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.

Geological risk assessment of Amtoudi Agadir in southern Morocco: a key case for sustainable cultural heritage

Medieval fortified granaries known as “agadirs” are very common in southern Morocco, being catalogued as world cultural heritage by United Nations. These Berber buildings (made of stones and tree trunks) usually located on rocky promontories, constitute historical testimonials related to the origin of Morocco, and, as tourist attractions, have a positive impact on the local economy. The sustainability of these ancient monuments requires geological-risk evaluations of the massif stability under the agadir with the proposal of stabilization measures, and an architectonic analysis with appropriate maintenance of the structural elements. An interdisciplinary study including climate, seismicity, hydrology, geology, geomorphology, geotechnical surveys of the massif, and diagnosis of the degradation of structural elements have been performed on the Amtoudi Agadir, selected as a case study. The main findings from this study are that the prevalent rocks used for construction (coming from the underlying substratum) are good-quality arkosic sandstones; the SW cliffs under the agadir are unstable under water saturation; some masonry walls are too thin and lack interlocking stones and mortar; and failures in the beams (due to flexure, fracture, and exhaustion in the resistance due to insect attacks or plant roots) are common. The basic risk assessment of ancient buildings of cultural heritage and their geologic substratum are needed especially in undeveloped areas with limited capacity to implement durable conservation policies. Therefore, recommendations have been provided to ensure the stability and maintenance of this important archaeological site.