Chaotic Neural Network with Legendre Function Self-feedback and Applications

The optimization performance of transiently chaotic neural network (TCNN) is affected by various factors such as chaotic characteristic, model parameters, and annealing function, and its capacity of global optimization is limited. It is demonstrated that the non-monotonic activation function can generate richer chaotic characteristic than the monotonic activation function in the TCNN model. Besides, the activation function involving neurobiological mechanism can not only reflect the rich brain activity in brain waves, but also enhance the non-linear dynamic characteristic, which may further improve the global optimization ability. Hence, a novel chaotic neuron model is proposed with the non-monotonic activation function based on the neurobiological mechanisms from the electroencephalogram. The electroencephalogram consists of five brain waves (i.e., , , , , and waves) which are defined by the quality and intensity of brain waves with different frequency bands ranging from 0.5 Hz to 100 Hz. The brain wave with a higher frequency and a lower amplitude represents a more active brain. Researches demonstrate that the five brain waves can be simplified into sinusoidal waves with different frequencies. Hence, a frequency conversion sinusoidal (FCS) function which has the consistent frequency range and features with brain waves is designed based on the above neurobiological mechanisms. Then a novel chaotic neuron model with non-monotonic activation function which is composed of the FCS function and sigmoid function, is proposed for richer chaotic dynamic characteristic. The reversed bifurcation and the Lyapunov exponent of the chaotic neuron are given and the dynamic system is analyzed, indicating that the proposed FCS neuron model owns richer chaotic dynamic characteristic than transiently chaotic neuron model due to its special non-monotonic activation function. Based on the neuron model, a novel transiently-chaotic neural networkfrequency conversion sinusoidal chaotic neural network (FCSCNN) is constructed and the basis of model parameter selection is provided as well. To validate the effectiveness of the proposed model, the FCSCNN is applied to nonlinear function optimization and 10-city, 30-city, 75-city traveling salesman problem. The experimental results show that 1) the FCSCNN has a good performance under the condition of moderate a, smaller cA(0) and 2(0); 2) on the basis of the appropriate model parameters, the FCSCNN has better global optimization ability and optimization accuracy than Hopfield neural network, TCNN, improved-TCNN due to its richer chaotic characteristic in complicated combinational optimization problem, especially in middle and large scale problem.

The paper reviews and introduces the state-of-the-art nature-inspired metaheuristic algorithms in optimization, including Firefly algorithm, PSO algorithms and ABC algorithm. By implementing these algorithms in Matlab, we will use worked examples to show how each algorithm works. Firefly algorithm is one of the evolutionary optimization algorithms, and is inspired by the flashing behaviour of fireflies in nature. There are many noisy non-linear mathematical optimization problems that can be effectively solved by Metaheuristic Algorithms. Mathematical optimization or programming is the study of such planning and design problems using mathematical tools. Nowadays, computer simulations become an indispensable tool for solving such optimization problems with various efficient search algorithms. Nature-inspired algorithms are among the most powerful algorithms for optimization. Firefly algorithm is one of the new metaheuristic algorithms for optimization problems. The algorithm is inspired by the flashing behaviour of fireflies. A Firefly Algorithm (FA) is a recent nature inspired optimization algorithm, which simulates the flash pattern and characteristics of fireflies. It is a powerful swarm intelligence algorithm inspired by the flash phenomenon of the fireflies. In this context, three types of meta-heuristics called Artificial bee Colony algorithm, Particle Swarm Optimization (PSO) and Firefly algorithms were devised to find optimal solutions of noisy non-linear continuous mathematical models. A series of computational experiments using each algorithm were conducted. The stimulation result of this experiment were analyzed and compared to the best solutions found so The Firefly algorithm in each noisy non linear optimization function seems to perform better and efficient.

Chaotic neural networks have been proved to be strong tools to solve the optimization problems. In order to escape the local minima, a new function is introduced into the self-feedback item of the transiently chaotic neural network, this paper proposes chaotic neural network with nonlinear function self-feedback by improving the self-feedback item of Chen and Aihara's network making that the network has new character which is different from linear self-feedback network. The figures of the reversed bifurcation and the maximal Lyapunov exponents of single neural unit were given, We constructed the energy function of chaotic neural network, and analyzed the sufficient condition for the networks to reach asymptotical stability and set the parameters of the networks for solving traveling salesman problem (TSP). The simulation results in solving 10-city TSP show that the new model is valid in solving optimization problems.

In order to install a chaotic neural network to solve the traveling salesman problem (TSP) on an integrated circuit, a chaotic neuron circuit is designed by using a current mode circuit. First, the characteristics of the output function of the neuron circuit are determined by numerical simulation so that they have excellent capability to solve the TSP. Next, the network is designed such that the chaotic neural network consisting of the designed neuron circuits can seek the optimum solution efficiently. Further, a circuit is proposed which realizes chaotic annealing in order to improve the convergence of the chaotic neural network. The ability of the proposed network using the annealing circuit to derive an optimum solution of the TSP is studied. From the simulation results, it is confirmed that the proposed chaotic neuron circuit is suitable for integration of the chaotic neural network to solve the TSP. © 1999 Scripta Technica, Electron Comm Jpn Pt 3, 82(9): 52–62, 1999

A Chaotic neural network model with Morlet wavelet function self-feedback is proposed by introducing Morlet wavelet function into self-feedback of chaotic neural network. The analyses of the optimization mechanism of the networks suggest that Morlet wavelet function self-feedback affects the original Hopfield energy function in the manner of the sum of the multiplications of Morlet wavelet function to the state, avoiding the network being trapped into the local minima. The energy function is constructed, and the sufficient condition for the networks to achieve asymptotical stability is analyzed and is used to instruct the parameter set of the networks for solving traveling salesman problem (TSP). Simulation researches on 10-city TSP indicate that the proposed networks can find the optimal solutions of combinatorial optimization problems.

The Traveling Salesman Problem (TSP) is a combinatorial optimization problem which has a plethora of real-world applications in various domains. Since large scale TSP problems are difficult to be solved by deterministic algorithms in a reasonable amount of time, heuristic and meta-heuristic algorithms such as swarm intelligence algorithms are usually used for solving TSP. These algorithms provide near-optimal solutions in a feasible amount of time. The Dragonfly Algorithm (DA) is a recent swarm intelligence algorithm and it has shown to have a higher performance as compared to other swarm intelligence algorithms in various applications. Since the original DA algorithm is proposed for solving continuous optimization problems, it cannot be used for solving TSP and although a binary version of the algorithm, called BDA, is also proposed, it is not suitable for solving TSP. Hence, in this paper, a variant of the DA algorithm is proposed. DA is adapted for solving TSP by adapting its equations and making use of the method of swap sequences to update the position of the artificial dragonflies in the search space. The algorithm is employed to solve a TSP problem which consists of four cities and the results show that the proposed algorithm is able to provide the optimal TSP path.

A novel chaotic-neuron model is presented by introducing the non-monotonous activation function which is composed of the Legendre function and the Sigmoid function.The reversed bifurcation of the chaotic neuron model is given and analyzed, meanwhile, how do parameters influence the network convergence speed is discussed.Based on the neuron model, the piecewise simulated annealing SLF chaotic neural network was made by introducing the simulated annealing idea, the model improve the convergence speed, at the same time, the precision of this network have not being influenced.The simulation experiment of function optimization and TSP problem verify the effectiveness of the segmented simulated annealing strategy.

The traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization. Several semidefinite programming relaxations have been proposed recently that exploit a variety of mathematical structures including, for example, algebraic connectivity, permutation matrices, and association schemes. The main results of this paper are twofold. First, de Klerk and Sotirov [de Klerk E, Sotirov R (2012) Improved semidefinite programming bounds for quadratic assignment problems with suitable symmetry. Math. Programming 133(1):75–91.] present a semidefinite program (SDP) based on permutation matrices and symmetry reduction; they show that it is incomparable to the subtour elimination linear program but generally dominates it on small instances. We provide a family of simplicial TSP instances that shows that the integrality gap of this SDP is unbounded. Second, we show that these simplicial TSP instances imply the unbounded integrality gap of every SDP relaxation of the TSP mentioned in the survey on SDP relaxations of the TSP in section 2 of Sotirov [Sotirov R (2012) SDP relaxations for some combinatorial optimization problems. Anjos MF, Lasserre JB, eds., Handbook on Semidefinite, Conic and Polynomial Optimization (Springer, New York), 795–819.]. In contrast, the subtour linear program performs perfectly on simplicial instances. The simplicial instances thus form a natural litmus test for future SDP relaxations of the TSP.

Discrete Spider Monkey Optimization for Travelling Salesman Problem

Construction of blind restoration model for super-resolution image based on chaotic neural network

Various neural network models have been proposed to tackle combinatorial optimization problems such as the travelling salesman problem (TSP). Existing learning-based TSP methods adopt a simple setting that the training and testing data are independent and identically distributed. However, the existing literature fails to solve TSP instances when training and testing data have different distributions. Concretely, we find that different training and testing distribution will result in more difficult TSP instances, i.e., the solution obtained by the model has a large gap from the optimal solution. To tackle this problem, in this work, we study learning-based TSP methods when training and testing data have different distributions using adaptive-hardness, i.e., how difficult a TSP instance can be for a solver. This problem is challenging because it is non-trivial to (1) define hardness measurement quantitatively; (2) efficiently and continuously generate sufficiently hard TSP instances upon model training; (3) fully utilize instances with different levels of hardness to learn a more powerful TSP solver. To solve these challenges, we first propose a principled hardness measurement to quantify the hardness of TSP instances. Then, we propose a hardness-adaptive generator to generate instances with different hardness. We further propose a curriculum learner fully utilizing these instances to train the TSP solver. Experiments show that our hardness-adaptive generator can generate instances ten times harder than the existing methods, and our proposed method achieves significant improvement over state-of-the-art models in terms of the optimality gap. The codes are publicly available.

The probabilistic travelling salesman problem (PTSP) is an extension of the well-known travelling salesman problem (TSP), which has been extensively studied in the field of combinatorial optimization. The goal of the TSP is to find the minimum length of a tour to all customers, given the distances between all pairs of customers whereas the objective of the PTSP is to minimize the expected length of the a priori tour where each customer requires a visit only with a given probability (Bertsimas, 1988; Bertsimas et al., 1990; Jaillet, 1985). The main difference between the PTSP and the TSP is that in the PTSP the probability of each node being visited is between 0.0 and 1.0 while in TSP the probability of each node being visited is 1.0. Due to the fact that the element of uncertainty not only exists, but also significantly affects the system performance in many real-world transportation and logistics applications, the results from the PTSP can provide insights into research in other probabilistic combinatorial optimization problems. Moreover, the PTSP can also be used to model many real-world applications in logistical and transportation planning, such as daily pickup-delivery services with stochastic demand, job sequencing involving changeover cost, design of retrieval sequences in a warehouse or in a cargo terminal operations, meals on wheels in senior citizen services, trip-chaining activities, vehicle routing problem with stochastic demand, and home delivery service under e-commerce (Bartholdi et al., 1983; Bertsimas et al., 1995; Campbell, 2006; Jaillet, 1988; Tang & Miller-Hooks, 2004). Early PTSP computational studies, dating from 1985, adopted heuristic approaches that were modified from the TSP (e.g., nearest neighbor, savings approach, spacefilling curve, radial sorting, 1-shift, and 2-opt exchanges) (Bartholdi & Platzman, 1988; Bertsimas, 1988; Bertsimas & Howell, 1993; Jaillet, 1985, 1987; Rossi & Gavioli, 1987). With its less than satisfactory performance in yielding solution quality, researchers in the recent years switch to metaheuristic methods, such as ant colony optimization (Bianchi, 2006; Branke & Guntsch, 2004), evolutionary algorithm (Liu et al., 2007), simulated annealing (Bowler et al., 2003), threshold accepting (Tang & Miller-Hooks, 2004) and scatter search (Liu, 2006, 2007, 2008). Because the genetic algorithm (GA), a conceptual framework of the population-based metaheuristic method, has been shown to yield promising outcomes for solving various complicated optimization problems in the past three decades (Back et al., 1997; Davis, 1991; O pe n A cc es s D at ab as e w w w .ite ch on lin e. co m

This work shows the application of Wang’s Recurrent Neural Network with the Winner Takes All (WTA) principle to solve the classic Operational Research problem called the Traveling Salesman Problem. The upgrade version proposed in this work for the ‘Winner Takes All’ principle is called soft, because the winning neuron is updated with only part of the activation values of the other competing neurons. The problems in the TSPLIB (Traveling Salesman Problem Library Reinelt, 1991) were used to compare the soft version with the ‘Winner Takes All’ hard version and they show improvement in the results using the 'Winner Takes All' soft version in most of the problems tested. The implementation of the technique proposed in this paper uses the parameters of Wang‘s Neural Network for the Assignment problem (Wang, 1992; Hung & Wang, 2003) using the ’Winner Takes All' principle to form Hamiltonian circuits (Siqueira et al. 2007) and can be used for both symmetric and asymmetric Traveling Salesman Problems. The 2-opt technique is used to improve the routes found with the proposed Neural Network, thus becoming a technique that is competitive to other Neural Networks. Other heuristic techniques have been developed recently to solve the Traveling Salesman Problem, and the work of Misevicius et al. (2005) shows the use of the ITS (Iterated Tabu Search) technique with a combination of intensification and diversification of solutions for the TSP. This technique is combined with the 5-opt and errors are almost zero in almost all of the TSPLIB problems tested. The work of Wang et al. (2007) shows the use of Particle Swarm to solve the TSP with the use of the fraction (quantum) principle to better guide the search for solutions. The authors make comparisons with Hill Climbing, Simulated Annealing and Tabu Search, and show in a 14-cities case that the results are better than those of the other techniques. In the area of Artificial Neural Networks, an interesting technique can be found in the work of Massutti & Castro (2009), who use a mechanism to stabilize winning neurons and the centroids of the groups of cities for the growing and pruning the network. The authors show modifications in the Rabnet’s (real-valued antibody network) parameters for the Traveling Salesman Problem and comparisons made with TSPLIB problems solved by other techniques show that the Rabnet has better results.

The classical travelling salesperson problem (TSP) models the movements of a salesperson travelling through a number of cities. The optimization problem is to choose the sequence in which to visit the cities in order to minimize the total distance travelled. This paper presents a generalized point-to-point motion-planning technique for multi-robot assemblysystems modelled as TSP-type optimization problems. However, in these augmented TSPs (TSP+), both the 'salesperson' (a robot with a tool) as well as the 'cities' (another robot with a workpiece) move. In addition to the sequencing of tasks, further planning is required to choose where the 'salesperson' (i.e., the tool) should rendezvous with each 'city' (i.e. the workpiece). The use of a genetic algorithm (GA) is chosen as the search engine for the solution of this TSP+ optimization problem. As an example area, the optimization of the electronic-component placement process is addressed. The simulation tools developed have been tested on five different component-placement system configurations. In the most generalized configuration, the placement robot meets the component delivery system at an optimal rendezvous location for the pick-up of the component and subsequently meets the printed-cirucit-board (on a mobile XY-table) at an optimal rendezvous location. In addition to the solution of the component-placement sequencing problem and the rendezvous-point planning problem, the collision-avoidance issue is addressed.

A novel model of chaotic neural network with double self-feedbacks composed of linear self-feedback and nonlinear Gauss-wavelet self-feedback is proposed in order to provide the network with both global searching ability and local characterizing ability. The single neuron with such double self-feedbacks can also exhibit complexly chaotic dynamic behaviors. Studies using the unified framework theory indicate that there exist two additional energy modifiers that can respectively provide the network with global searching ability and local characterizing ability to help the network to find globally optimal or near-optimal solutions. Although the network has complex self-feedbacks, it still can reach asymptotical stability. The simulation results on traveling salesman problems (TSP) show that the network with the double self-feedbacks has a higher probability of obtaining a global optimization solution compared with that with linear self-feedback or nonlinear Gauss-wavelet self-feedback.