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.

Motzkin decomposition of closed convex sets via truncation

A nonempty set F is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set C with a closed convex cone D. In that case, the sets C and D are called compact and conic components of F. This paper provides new characterizations of the Motzkin decomposable sets involving truncations of F (i.e., intersections of FF with closed halfspaces), when F contains no lines, and truncations of the intersection F̂ of F with the orthogonal complement of the lineality of F, otherwise. In particular, it is shown that a nonempty closed convex set F is Motzkin decomposable if and only if there exists a hyperplane H parallel to the lineality of F such that one of the truncations of F̂ induced by H is compact whereas the other one is a union of closed halflines emanating from H. Thus, any Motzkin decomposable set F can be expressed as F=C+D, where the compact component C is a truncation of F̂. These Motzkin decompositions are said to be of type T when F contains no lines, i.e., when C is a truncation of F. The minimality of this type of decompositions is also discussed.

On the stability of the Motzkin representation of closed convex sets

A set is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set with a closed convex cone. This paper analyzes the continuity properties of the set-valued mapping associating to each couple (C,D) formed by a compact convex set C and a closed convex cone D its Minkowski sum C + D. The continuity properties of other related mappings are also analyzed.

Activation of a spherical carbon for toluene adsorption at low concentration

This paper complements a previous one [1] about toluene adsorption on a commercial spherical activated carbon and on samples obtained from it by CO2 or steam activation. The present paper deals with the activation of a commercial spherical carbon (SC) having low porosity and high bed density (0.85 g/cm3) using the same procedure. Our results show that SC can be well activated with CO2 or steam. The increase in the burn-off percentage leads to an increase in the gravimetric adsorption capacity (more intensively for CO2) and a decrease in bed density (more intensively for CO2). However, for similar porosity developments similar bed densities are achieved for CO2 and steam. Especial attention is paid to differences between both activating agents, comparing samples having similar or different activation rates, showing that CO2 generates more narrow porosity and penetrates more inside the spherical particles than steam. Steam activates more from the outside to the interior of the spheres and hence produces larger spheres size reductions. With both activation agents and with a suitable combination of porosity development and bed density, quite high volumetric adsorption values of toluene (up to 236 g toluene/L) can be obtained even using a low toluene concentration (200 ppmv).

Spherical activated carbon as an enhanced support for TiO2/AC photocatalysts

Titanium dioxide nanoparticles prepared in situ by sol–gel method were supported on a spherical activated carbon to prepare TiO2/AC hybrid photocatalysts for the oxidation of gaseous organic compounds. Additionally, a granular activated carbon was studied for comparison purposes. In both types of TiO2/AC composites the effect of different variables (i.e., the thermal treatment conditions used during the preparation of these materials) and the UV-light wavelength used during photocatalytic oxidation were analyzed. The prepared materials were deeply characterized (by gas adsorption, TGA, XRD, SEM and photocatalytic propene oxidation). The obtained results show that the carbon support has an important effect on the properties of the deposited TiO2 and, therefore, on the photocatalytic activity of the resulting TiO2/AC composites. Thus, the hybrid materials prepared over the spherical activated carbon show better results than those prepared over the granular one; a good TiO2 coverage with a high crystallinity of the deposited titanium dioxide, which just needs an air oxidation treatment at low-moderate temperature (350–375 °C) to present high photoactivity, without the need of additional inert atmosphere treatments. Additionally, these materials are more active at 365 nm than at 257.7 nm UV radiation, opening the possibility of using solar light for this application.

Spherical carbons: Synthesis, characterization and activation processes

Spherical carbons have been prepared through hydrothermal treatment of three carbohydrates (glucose, saccharose and cellulose). Preparation variables such as treatment time, treatment temperature and concentration of carbohydrate have been analyzed to obtain spherical carbons. These spherical carbons can be prepared with particle sizes larger than 10 μm, especially from saccharose, and have subsequently been activated using different activation processes (H3PO4, NaOH, KOH or physical activation with CO2) to develop their textural properties. All these spherical carbons maintained their spherical morphology after the activation process, except when KOH/carbon ratios higher than 4/1 were used, which caused partial destruction of the spheres. The spherical activated carbons develop interesting textural properties with the four activating agents employed, reaching surface areas up to 3100 m2/g. Comparison of spherical activated carbons obtained with the different activating agents, taking into account the yields obtained after the activation process, shows that phosphoric acid activation produces spherical activated carbons with higher developed surface areas. Also, the spherical activated carbons present different oxygen groups’ content depending on the activating agent employed (higher surface oxygen groups content for chemical activation than for physical activation).

Factors governing the adsorption of ethanol on spherical activated carbons

Ethanol adsorption on different activated carbons (mostly spherical ones) was investigated covering the relative pressure range from 0.001 to 1. Oxygen surface contents of the ACs were modified by oxidation (in HNO3 solution or air) and/or by thermal treatment in N2. To differentiate the concomitant effects of porosity and oxygen surface chemistry on ethanol adsorption, different sets of samples were used to analyze different relative pressure ranges (below 1000 ppmv concentration and close to unity). To see the effect of oxygen surface chemistry, selected samples having similar porosity but different oxygen contents were studied in the low relative pressure range. At low ethanol concentration (225 ppmv) adsorption is favored in oxidized samples, remarking the effect of the oxidizing treatment used (HNO3 is more effective than air) and the type of oxygen functionalities created (carboxyl and anhydride groups are more effective than phenolic, carbonyl and derivatives). To analyze the high relative pressure range, spherical and additional ACs were used. As the relative pressure of ethanol increases, the effect of oxygen-containing surface groups decreases and microporosity becomes the most important variable affecting the adsorption of ethanol.

Primal Attainment in Convex Infinite Optimization Duality

This article provides results guarateeing that the optimal value of a given convex infinite optimization problem and its corresponding surrogate Lagrangian dual coincide and the primal optimal value is attainable. The conditions ensuring converse strong Lagrangian (in short, minsup) duality involve the weakly-inf-(locally) compactness of suitable functions and the linearity or relative closedness of some sets depending on the data. Applications are given to different areas of convex optimization, including an extension of the Clark-Duffin Theorem for ordinary convex programs.

Comparative study of RPSALG algorithm for convex semi-infinite programming

The Remez penalty and smoothing algorithm (RPSALG) is a unified framework for penalty and smoothing methods for solving min-max convex semi-infinite programing problems, whose convergence was analyzed in a previous paper of three of the authors. In this paper we consider a partial implementation of RPSALG for solving ordinary convex semi-infinite programming problems. Each iteration of RPSALG involves two types of auxiliary optimization problems: the first one consists of obtaining an approximate solution of some discretized convex problem, while the second one requires to solve a non-convex optimization problem involving the parametric constraints as objective function with the parameter as variable. In this paper we tackle the latter problem with a variant of the cutting angle method called ECAM, a global optimization procedure for solving Lipschitz programming problems. We implement different variants of RPSALG which are compared with the unique publicly available SIP solver, NSIPS, on a battery of test problems.

New glimpses on convex infinite optimization duality

Given a convex optimization problem (P) in a locally convex topological vector space X with an arbitrary number of constraints, we consider three possible dual problems of (P), namely, the usual Lagrangian dual (D), the perturbational dual (Q), and the surrogate dual (Δ), the last one recently introduced in a previous paper of the authors (Goberna et al., J Convex Anal 21(4), 2014). As shown by simple examples, these dual problems may be all different. This paper provides conditions ensuring that inf(P)=max(D), inf(P)=max(Q), and inf(P)=max(Δ) (dual equality and existence of dual optimal solutions) in terms of the so-called closedness regarding to a set. Sufficient conditions guaranteeing min(P)=sup(Q) (dual equality and existence of primal optimal solutions) are also provided, for the nominal problems and also for their perturbational relatives. The particular cases of convex semi-infinite optimization problems (in which either the number of constraints or the dimension of X, but not both, is finite) and linear infinite optimization problems are analyzed. Finally, some applications to the feasibility of convex inequality systems are described.

Constraint qualifications in convex vector semi-infinite optimization

Convex vector (or multi-objective) semi-infinite optimization deals with the simultaneous minimization of finitely many convex scalar functions subject to infinitely many convex constraints. This paper provides characterizations of the weakly efficient, efficient and properly efficient points in terms of cones involving the data and Karush–Kuhn–Tucker conditions. The latter characterizations rely on different local and global constraint qualifications. The results in this paper generalize those obtained by the same authors on linear vector semi-infinite optimization problems.

Corrigendum to “The envelope theorem for multiobjective convex programming via contingent derivatives” [J. Math. Anal. Appl. 372 (1) (2010) 197–207]

The aim of this note is to formulate an envelope theorem for vector convex programs. This version corrects an earlier work, “The envelope theorem for multiobjective convex programming via contingent derivatives” by Jiménez Guerra et al. (2010) [3]. We first propose a necessary and sufficient condition allowing to restate the main result proved in the alluded paper. Second, we introduce a new Lagrange multiplier in order to obtain an envelope theorem avoiding the aforementioned error.

Sensitivity Analysis in Convex Optimization through the Circatangent Derivative

The main goal of this paper is to analyse the sensitivity of a vector convex optimization problem according to variations in the right-hand side. We measure the quantitative behavior of a certain set of Pareto optimal points characterized to become minimum when the objective function is composed with a positive function. Its behavior is analysed quantitatively using the circatangent derivative for set-valued maps. Particularly, it is shown that the sensitivity is closely related to a Lagrange multiplier solution of a dual program.

Fabrication of Mg foams for biomedical applications by means of a replica method based upon spherical carbon particles

Interest in Mg foams is increasing due to their potential use as biomaterials. Fabrication methods determine to a great extent their structure and, in some cases, may pollute the foam. In this work Mg foams are fabricated by a replica method that uses as skeleton packed spheres of active carbon, a material widely utilized in medicine. After Mg infiltration, carbon particles are eliminated by an oxidizing heat treatment. The latter covers Mg with MgO which improves performance. In particular, oxidation retards degradation of the foam, as the polarization curves of the Mg foam with and without oxide indicate. The sphericity and regularity of C particles allows control of the structure of the produced open-cell foams.