohne, comprises the
attainment of the infimum as well as a minimality property. The main result is
a Minty type variational inequality for set optimization problems which
provides a sufficient optimality condition under lower semicontinuity
assumptions and a necessary condition under appropriate generalized convexity
assumptions. The variational inequality is based on a new Dini directional
derivative for set-valued functions which is defined in terms of a "lattice
difference quotient": A residual operation in a lattice of sets replaces the
inverse addition in linear spaces. Relationships to families of scalar problems
are pointed out and used for proofs: The appearance of improper scalarizations
poses a major difficulty which is dealt with by extending known scalar results
such as Diewert's theorem to improper functions
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