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 ∞(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 Cánovas et al. (SIAM J. Optim. 20, 1504–1526, 2009) 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.

Calmness of the Feasible Set Mapping for Linear Inequality Systems

In this paper we deal with parameterized linear inequality systems in the n-dimensional Euclidean space, whose coefficients depend continuosly on an index ranging in a compact Hausdorff space. The paper is developed in two different parametric settings: the one of only right-hand-side perturbations of the linear system, and that in which both sides of the system can be perturbed. Appealing to the backgrounds on the calmness property, and exploiting the specifics of the current linear structure, we derive different characterizations of the calmness of the feasible set mapping, and provide an operative expresion for the calmness modulus when confined to finite systems. In the paper, the role played by the Abadie constraint qualification in relation to calmness is clarified, and illustrated by different examples. We point out that this approach has the virtue of tackling the calmness property exclusively in terms of the system’s data.

Intimate partner violence among women in Spain: the impact of regional-level male unemployment and income inequality

Background: Intimate partner violence (IPV) against women is a complex worldwide public health problem. There is scarce research on the independent effect on IPV exerted by structural factors such as labour and economic policies, economic inequalities and gender inequality. Objective: To analyse the association, in Spain, between contextual variables of regional unemployment and income inequality and individual women’s likelihood of IPV, independently of the women’s characteristics. Method: We conducted multilevel logistic regression to analyse cross-sectional data from the 2011 Spanish Macrosurvey of Gender-based Violence which included 7898 adult women. The first level of analyses was the individual women’ characteristics and the second level was the region of residence. Results: Of the survey participants, 12.2% reported lifetime IPV. The region of residence accounted for 3.5% of the total variability in IPV prevalence. We determined a direct association between regional male long-term unemployment and IPV likelihood (P = 0.007) and between the Gini Index for the regional income inequality and IPV likelihood (P < 0.001). Women residing in a region with higher gender-based income discrimination are at a lower likelihood of IPV than those residing in a region with low gender-based income discrimination (odds ratio = 0.64, 95% confidence intervals: 0.55–0.75). Conclusions: Growing regional unemployment rates and income inequalities increase women’s likelihood of IPV. In times of economic downturn, like the current one in Spain, this association may translate into an increase in women’s vulnerability to IPV.