Quantitative Stability of Variational Systems II. A Framework for Nonlinear Conditioning

@article{Attouch1993QuantitativeSO,
  title={Quantitative Stability of Variational Systems II. A Framework for Nonlinear Conditioning},
  author={H{\'e}dy Attouch and Roger J.-B. Wets},
  journal={SIAM J. Optim.},
  year={1993},
  volume={3},
  pages={359-381}
}
Stability results of Lipschitz and Holder type are obtained for the solutions and optimal values of optimization problems when perturbations are measured in terms of the $\rho $-epi-distance. 

Figures from this paper

Quantitative stability of variational systems: III.ε-approximate solutions

We prove that theε-optimal solutions of convex optimization problems are Lipschitz continuous with respect to data perturbations when these are measured in terms of the epi-distance. A similar

Stability results of a class of well-posed optimization problems

We study here stability properties of a class of well-posed optimization problems under suitable variational convergences. Moreover we investigate the stability for uniformly well-posed optimization

Quantitative stability of variational systems. I. The epigraphical distance

This paper proposes a global measure for the distance between the elements of a variational system (parametrized families of optimization problems).

Characterizations of Lipschitz Stability in Optimization

We show that the following local Lipschitz properties of solutions to generalized equations under canonical perturbations are invariant under smooth approximations: the pseudo-Lipschitz property, the

On Quantitative Stability for C1,1 Programs

This paper concerns the quantitative stability behavior of optimal solutions to a C 1,1 program depending on a parameter vector. For example, given a strict local minimizer of order 2 which satisfies

Solution Sensitivity of Variational Inequalities

Our goal is to show that the continuity of the solutions of monotone variational inequalities with respect to perturbations is independent from the geometrical properties of reflexive Banach spaces.

Well-Posedness and Optimization under Perturbations

Estimates of the size of sets of approximate solutions are obtained for well-posed optimization problems in a Banach space, and extended to problems subject to perturbations of a general form. An

On weak sharp minima in vector optimization with applications to parametric problems

The concepts of weak sharp solutions to vector optimization problems are discussed and as an application the application provides sufficient conditions for stability of solutions in perturbedvector optimization problems.

Holder continuity of solutions to a parametric variational inequality

We prove a Hölder continuity property of the locally unique solution to a parametric variational inequality without assuming differentiability of the given data.

Hölder Continuity of a Parametric Generalized Variational Inequality

By using the classic metric projection method, we obtain sufficient conditions for Holder continuity of the nonunique solution mapping for a parametric generalized variational inequality with respect
...

References

SHOWING 1-10 OF 58 REFERENCES

Quantitative stability of variational systems: III.ε-approximate solutions

We prove that theε-optimal solutions of convex optimization problems are Lipschitz continuous with respect to data perturbations when these are measured in terms of the epi-distance. A similar

Lipschitzian Stability in Optimization: The Role of Nonsmooth Analysis

The motivations of nonsmooth analysis are discussed. Applications are given to the sensitivity of optimal values, the interpretation of Lagrange multipliers, and the stability of constraint systems

Quantitative stability of variational systems. I. The epigraphical distance

This paper proposes a global measure for the distance between the elements of a variational system (parametrized families of optimization problems).

Global Regularity Theorems

For a system of nondifferentiable convex inequalities and linear equalities, the best bound is given for the quotient of the distance of an infeasible point and the norm of the residual vector.

Lipschitzian Stability of Epsilon-Approximate Solutions in Convex Optimization

We prove that the epsilon-optimal solutions are Lipschitzian with respect to data perturbations when these are measured in terms of the epigraphical distance.

Stability in Nonlinear Programming

This paper establishes necessary and sufficient conditions for constraint set stability requiring neither convex constraint functions not convex constraint sets. These conditions then lead to a

Perturbations, Approximations, and Sensitivity Analysis of Optimal Control Systems

Estimates of the solutions of abstract optimization problems.- Regular perturbations.- Singular perturbations.- Finite - Difference approximations.- Sensitivity analysis of the open - Loop control

Stability Theory for Systems of Inequalities. Part I: Linear Systems

This paper deals with the stability of systems of linear inequalities in partially ordered Banach spaces when the data are subjected to small perturbations. We show that a certain condition is

Convergence of minima in convergence spaces 1

Sufficient and necessary conditions for the persistence and the adherence of minima in general convergence spaces are provided and specialized in the cases of topological and sequential convergences,

The Generalized Gradient of a Marginal Function in Mathematical Programming

The marginal function of a vertically perturbed nonlinear mathematical program with equality and inequality constraints is considered, locally Lipschitz and estimates for its generalized gradient are obtained.
...