15 research outputs found
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
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
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
Error bounds for vector-valued funtions on metric spaces
In this paper, we attempt to extend the definition and existing local error bound criteria to vector-valued functions, or more generally, to functions taking values in a normed linear space. Some new primal space derivative-like objects – slopes – are introduced and a classification scheme of error bound criteria is presented