A numerical algorithm for finding solutions of a generalized Nash equilibrium problem

A family of nonempty closed convex sets is built by using the data of the Generalized Nash equilibrium problem (GNEP). The sets are selected iteratively such that the intersection of the selected sets contains solutions of the GNEP. The algorithm introduced by Iusem-Sosa (2003) is adapted to obtain solutions of the GNEP. Finally some numerical experiments are given to illustrate the numerical behavior of the algorithm.

Emotional attention as a modulatory system of perception

Report for the scientific sojourn carried out at the University Medical Center, Swiss, from 2010 to 2012. Abundant evidence suggests that negative emotional stimuli are prioritized in the perceptual systems, eliciting enhanced neural responses in early sensory regions as compared with neutral information. This facilitated detection is generally paralleled by larger neural responses in early sensory areas, relative to the processing of neutral information. In this sense, the amygdala and other limbic regions, such as the orbitofrontal cortex, may play a critical role by sending modulatory projections onto the sensory cortices via direct or indirect feedback.The present project aimed at investigating two important issues regarding these mechanisms of emotional attention, by means of functional magnetic resonance imaging. In Study I, we examined the modulatory effects of visual emotion signals on the processing of task-irrelevant visual, auditory, and somatosensory input, that is, the intramodal and crossmodal effects of emotional attention. We observed that brain responses to auditory and tactile stimulation were enhanced during the processing of visual emotional stimuli, as compared to neutral, in bilateral primary auditory and somatosensory cortices, respectively. However, brain responses to visual task-irrelevant stimulation were diminished in left primary and secondary visual cortices in the same conditions. The results also suggested the existence of a multimodal network associated with emotional attention, presumably involving mediofrontal, temporal and orbitofrontal regions Finally, Study II examined the different brain responses along the low-level visual pathways and limbic regions, as a function of the number of retinal spikes during visual emotional processing. The experiment used stimuli resulting from an algorithm that simulates how the visual system perceives a visual input after a given number of retinal spikes. The results validated the visual model in human subjects and suggested differential emotional responses in the amygdala and visual regions as a function of spike-levels. A list of publications resulting from work in the host laboratory is included in the report.

Emotional attention as a modulatory system of perception

Report for the scientific sojourn carried out at the University Medical Center, Swiss, from 2010 to 2012. Abundant evidence suggests that negative emotional stimuli are prioritized in the perceptual systems, eliciting enhanced neural responses in early sensory regions as compared with neutral information. This facilitated detection is generally paralleled by larger neural responses in early sensory areas, relative to the processing of neutral information. In this sense, the amygdala and other limbic regions, such as the orbitofrontal cortex, may play a critical role by sending modulatory projections onto the sensory cortices via direct or indirect feedback.The present project aimed at investigating two important issues regarding these mechanisms of emotional attention, by means of functional magnetic resonance imaging. In Study I, we examined the modulatory effects of visual emotion signals on the processing of task-irrelevant visual, auditory, and somatosensory input, that is, the intramodal and crossmodal effects of emotional attention. We observed that brain responses to auditory and tactile stimulation were enhanced during the processing of visual emotional stimuli, as compared to neutral, in bilateral primary auditory and somatosensory cortices, respectively. However, brain responses to visual task-irrelevant stimulation were diminished in left primary and secondary visual cortices in the same conditions. The results also suggested the existence of a multimodal network associated with emotional attention, presumably involving mediofrontal, temporal and orbitofrontal regions Finally, Study II examined the different brain responses along the low-level visual pathways and limbic regions, as a function of the number of retinal spikes during visual emotional processing. The experiment used stimuli resulting from an algorithm that simulates how the visual system perceives a visual input after a given number of retinal spikes. The results validated the visual model in human subjects and suggested differential emotional responses in the amygdala and visual regions as a function of spike-levels. A list of publications resulting from work in the host laboratory is included in the report.

Condensation of polyhedric structures onto soap films

We study the existence of solutions to general measure-minimization problems over topological classes that are stable under localized Lipschitz homotopy, including the standard Plateau problem without the need for restrictive assumptions such as orientability or even rectifiability of surfaces. In case of problems over an open and bounded domain we establish the existence of a “minimal candidate”, obtained as the limit for the local Hausdorff convergence of a minimizing sequence for which the measure is lower-semicontinuous. Although we do not give a way to control the topological constraint when taking limit yet— except for some examples of topological classes preserving local separation or for periodic two-dimensional sets — we prove that this candidate is an Almgren-minimal set. Thus, using regularity results such as Jean Taylor’s theorem, this could be a way to find solutions to the above minimization problems under a generic setup in arbitrary dimension and codimension.

Generic optimality conditions for semi-algebraic convex programs

We consider linear optimization over a nonempty convex semi-algebraic feasible region F. Semidefinite programming is an example. If F is compact, then for almost every linear objective there is a unique optimal solution, lying on a unique \active" manifold, around which F is \partly smooth", and the second-order sufficient conditions hold. Perturbing the objective results in smooth variation of the optimal solution. The active manifold consists, locally, of these perturbed optimal solutions; it is independent of the representation of F, and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes. These results extend to unbounded sets F.

Lipschitz harmonic capacity and Bilipschitz images of cantor sets

For bilipschitz images of Cantor sets in Rd we estimate the Lipschitz harmonic capacity and show this capacity is invariant under bilipschitz homeomorphisms.

Permutations defining convex permutominoes

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Characterizations of Lojasiewicz inequalities and applications

The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame geometry. This paper provides alternative characterizations of this type of inequalities for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In a metric context, we show that a generalized form of the Lojasiewicz inequality (hereby called the Kurdyka- Lojasiewicz inequality) relates to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by -∂f are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the Kurdyka- Lojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines -a concept linked to the location of the less steepest points at the level sets of f- and integrability conditions are given. In the convex case these results are significantly reinforced, allowing in particular to establish the asymptotic equivalence of discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C2 function in R2 is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka- Lojasiewicz inequality.

Maximal operators and differentiation theorems for sparse sets

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Calderón-Zygmund capacities and Wolff potentials on Cantor sets

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On stability of solutions to systems of convex inequalities

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On the stability of the optimal value and the optimal set in optimization problems

The paper develops a stability theory for the optimal value and the optimal set mapping of optimization problems posed in a Banach space. The problems considered in this paper have an arbitrary number of inequality constraints involving lower semicontinuous (not necessarily convex) functions and one closed abstract constraint set. The considered perturbations lead to problems of the same type as the nominal one (with the same space of variables and the same number of constraints), where the abstract constraint set can also be perturbed. The spaces of functions involved in the problems (objective and constraints) are equipped with the metric of the uniform convergence on the bounded sets, meanwhile in the space of closed sets we consider, coherently, the Attouch-Wets topology. The paper examines, in a unified way, the lower and upper semicontinuity of the optimal value function, and the closedness, lower and upper semicontinuity (in the sense of Berge) of the optimal set mapping. This paper can be seen as a second part of the stability theory presented in [17], where we studied the stability of the feasible set mapping (completed here with the analysis of the Lipschitz-like property).

A way to play bankruptcy problems.

The commitment among agents has always been a difficult task, especially when they have to decide how to distribute the available amount of a scarce resource among all. On the one hand, there are a multiplicity of possible ways for assigning the available amount; and, on the other hand, each agent is going to propose that distribution which provides her the highest possible award. In this paper, with the purpose of making this agreement easier, firstly we use two different sets of basic properties, called Commonly Accepted Equity Principles, to delimit what agents can propose as reasonable allocations. Secondly, we extend the results obtained by Chun (1989) and Herrero (2003), obtaining new characterizations of old and well known bankruptcy rules. Finally, using the fact that bankruptcy problems can be analyzed from awards and losses, we define a mechanism which provides a new justification of the convex combinations of bankruptcy rules. Keywords: Bankruptcy problems, Unanimous Concessions procedure, Diminishing Claims mechanism, Piniles’ rule, Constrained Egalitarian rule. JEL classification: C71, D63, D71.

Beurling-Landau densities of weighted Fekete sets and correlation kernel estimates

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Onto-Trauma : Ontologia d'aprenentatge en l'àrea de la cirurgia traumatològica

Treball en què es crea una ontologia per simular totes les etapes d'una operació traumatològica, i usa aquesta ontologia per crear un prototip de wiki semàntica.