An analysis of various elastic net algorithms

The Elastic Net Algorithm (ENA) for solving the Traveling Salesman Problem is analyzed applying statistical mechanics. Using some general properties of the free energy function of stochastic Hopfield Neural Networks, we argue why Simic's derivation of the ENA from a Hopfield network is incorrect. However, like the Hopfield-Lagrange method, the ENA may be considered a specific dynamic penalty method, where, in this case, the weights of the various penalty terms decrease during execution of the algorithm. This view on the ENA corresponds to the view resulting from the theory on `deformable templates', where the term stochastic penalty method seems to be most appropriate. Next, the ENA is analyzed both on the level of the energy function as well as on the level of the motion equations. It will be proven and shown experimentally, why a non-feasible solution is sometimes found. It can be caused either by a too rapid lowering of the temperature parameter (which is avoidable), or by a peculiar property of the algorithm,namely, that of adhering to equidistance of the elastic net points. Thereupon, an alternative, Non-equidistant Elastic Net Algorithm (NENA) is presented and analyzed. It has a correct distance measure and it is hoped to guarantee feasibility in a more natural way. For small problem instances, this conjecture is confirmed experimentally. However, trying larger problem instances, the pictures changes: our experimental results show that the elastic net points appear to become `lumpy' which may cause non-feasibility again. Moreover, in cases both algorithms yield a feasible solution, the quality of the solution found by the NENA is often slightly worse than the one found by the original ENA. This motivated us to try an Hybrid Elastic Net Algorithm (HENA), which starts using the ENA and, after having found a good approximate solution, switches to the NENA in order to guarantee feasibility too. In practice, the ENA and HENA perform more or less the same. Up till now, we did not find parameters of the HENA, which invariably guarantee the desired feasibility of solutions

Coverage, Continuity and Visual Cortical Architecture

The primary visual cortex of many mammals contains a continuous representation of visual space, with a roughly repetitive aperiodic map of orientation preferences superimposed. It was recently found that orientation preference maps (OPMs) obey statistical laws which are apparently invariant among species widely separated in eutherian evolution. Here, we examine whether one of the most prominent models for the optimization of cortical maps, the elastic net (EN) model, can reproduce this common design. The EN model generates representations which optimally trade of stimulus space coverage and map continuity. While this model has been used in numerous studies, no analytical results about the precise layout of the predicted OPMs have been obtained so far. We present a mathematical approach to analytically calculate the cortical representations predicted by the EN model for the joint mapping of stimulus position and orientation. We find that in all previously studied regimes, predicted OPM layouts are perfectly periodic. An unbiased search through the EN parameter space identifies a novel regime of aperiodic OPMs with pinwheel densities lower than found in experiments. In an extreme limit, aperiodic OPMs quantitatively resembling experimental observations emerge. Stabilization of these layouts results from strong nonlocal interactions rather than from a coverage-continuity-compromise. Our results demonstrate that optimization models for stimulus representations dominated by nonlocal suppressive interactions are in principle capable of correctly predicting the common OPM design. They question that visual cortical feature representations can be explained by a coverage-continuity-compromise.Comment: 100 pages, including an Appendix, 21 + 7 figure

Channel allocation in elastic optical networks using traveling salesman problem algorithms

Elastic optical networks have been proposed to support high data rates in metro and core networks. However, frequency allocation of the channels (i.e., channel ordering) in such networks is a challenging problem. This requires arranging the optical channels within the frequency grid with the objective of ensuring a minimum signal-to-noise ratio (SNR). An optimal arrangement results in the highest SNR margin for the entire network. However, determining the optimal arrangement requires an exhaustive search through all possible arrangements (permutations) of the channels. The search space increases exponentially with the number of channels. This discourages an algorithm employing an exhaustive search for the optimal frequency allocation. We utilize the Gaussian noise (GN) model to formulate the frequency allocation (channel ordering) problem as a variant of the traveling salesman problem (TSP) using graph theory. Thereafter, we utilize graph-theoretic tools for the TSP from the existing literature to solve the channel ordering problem. Performance figures obtained for the proposed scheme show that it is marginally inferior to the optimal search (through all possible permutations) and outperforms any random allocation scheme. Moreover, the proposed scheme is implementable for a scenario with a large number of channels. In comparison, an exhaustive search with the GN model and split-step Fourier method simulations are shown to be feasible for a small number of channels only. It is also illustrated that the SNR decreases with an increase in bandwidth when the frequency separation is high

Traveling Salesman Problem

This book is a collection of current research in the application of evolutionary algorithms and other optimal algorithms to solving the TSP problem. It brings together researchers with applications in Artificial Immune Systems, Genetic Algorithms, Neural Networks and Differential Evolution Algorithm. Hybrid systems, like Fuzzy Maps, Chaotic Maps and Parallelized TSP are also presented. Most importantly, this book presents both theoretical as well as practical applications of TSP, which will be a vital tool for researchers and graduate entry students in the field of applied Mathematics, Computing Science and Engineering