Quantitative Stability of Linear Infinite Inequality Systems under Block Perturbations with Applications to Convex Systems

The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set $J$. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is $l_{\infty}(J)$. Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel-Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system's data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of [3] developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system's coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case

Quantitative Stability and Optimality Conditions in Convex Semi-Infinite and Infinite Programming

This paper concerns parameterized convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional Banach (resp. finite-dimensional) spaces and that are indexed by an arbitrary fixed set T . Parameter perturbations on the right-hand side of the inequalities are measurable and bounded, and thus the natural parameter space is $l_{\infty}(T)$. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map, which involves only the system data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. On one hand, in this way we extend to the convex setting the results of [4] developed in the linear framework under the boundedness assumption on the system coefficients. On the other hand, in the case when the decision space is reflexive, we succeed to remove this boundedness assumption in the general convex case, establishing therefore results new even for linear infinite and semi-infinite systems. The last part of the paper provides verifiable necessary optimality conditions for infinite and semi-infinite programs with convex inequality constraints and general nonsmooth and nonconvex objectives. In this way we extend the corresponding results of [5] obtained for programs with linear infinite inequality constraints

Robust Stability and Optimality Conditions for Parametric Infinite and Semi-Infinite Programs

This paper primarily concerns the study of parametric problems of infinite and semi-infinite programming, where functional constraints are given by systems of infinitely many linear inequalities indexed by an arbitrary set T, where decision variables run over Banach (infinite programming) or finite-dimensional (semi-infinite case) spaces, and where objectives are generally described by nonsmooth and nonconvex cost functions. The parameter space of admissible perturbations in such problems is formed by all bounded functions on T equipped with the standard supremum norm. Unless the index set T is finite, this space is intrinsically infinite-dimensional (nonreflexive and nonseparable) of the l(infinity)-type. By using advanced tools of variational analysis and generalized differentiation and largely exploiting underlying specific features of linear infinite constraints, we establish complete characterizations of robust Lipschitzian stability (with computing the exact bound of Lipschitzian moduli) for parametric maps of feasible solutions governed by linear infinite inequality systems and then derive verifiable necessary optimality conditions for the infinite and semi-infinite programs under consideration expressed in terms of their initial data. A crucial part of our analysis addresses the precise computation of coderivatives and their norms for infinite systems of parametric linear inequalities in general Banach spaces of decision variables. The results obtained are new in both frameworks of infinite and semi-infinite programming

Variational Analysis in Semi-Infinite and Infinite Programming, II: Necessary Optimality Conditions

This paper concerns applications of advanced techniques of variational analysis and generalized differentiation to problems of semi-infinite and infinite programming with feasible solution sets defined by parameterized systems of infinitely many linear inequalities of the type intensively studied in the preceding development [5] from our viewpoint of robust Lipschitzian stability. We present meaningful interpretations and practical examples of such models. The main results establish necessary optimality conditions for a broad class of semi-infinite and infinite programs, where objectives are generally described by nonsmooth and nonconvex functions on Banach spaces and where infinite constraint inequality systems are indexed by arbitrary sets. The results obtained are new in both smooth and nonsmooth settings of semi-infinite and infinite programming

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-

An Integrated Approach to Flood Risk Management: A Case Study of Navaluenga (Central Spain)

Flood risk management decisions require the rational assessment of mitigation strategies. This is a complex decision-making process involving many uncertainties. This paper presents a case study where a cost-benefit based methodology is used to define the best intervention measures for flood-risk mitigation in central Spain. Based on different flood hazard scenarios, several structural measures considered by the local Basin Water Authority and others defined by engineering criteria were checked for operability. Non-systematic data derived from dendrogeomorphological analysis of riparian trees were included in the flood frequency analysis. Flood damage was assessed by means of depth-damage functions, and flooded urban areas were obtained by applying a hydraulic model. The best defense strategies were obtained by a cost-benefit procedure, where uncertainties derived from each analytical process were incorporated based on a stochastic approach to estimate expected economic losses. The results showed that large structural solutions are not economically viable when compared with other smaller structural measures, presumably because of the pre-established location of dams in the upper part of the basin which do not laminate the flow generated by the surrounding catchment to Navalueng

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

Projection-based local and global Lipschitz moduli of the optimal value in linear programming

In this paper, we use a geometrical approach to sharpen a lower bound given in [5] for the Lipschitz modulus of the optimal value of (finite) linear programs under tilt perturbations of the objective function. The key geometrical idea comes from orthogonally projecting general balls on linear subspaces. Our new lower bound provides a computable expression for the exact modulus (as far as it only depends on the nominal data) in two important cases: when the feasible set has extreme points and when we deal with the Euclidean norm. In these two cases, we are able to compute or estimate the global Lipschitz modulus of the optimal value function in different perturbations frameworks.This research has been partially supported by Grants PGC2018-097960-B-C21 and PID2020-116694GB-I00 from MICINN, Spain, and ERDF, ‘A way to make Europe,’ European Union

Estudio experimental del comportamiento a compresión de probetas de hormigón de resistencias bajas y medias confinadas con tejidos de fibras de carbono y con defectos muy importantes de ejecución

This behaviour of low- and medium-strength concrete specimens confined with carbon fibre-reinforced polymer (CFRP) was analysed in three loading cycles. In some cases, stress levels were achieved that produced intemal microcracks, which allowed residual rigidity and the behaviour of completely microcraked concrete specimens to be studied. The specimens were subsequently tested to compression to the fracture point. Specimens reinforced in accordance with no manufacturing defects (100% CFRP reinforcement) and major manufacturing defects (50% CFRP reinforcement) were assessed for effectiveness and behaviour of the confined elements in less than ideal conditions. Results show that confinement was higher in low-resistance concretes, that the behaviour of reinforced specimens was unaffected by defective implementation conditions and that the reinforced specimens were less rigid than the non-reinforced specimens when tested up to 40% of ultimate fracture strength.En este trabajo se estudia el comportamiento de hormigones de resistencias bajas y medias confinados con CFRP. Se han realizado 3 ciclos de carga llegando en algunos casos a niveles de tensión que han microfisurado internamente el hormigón, lo que ha permitido estudiar la rigidez residual y el comportamiento de probetas confinadas con el hormigón totalmente microfisurado. Posteriormente todas las probetas se han ensayado a compresión hasta rotura. Los refuerzos se han realizado con buenas condiciones de ejecución y simulando grandes defectos para poder evaluar la eficacia de los elementos confinados cuando las condiciones de ejecución no son las correctas. Los resultados muestran que el efecto de confinamiento es superior en hormigones poco resistentes, el comportamiento de las probetas reforzadas es poco sensible a grandes defectos de ejecución y su rigidez es inferior al de las probetas originales cuando se ensayan hasta el 40% de la tensión de rotura