H\"older Error Bounds and H\"older Calmness with Applications to Convex Semi-Infinite Optimization

Using techniques of variational analysis, necessary and sufficient subdifferential conditions for H\"older error bounds are investigated and some new estimates for the corresponding modulus are obtained. As an application, we consider the setting of convex semi-infinite optimization and give a characterization of the H\"older calmness of the argmin mapping in terms of the level set mapping (with respect to the objective function) and a special supremum function. We also estimate the H\"older calmness modulus of the argmin mapping in the framework of linear programming.Comment: 25 page

Calmness modulus of linear semi-infinite programs

Our main goal is to compute or estimate the calmness modulus of the argmin mapping of linear semi-infinite optimization problems under canonical perturbations, i.e., perturbations of the objective function together with continuous perturbations of the right-hand side of the constraint system (with respect to an index ranging in a compact Hausdorff space). Specifically, we provide a lower bound on the calmness modulus for semi-infinite programs with unique optimal solution which turns out to be the exact modulus when the problem is finitely constrained. The relationship between the calmness of the argmin mapping and the same property for the (sub)level set mapping (with respect to the objective function), for semi-infinite programs and without requiring the uniqueness of the nominal solution, is explored, too, providing an upper bound on the calmness modulus of the argmin mapping. When confined to finitely constrained problems, we also provide a computable upper bound as it only relies on the nominal data and parameters, not involving elements in a neighborhood. Illustrative examples are provided.This research has been partially supported by grants MTM2011-29064-C03 (02-03) from MINECO, Spain, ACOMP/2013/062 from Generalitat Valenciana, Spain, grant C10E08 from ECOS-SUD, and grant DP110102011 from the Australian Research Council

On Hölder calmness of minimizing sets

We present conditions for Hölder calmness and upper Hölder continuity of optimal solution sets to perturbed optimization problems in finite dimensions. Studies on Hölder type stability were a popular subject in variational analysis already in the 1980s and 1990s, and have become a revived interest in the last decade. In this paper, we focus on conditions for Hölder calmness of the argmin mapping in the case of non-isolated minima. We recall known ideas and results in this context for general as well as special parametric programs, refine them and discuss particular settings, including nonlinear programs and convex semi-infinite optimization problems

Outer limit of subdifferentials and calmness moduli in linear and nonlinear programming

With a common background and motivation, the main contributions of this paper are developed in two different directions. Firstly, we are concerned with functions which are the maximum of a finite amount of continuously differentiable functions of n real variables, paying attention to the case of polyhedral functions. For these max-functions, we obtain some results about outer limits of subdifferentials, which are applied to derive an upper bound for the calmness modulus of nonlinear systems. When confined to the convex case, in addition, a lower bound on this modulus is also obtained. Secondly, by means of a KKT index set approach, we are also able to provide a point-based formula for the calmness modulus of the argmin mapping of linear programming problems without any uniqueness assumption on the optimal set. This formula still provides a lower bound in linear semi-infinite programming. Illustrative examples are given

Critical objective size and calmness modulus in linear programming

This paper introduces the concept of critical objective size associated with a linear program in order to provide operative point-based formulas (only involving the nominal data, and not data in a neighborhood) for computing or estimating the calmness modulus of the optimal set (argmin) mapping under uniqueness of nominal optimal solution and perturbations of all coefficients. Our starting point is an upper bound on this modulus given in \cite{CHPTmp}. In this paper we prove that this upper bound is attained if and only if the norm of the objective function coefficient vector is less than or equal to the critical objective size. This concept also allows us to obtain operative lower bounds on the calmness modulus. We analyze in detail an illustrative example in order to explore some strategies that can improve the referred upper and lower bounds

On calmness of the optimal value function

The paper is devoted to the calmness from below/from above for the optimal value function of parametric optimization problems, where we are specifically interested in perturbed semi-infinite programs. A main intention is to revisit classical results and to derive refinements of them. In particular, we show in the context of semi-infinite optimization that calmness from below for j holds under quasiconvexity of the data functions and compactness of the solution set, which extends results on the lower semicontinuity of Illustrative examples are given, which demonstrate the significance of the imposed assumptions even in the case of linear and quadratic programs