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

Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric

This paper analyzes the Lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in Rn. To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of Rn+1. In this framework the size of perturbations is measured by means of the (extended) Hausdorff distance. A direct antecedent, extensively studied in the literature, comes from considering the parameter space of all linear systems with a fixed index set, T, where the Chebyshev (extended) distance is used to measure perturbations. In the present work we propose an appropriate indexation strategy which allows us to establish the equality of the Lipschitz moduli of the feasible set mappings in both parametric contexts, as well as to benefit from existing results in the Chebyshev setting for transferring them to the Hausdorff one. In a second stage, the possibility of perturbing directly the set of coefficient vectors of a linear system leads to new contributions on the Lipschitz behavior of convex systems via linearization techniques. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society. Correction: The article “Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric”, written by Beer,G., Cánovas, M.J., López, M.A., Parra, J.was originally published Online First without Open Access. After publication in volume 189, issue 1–2, page 75–98 the author decided to opt for Open Choice and to make the article an Open Access publication. Therefore, the copyright of the article has been changed to © The Author(s) 2020 and the article is forthwith distributed under the terms of the Creative Commons Attribution 4.0 International License. https://doi.org/10.1007/s10107-021-01751-

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

Error bounds revisited

We propose a unifying general framework of quantitative primal and dual sufficient and necessary error bound conditions covering linear and nonlinear, local and global settings. The function is not assumed to possess any particular structure apart from the standard assumptions of lower semicontinuity in the case of sufficient conditions and (in some cases) convexity in the case of necessary conditions. We expose the roles of the assumptions involved in the error bound assertions, in particular, on the underlying space: general metric, normed, Banach or Asplund. Employing special collections of slope operators, we introduce a succinct form of sufficient error bound conditions, which allows one to combine in a single statement several different assertions: nonlocal and local primal space conditions in complete metric spaces, and subdifferential conditions in Banach and Asplund spaces. © 2022 Informa UK Limited, trading as Taylor & Francis Group

Calmness of the Optimal Value in Linear Programming

This research has been partially supported by grant MTM2014-59179-C2-2-P from MINECO, Spain, and FEDER "Una manera de hacer Europa," European Union

Geometry in structured optimisation problems

In this thesis, we start by providing some background knowledge on importance of convex analysis. Then, we will be looking at the Demyanov-Ryabova conjecture. This conjecture claims that if we convert between finite families of upper and lower exhausters with the given convertor function, the process will reach a cycle of length at most two. We will show that the conjecture is true in the afflinely independent special case, and also provide an equivalent algebraic reformulation of the conjecture. After that, we will generalise the outer subdifferential construction for max type functions to pointwise minima of regular Lipschitz functions. We will also answer an open question about the relation between the outer subdifferential of the support of a regular function and the end set of its subdifferential. Lastly, we will address the question of what kind of dimensional patterns are possible for the faces of general closed convex sets.  We show that for any finite increasing sequence of positive integers, there exist convex compact sets which only have faces with dimensions from  this prescribed  sequence. We will also discuss another approach to dimensionality by considering  the dimension of the union of all faces of the same dimension. We will demonstrate that the problems arising from this approach are highly nontrivial by providing some examples of convex sets where the sets of extreme points have fractal dimensions

Perturbation of error bounds

Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi: 10.1137/100782206) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions 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.The research is supported by the Australian Research Council: project DP160100854; EDF and the Jacques Hadamard Mathematical Foundation: Gaspard Monge Program for Optimization and Operations Research. The research of the second and third authors is also supported by MINECO of Spain and FEDER of EU: Grant MTM2014-59179-C2-1-P

Lipschitz lower semicontinuity moduli for linear inequality systems

The paper is focussed on the Lipschitz lower semicontinuity of the feasible set mapping for linear (finite and infinite) inequality systems in three different perturbation frameworks: full, right-hand side and left-hand side perturbations. Inspired by [14], we introduce the Lipschitz lower semicontinuity-star as an intermediate notion between the Lipschitz lower semicontinuity and the well-known Aubin property. We provide explicit point-based formulae for the moduli (best constants) of all three Lipschitz properties in all three perturbation settings