Stability of Error Bounds for Convex Constraint Systems in Banach Spaces (original) (raw)
Related papers
arXiv (Cornell University), 2023
This article is devoted to the stability of error bounds (local and global) for semi-infinite convex constraint systems in Banach spaces. We provide primal characterizations of the stability of local and global error bounds when systems are subject to small perturbations. These characterizations are given in terms of the directional derivatives of the functions that enter into the definition of these systems. It is proved that the stability of error bounds is essentially equivalent to verifying that the optimal values of several minimax problems, defined in terms of the directional derivatives of the functions defining these systems, are outside of some neighborhood of zero. Moreover, such stability only requires that all component functions entering the system have the same linear perturbation. When these stability results are applied to the sensitivity analysis of Hoffman's constants for semi-infinite linear systems, primal criteria for Hoffman's constants to be uniformly bounded under perturbations of the problem data are obtained.
Characterizations of Stability of Error Bounds for Convex Inequality Constraint Systems
2021
In this paper, we mainly study error bounds for a single convex inequality and semi-infinite convex constraint systems, and give characterizations of stability of error bounds via directional derivatives. For a single convex inequality, it is proved that the stability of local error bounds under small perturbations is essentially equivalent to the non-zero minimun of the directional derivative at a reference point over the sphere, and the stability of global error bounds is proved to be equivalent to the strictly positive infimum of the directional derivatives, at all points in the boundary of the solution set, over the sphere as well as some mild constraint qualification. When these results are applied to semi-infinite convex constraint systems, characterizations of stability of local and global error bounds under small perturbations are also provided. In particular such stability of error bounds is proved to only require that all component functions in semi-infinite convex constra...
Characterizations for Stability of Error Bounds for Convex Inequalities Constraint Systems
HAL (Le Centre pour la Communication Scientifique Directe), 2021
In this paper, we mainly study error bounds for a single convex inequality and semi-infinite convex constraint systems, and give characterizations for stability of error bounds via directional derivatives. For a single convex inequality, it is proved that the stability of local error bounds under small perturbations is essentially equivalent to the non-zero minimun of the directional derivative at a reference point over the sphere, and the stability of global error bounds is proved to be equivalent to the strictly positive infimum of the absolute directional derivatives, at all points in the boundary of the solution set, over the sphere as well as some mild constraint qualification. When these results are applied to semi-infinite convex constraint systems, characterizations for the stability of local and global error bounds under small perturbations are also provided. In particular such stability of error bounds is proved to only require that all component functions in semi-infinite convex constraint systems have the same linear perturbation. Our work demonstrates that verifying the stability of error bounds for convex inequalities constraint systems is, to some degree, equivalent to solving the convex optimization (defined by directional derivatives) over the sphere.
Sufficient Conditions for Error Bounds
SIAM Journal on Optimization, 2002
For a lower semicontinuous (l.s.c.) inequality system on a Banach space, it is shown that error bounds hold, provided every element in an abstract subdifferential of the constraint function at each point outside the solution set is norm bounded away from zero. A sufficient condition for a global error bound to exist is also given for an l.s.c. inequality system on a real normed linear space. It turns out that a global error bound closely relates to metric regularity, which is useful for presenting sufficient conditions for an l.s.c. system to be regular at sets. Under the generalized Slater condition, a continuous convex system on R n is proved to be metrically regular at bounded sets.
Solution Stability of a Linearly Perturbed Constraint System and Applications
Set-Valued and Variational Analysis, 2017
Linear complementarity problems and affine variational inequalities have been intensively investigated by different methods. Recently, some authors have shown that solution stability of these problems with respect to total perturbations can be effectively studied via a generalized linear constraint system. The present paper focuses on characterizing stability properties of the solution map of a linearly perturbed generalized linear constraint system. The obtained results lead to several stability conditions for parametric linear complementarity problems and affine variational inequalities in explicit forms. Keywords Generalized linear constraint system • Linear complementarity problem • Affine variational inequality • Solution map • Lipschitz-like property • Metric regularity • Uniform local error bound Mathematics Subject Classification (2010) 49J40 • 49J53 • 49K40 • 90C31 • 90C33 J.-C. Yao was partially supported by the Grant MOST 105-2115-M-039-002-MY3.
First-Order and Second-Order Conditions for Error Bounds
SIAM Journal on Optimization, 2004
For a lower semicontinuous function f on a Banach space X, we study the existence of a positive scalar µ such that the distance function d S associated with the solution set S of f (x) ≤ 0 satisfies d S (x) ≤ µ max{f (x), 0} for each point x in a neighborhood of some point x 0 in X with f (x) < for some 0 < ≤ +∞. We give several sufficient conditions for this in terms of an abstract subdifferential and the Dini derivatives of f. In a Hilbert space we further present some second-order conditions. We also establish the corresponding results for a system of inequalities, equalities, and an abstract constraint set.
SIAM Journal on Control and Optimization, 1996
Using a directional form of constraint qualification weaker than Robinson's, we derive an implicit function theorem for inclusions and use it for firstand second-order sensitivity analyses of the value function in perturbed constrained optimization. We obtain H61der and Lipschitz properties and, under a no-gap condition, first-order expansions for exact and approximate solutions. As an application, differentiability properties of metric projections in Hilbert spaces are obtained, using a condition generalizing polyhedricity. We also present in the appendix a short proof of a generalization of the convex duality theorem in Banach spaces.
Mathematical Programming
Our aim in the current article is to extend the developments in Kruger, Ngai & Théra, SIAM J. Optim. 20(6), 3280-3296 (2010) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by proper lower semicontinuous under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the 'radius of error bounds'. The definitions and characterizations are illustrated by examples.
Sufficient Conditions for Error Bounds and Applications
Applied Mathematics and Optimization, 2004
Our aim in this paper is to present sufficient conditions for error bounds in terms of Fréchet and limiting Fréchet subdifferentials in general Banach spaces. This allows us to develop sufficient conditions in terms of the approximate subdifferential for systems of the form (x, y) ∈ C × D, g(x, y, u) = 0, where g takes values in an infinite dimensional space and u plays the role of a parameter. This symmetric structure offers us the choice to impose conditions either on C or D. We use these results to prove nonemptiness and weak-star compactness of Fritz-John and Karush-Kuhn-Tucker multiplier sets, to establish Lipschitz continuity of the value function and to compute its subdifferential and finally to obtain results on local controllability in control problems of nonconvex unbounded differential inclusions.