What is the price of anarchy when unsplittable demands are routed selfishly in general networks with load-dependent edge delays? Motivated by this question we generalize the model of [14] to the case of weighted congestion games. We show that varying demands of users crucially affect the nature of these games, which are no longer isomorphic to exact potential games, even for very simple instances. Indeed we construct examples where even a single-commodity (weighted) network congestion game may have no pure Nash equilibrium.

Computing the chromatic number of a graph is an NP-hard problem. For random graphs and some other classes of graphs, estimators of the expected chromatic number have been well studied. In this paper, a new 0--1 integer programming formulation for the graph coloring problem is presented. The proposed new formulation is used to develop a method that generates graphs of known chromatic number by using the KKT optimality conditions of a related continuous nonlinear program.

The backpropagation algorithm is an iterative gradient descent algorithm designed to train multilayer neural networks. Despite its popularity and eectiveness, the orthogonal steps (zigzagging) near the optimum point slows down the convergence of this algorithm. To overcome the ineciency of zigzagging in the conventional backpropagation algorithm, one of the authors earlier proposed the use of a deecting gradient technique to improve the convergence of backpropagation learning algorithm. The proposed method is called Partan backpropagation learning algorithm[3]. The convergence time of multilayer networks has further improved through dynamic adaptation of their learning rates[6]. In this paper, an extension to the dynamic parallel tangent learning algorithm is proposed. In the proposed algorithm, each connection has its own learning as well as acceleration rate. These individual rates are dynamically adapted as the learning proceeds. Simulation studies are carried out on dierent learning problems. Faster rate of convergence is achieved for all problems used in the simulations. Keywords: Articial neural networks, Backpropagation, Gradient descent, Parallel tangent, Dynamic parallel tangent. 1

Abstract Indisputably Normalized Cuts is one of the most popular segmentation algorithms in pattern recognition and computer vision. It has been applied to a wide range of segmentation tasks with great success. A number of extensions to this approach have also been proposed, including ones that can deal with multiple classes or that can incorporate a priori information in the form of grouping constraints. However, what is common for all these methods is that they are noticeably limited in the type of constraints that can be incorporated and can only address segmentation problems on a very specific form. In this paper, we present a reformulation of Normalized Cut segmentation that in a unified way can handle linear equality constraints for an arbitrary number of classes. This is done by restating the problem and showing how linear constraints can be enforced exactly in the optimization scheme through duality. This allows us to add group priors, for example, that certain pixels should belong to a given class. In addition, it provides a principled way to perform multi-class segmentation for tasks like interactive segmentation. The method has been tested on real data showing good performance and improvements compared to standard normalized cuts.

Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality

Iterant recombination with one-norm minimization for multilevel