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

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

Revisiting the dynamic of Q-deformed logistic maps

We consider the logistic family and apply the $q$-deformation $\phi_q(x)=\frac{1-q^x}{1-q}$. We study the stability regions of the fixed points of the $q$-deformed logistic map and the regions where the dynamic is complex through topological entropy and Lyapunov exponents. Our results show that the dynamic of this deformed family is richer than that of the $q$-deformed family studied in [8].Comment: 23 pages, 52 figur

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)

An Upper Limit on the Mass of the Circumplanetary Disk for DH Tau b

Indexación: Scopus.DH Tau is a young (sim;1 Myr) classical T Tauri star. It is one of the few young PMS stars known to be associated with a planetary mass companion, DH Tau b, orbiting at large separation and detected by direct imaging. DH Tau b is thought to be accreting based on copious Ha emission and exhibits variable Paschen Beta emission. NOEMA observations at 230 GHz allow us to place constraints on the disk dust mass for both DH Tau b and the primary in a regime where the disks will appear optically thin. We estimate a disk dust mass for the primary, DH Tau A of 17.2 ± 1.7 MÅ, which gives a disk to star mass ratio of 0.014 (assuming the usual gas to dust mass ratio of 100 in the disk). We find a conservative disk dust mass upper limit of 0.42M⊕ for DH Tau b, assuming that the disk temperature is dominated by irradiation from DH Tau b itself. Given the environment of the circumplanetary disk, variable illumination from the primary or the equilibrium temperature of the surrounding cloud would lead to even lower disk mass estimates. A MCFOST radiative transfer model, including heating of the circumplanetary disk by DH Tau b and DH Tau A, suggests that a mass-averaged disk temperature of 22 K is more realistic, resulting in a dust disk mass upper limit of 0.09M⊕ for DH Tau b. We place DH Tau b in context with similar objects and discuss the consequences for planet formation models.http://iopscience.iop.org/article/10.3847/1538-3881/aa74cd/met

On topological sequence entropy of circle maps

[EN] We classify completely continuous circle maps from the point of view of topological sequence entropy. This improves a result of Roman Hric.This paper has been partially supported by the grant D.G.I.C.Y.T. PB98-0374-C03-01Cánovas, JS. (2001). On topological sequence entropy of circle maps. Applied General Topology. 2(1):1-7. https://doi.org/10.4995/agt.2001.3010SWORD172

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