Learning Convex Partitions and Computing Game-theoretic Equilibria from Best Response Queries

Suppose that an $m$-simplex is partitioned into $n$ convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance $\epsilon$ from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant $m$ uses $poly(n, \log \left( \frac{1}{\epsilon} \right))$ queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant $n$ uses $poly(m, \log \left( \frac{1}{\epsilon} \right))$ queries. We show via Kakutani's fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies. We also partially extend our results to games with multiple players, establishing further query complexity bounds for computing approximate well-supported equilibria in this setting.Comment: 38 pages, 7 figures, second version strengthens lower bound in Theorem 6, adds footnotes with additional comments and fixes typo

Weak lensing goes bananas: What flexion really measures

In weak gravitational lensing, the image distortion caused by shear measures the projected tidal gravitational field of the deflecting mass distribution. To lowest order, the shear is proportional to the mean image ellipticity. If the image sizes are not small compared to the scale over which the shear varies, higher-order distortions occur, called flexion. For ordinary weak lensing, the observable quantity is not the shear, but the reduced shear, owing to the mass-sheet degeneracy. Likewise, the flexion itself is unobservable. Rather, higher-order image distortions measure the reduced flexion, i.e., derivatives of the reduced shear. We derive the corresponding lens equation in terms of the reduced flexion and calculate the resulting relation between brightness moments of source and image. Assuming an isotropic distribution of source orientations, estimates for the reduced shear and flexion are obtained; these are then tested with simulations. In particular, the presence of flexion affects the determination of the reduced shear. The results of these simulations yield the amount of bias of the estimators, as a function of the shear and flexion. We point out and quantify a fundamental limitation of the flexion formalism, in terms of the product of reduced flexion and source size. If this product increases above the derived threshold, multiple images of the source are formed locally, and the formalism breaks down. Finally, we show how a general (reduced) flexion field can be decomposed into its four components: two of them are due to a shear field, carrying an E- and B-mode in general. The other two components do not correspond to a shear field; they can also be split up into corresponding E- and B-modes.Comment: 17 pages, 6 figures, submitted to A&

Estimate of dark halo ellipticity by lensing flexion

Aims. The predictions of the ellipticity of the dark matter halos from models of structure formation are notoriously difficult to test with observations. A direct measurement would give important constraints on the formation of galaxies, and its effect on the dark matter distribution in their halos. Here we show that galaxy-galaxy flexion provides a direct and potentially powerful method for determining the ellipticity of (an ensemble of) elliptical lenses. Methods. We decompose the spin-1 flexion into a radial and a tangential component. Using the ratio of tangential-to- radial flexion, which is independent of the radial mass profile, the mass ellipticity can be estimated. Results. An estimator for the ellipticity of the mass distribution is derived and tested with simulations. We show that the estimator is slightly biased. We quantify this bias, and provide a method to reduce it. Furthermore, a parametric fitting of the flexion ratio and orientation provides another estimate for the dark halo ellipticity, which is more accurate for individual lenses Overall, galaxy-galaxy flexion appears as a powerful tool for constraining the ellipticity of mass distributions.Comment: 6 pages,5 figures, submitted to AA, comments welcom

A directed mutation operator for real coded genetic algorithms

Copyright @ Springer-Verlag Berlin Heidelberg 2010.Developing directed mutation methods has been an interesting research topic to improve the performance of genetic algorithms (GAs) for function optimization. This paper introduces a directed mutation (DM) operator for GAs to explore promising areas in the search space. In this DM method, the statistics information regarding the fitness and distribution of individuals over intervals of each dimension is calculated according to the current population and is used to guide the mutation of an individual toward the neighboring interval that has the best statistics result in each dimension. Experiments are carried out to compare the proposed DM technique with an existing directed variation on a set of benchmark test problems. The experimental results show that the proposed DM operator achieves a better performance than the directed variation on most test problems

Approximate Well-supported Nash Equilibria below Two-thirds

In an epsilon-Nash equilibrium, a player can gain at most epsilon by changing his behaviour. Recent work has addressed the question of how best to compute epsilon-Nash equilibria, and for what values of epsilon a polynomial-time algorithm exists. An epsilon-well-supported Nash equilibrium (epsilon-WSNE) has the additional requirement that any strategy that is used with non-zero probability by a player must have payoff at most epsilon less than the best response. A recent algorithm of Kontogiannis and Spirakis shows how to compute a 2/3-WSNE in polynomial time, for bimatrix games. Here we introduce a new technique that leads to an improvement to the worst-case approximation guarantee

Accelerated Particle Swarm Optimization and Support Vector Machine for Business Optimization and Applications

Business optimization is becoming increasingly important because all business activities aim to maximize the profit and performance of products and services, under limited resources and appropriate constraints. Recent developments in support vector machine and metaheuristics show many advantages of these techniques. In particular, particle swarm optimization is now widely used in solving tough optimization problems. In this paper, we use a combination of a recently developed Accelerated PSO and a nonlinear support vector machine to form a framework for solving business optimization problems. We first apply the proposed APSO-SVM to production optimization, and then use it for income prediction and project scheduling. We also carry out some parametric studies and discuss the advantages of the proposed metaheuristic SVM.Comment: 12 page

Determination of the intrinsic behavior of polymers using digital image correlation combined with video-monitored testing

AbstractThree methods for the determination of the large-strain behavior of ductile polymers are compared in both tension and compression. Each method relies on some (non-contact) measurement of the strain and some approximations in the calculation of stress. The strain measurement techniques include digital image correlation (DIC) and two techniques of video-based extensometry: marker tracking and area variation monitoring. Since the specimens are inevitably subject to structural plastic instabilities (necking in tension, barreling in compression) the strain and stress states are no longer uniform in the gauge section after the peak load. Under such circumstances, it is demonstrated that the three experimental methods can lead to significant differences. It is inferred from the comparative analysis that the method based on vertical marker tracking is not reliable. Validated by DIC, video-based area variation is shown to be a simple alternative way to obtain an excellent estimate of the intrinsic true stress–strain behavior of the polymer

Genetic algorithms with self-organizing behaviour in dynamic environments

Copyright @ 2007 Springer-VerlagIn recent years, researchers from the genetic algorithm (GA) community have developed several approaches to enhance the performance of traditional GAs for dynamic optimization problems (DOPs). Among these approaches, one technique is to maintain the diversity of the population by inserting random immigrants into the population. This chapter investigates a self-organizing random immigrants scheme for GAs to address DOPs, where the worst individual and its next neighbours are replaced by random immigrants. In order to protect the newly introduced immigrants from being replaced by fitter individuals, they are placed in a subpopulation. In this way, individuals start to interact between themselves and, when the fitness of the individuals are close, one single replacement of an individual can affect a large number of individuals of the population in a chain reaction. The individuals in a subpopulation are not allowed to be replaced by individuals of the main population during the current chain reaction. The number of individuals in the subpopulation is given by the number of individuals created in the current chain reaction. It is important to observe that this simple approach can take the system to a self-organization behaviour, which can be useful for GAs in dynamic environments.Financial support was obtained from FAPESP (Proc. 04/04289-6)

COVNET : A cooperative coevolutionary model for evolving artificial neural networks

This paper presents COVNET, a new cooperative coevolutionary model for evolving artificial neural networks. This model is based on the idea of coevolving subnetworks. that must cooperate to form a solution for a specific problem, instead of evolving complete networks. The combination of this subnetwork is part of a coevolutionary process. The best combinations of subnetworks must be evolved together with the coevolution of the subnetworks. Several subpopulations of subnetworks coevolve cooperatively and genetically isolated. The individual of every subpopulation are combined to form whole networks. This is a different approach from most current models of evolutionary neural networks which try to develop whole networks. COVNET places as few restrictions as possible over the network structure, allowing the model to reach a wide variety of architectures during the evolution and to be easily extensible to other kind of neural networks. The performance of the model in solving three real problems of classification is compared with a modular network, the adaptive mixture of experts and with the results presented in the bibliography. COVNET has shown better generalization and produced smaller networks than the adaptive mixture of experts and has also achieved results, at least, comparable with the results in the bibliography

The K\"ahler-Ricci flow with positive bisectional curvature

We show that the K\"ahler-Ricci flow on a manifold with positive first Chern class converges to a K\"ahler-Einstein metric assuming positive bisectional curvature and certain stability conditions.Comment: 15 page