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 and continuous metric subregularity for linear inequality systems

This paper introduces two new variational properties, robust and continuous metric subregularity, for finite linear inequality systems under data perturbations. The motivation of this study goes back to the seminal work by Dontchev, Lewis, and Rockafellar (2003) on the radius of metric regularity. In contrast to the metric regularity, the unstable continuity behavoir of the (always finite) metric subregularity modulus leads us to consider the aforementioned properties. After characterizing both of them, the radius of robust metric subregularity is computed and some insights on the radius of continuous metric subregularity are provided.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This research has been partially supported by Grant PGC2018-097960-B-C2(1,2) from MICINN, Spain, and ERDF, “A way to make Europe”, European Union, and Grant PROMETEO/2021/063 from Generalitat Valenciana, Spain

Variational Analysis in Semi-Infinite and Infinite Programming, I: Stability of Linear Inequality Systems of Feasible Solutions

This paper concerns applications of advanced techniques of variational analysis and generalized differentiation to parametric problems of semi-infinite and infinite programming, where decision variables run over finite-dimensional and infinite-dimensional spaces, respectively. Part I is primarily devoted to the study of robust Lipschitzian stability of feasible solutions maps for such problems described by parameterized systems of infinitely many linear inequalities in Banach spaces of decision variables indexed by an arbitrary set T. The parameter space of admissible perturbations under consideration 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∞ type. By using advanced tools of variational analysis and exploiting specific features of linear infinite systems, we establish complete characterizations of robust Lipschitzian stability entirely via their initial data with computing the exact bound of Lipschitzian moduli. A crucial part of our analysis addresses the precise computation of the coderivative of the feasible set mapping and its norm. The results obtained are new in both semi-infinite and infinite frameworks. (A correction to the this article has been appended at the end of the pdf file.)This research was partially supported by grants MTM2005-08572-C03 (01-02) from MEC (Spain) and FEDER (EU), MTM2008-06695-C03 (01-02) from MICINN (Spain), and ACOMP/2009/047&133 from Generalitat Valenciana (Spain); National Science Foundation (USA) under grant DMS-0603846

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 [Cánovas et al., SIAM J. Optim., 20 (2009), pp. 1504–1526] from the viewpoint of robust Lipschitzian stability. The main results establish necessary optimality conditions for broad classes 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. We illustrate our model and results by considering a practically meaningful model of water resource optimization via systems of reservoirs.This research was partially supported by grants MTM2008-06695-C03 (01-02) from MICINN (Spain), ACOMP/2009/047&133, and ACOMP/2010/269 from Generatitat Valenciana (Spain)

A census of ρ\rho Oph candidate members from Gaia DR2

The Ophiuchus cloud complex is one of the best laboratories to study the earlier stages of the stellar and protoplanetary disc evolution. The wealth of accurate astrometric measurements contained in the Gaia Data Release 2 can be used to update the census of Ophiuchus member candidates. We seek to find potential new members of Ophiuchus and identify those surrounded by a circumstellar disc. We constructed a control sample composed of 188 bona fide Ophiuchus members. Using this sample as a reference we applied three different density-based machine learning clustering algorithms (DBSCAN, OPTICS, and HDBSCAN) to a sample drawn from the Gaia catalogue centred on the Ophiuchus cloud. The clustering analysis was applied in the five astrometric dimensions defined by the three-dimensional Cartesian space and the proper motions in right ascension and declination. The three clustering algorithms systematically identify a similar set of candidate members in a main cluster with astrometric properties consistent with those of the control sample. The increased flexibility of the OPTICS and HDBSCAN algorithms enable these methods to identify a secondary cluster. We constructed a common sample containing 391 member candidates including 166 new objects, which have not yet been discussed in the literature. By combining the Gaia data with 2MASS and WISE photometry, we built the spectral energy distributions from 0.5 to 22\microm for a subset of 48 objects and found a total of 41 discs, including 11 Class II and 1 Class III new discs. Density-based clustering algorithms are a promising tool to identify candidate members of star forming regions in large astrometric databases. If confirmed, the candidate members discussed in this work would represent an increment of roughly 40% of the current census of Ophiuchus.Comment: A&A, Accepted. Abridged abstrac

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

Optimum nitride concentration in multiband III-N-V alloys for high efficiency ideal solar cells

III-Nx–V1−x highly mismatched alloys HMAs have been proposed as promising material candidates for the development of high efficiency solar cells. According to the band anticrossing model, these alloys present a multiband character with an intermediate band within the otherwise fundamental bandgap that gives them the ability of improving the efficiency by means of below-bandgap photon absorption. The efficiency of GaNxAs1−x, GaNxP1−x, and their quaternaries InyGa1−yNxAs1−x and GaNxP1−x−yAsy is estimated theoretically versus nitrogen content in this letter. Low nitrogen content in the range of 1%–3.5% in the HMAs analyzed leads to theoretical efficiencies above 60%