Population equilibrium with support in evolutionary matrix games

In this paper we consider symmetric bimatrix games [A, A^T]. We use a matrix operator s(A), defined as the sum of the cofactors of the given matrix A, for finding the population equilibrium and its fitness in evolutionarily matrix games with all supported strategies, and to complete Bishop-Cannings Theorem.     

A novel particle swarm optimization algorithm based on particle migration

Inspired by the migratory behavior in the nature, a novel particle swarm optimization algorithm based on particle migration (MPSO) is proposed in this work. In this new algorithm, the population is randomly partitioned into several sub-swarms, each of which is made to evolve based on particle swarm optimization with time varying inertia weight and acceleration coefficients (LPSO-TVAC). At periodic stage in the evolution, some particles migrate from one complex to another to enhance the diversity of the population and avoid premature convergence. It further improves the ability of exploration and exploitation. Simulations for benchmark test functions illustrate that the proposed algorithm possesses better ability to find the global optima than other variants and is an effective global optimization tool.     

Weak solutions of the interior boundary value problems of plane Cosserat elasticity

In this paper we formulate interior Dirichlet and Neumann boundary value problems of plane Cosserat elasticity in Sobolev spaces, show that these problems are well-posed and find the corresponding weak solutions in the form of integral potentials. Received: April 7, 2005     

Collusive game solutions via optimization

A Nash-based collusive game among a finite set of players is one in which the players coordinate in order for each to gain higher payoffs than those prescribed by the Nash equilibrium solution. In this paper, we study the optimization problem of such a collusive game in which the players collectively maximize the Nash bargaining objective subject to a set of incentive compatibility constraints. We present a smooth reformulation of this optimization problem in terms of a nonlinear complementarity problem. We establish the convexity of the optimization problem in the case where each player's strategy set is unidimensional. In the multivariate case, we propose upper and lower bounding procedures for the collusive optimization problem and establish convergence properties of these procedures. Computational results with these procedures for solving some test problems are reported. It is with great honor that we dedicate this paper to Professor Terry Rockafellar on the occasion of his 70th birthday. Our work provides another example showing how Terry's fundamental contributions to convex and variational analysis have impacted the computational solution of applied game problems. This author's research was partially supported by the National Science Foundation under grant ECS-0080577. This author's research was partially supported by the National Science Foundation under grant CCR-0098013.     

Floating boundary particle swarm optimization algorithm

A new modification to the particle swarm optimization (PSO) algorithm is proposed aiming to make the algorithm less sensitive to selection of the initial search domain. To achieve this goal, we release the boundaries of the search domain and enable each boundary to drift independently, guided by the number of collisions with particles involved in the optimization process. The gradual modification of the active search domain range enables us to prevent particles from revisiting less promising regions of the search domain and also to explore the areas located outside the initial search domain. With time, the search domain shrinks around a region holding a global extremum. This helps improve the quality of the final solution obtained. It also makes the algorithm less sensitive to initial choice of the search domain ranges. The effectiveness of the proposed Floating Boundary PSO (FBPSO) is demonstrated using a set of standard test functions. To control the performance of the algorithm, new parameters are introduced. Their optimal values are determined through numerical examples.     

A rank based particle swarm optimization algorithm with dynamic adaptation

The particle swarm optimization (PSO) technique is a powerful stochastic evolutionary algorithm that can be used to find the global optimum solution in a complex search space. This paper presents a variation on the standard PSO algorithm called the rank based particle swarm optimizer, or PSO_rank, employing cooperative behavior of the particles to significantly improve the performance of the original algorithm. In this method, in order to efficiently control the local search and convergence to global optimum solution, the γ best particles are taken to contribute to the updating of the position of a candidate particle. The contribution of each particle is proportional to its strength. The strength is a function of three parameters: strivness, immediacy and number of contributed particles. All particles are sorted according to their fitness values, and only the γ best particles will be selected. The value of γ decreases linearly as the iteration increases. A time-varying inertia weight decreasing non-linearly is introduced to improve the performance. PSO_rank is tested on a commonly used set of optimization problems and is compared to other variants of the PSO algorithm presented in the literature. As a real application, PSO_rank is used for neural network training. The PSO_rank strategy outperformed all the methods considered in this investigation for most of the functions. Experimental results show the suitability of the proposed algorithm in terms of effectiveness and robustness.     

Reservoir flood control operation based on chaotic particle swarm optimization algorithm

Reservoir flood control operation is a complex engineering optimization problem with a large number of constraints. In order to solve this problem, a chaotic particle swarm optimization (CPSO) algorithm based on the improved logistic map is presented, which uses the discharge flow process as the decision variables combined with the death penalty function. According to the principle of maximum eliminating flood peak, a novel flood control operation model has been established with the goal of minimum standard deviation of the discharge flow process. At the same time, a piecewise linear interpolation function (PLIF) is applied to deal with the constraints for solving objective function. The performance of the proposed model and method is evaluated on two typical floods of Three Gorges reservoir. In comparison with existing models and other algorithms, the proposed model and algorithm can generate better solutions with the minimal flood peak discharge and the maximal peak-clipping rate for reservoir flood control operation.     

Improved particle swarm algorithm for hydrological parameter optimization

In this paper, a new method named MSSE-PSO (master-slave swarms shuffling evolution algorithm based on particle swarm optimization) is proposed. Firstly, a population of points is sampled randomly from the feasible space, and then partitioned into several sub-swarms (one master swarm and other slave swarms). Each slave swarm independently executes PSO or its variants, including the update of particles’ position and velocity. For the master swarm, the particles enhance themselves based on the social knowledge of master swarm and that of slave swarms. At periodic stage in the evolution, the master swarm and the whole slave swarms are forced to mix, and points are then reassigned to several sub-swarms to ensure the share of information. The process is repeated until a user-defined stopping criterion is reached. The tests of numerical simulation and the case study on hydrological model show that MSSE-PSO remarkably improves the accuracy of calibration, reduces the time of computation and enhances the performance of stability. Therefore, it is an effective and efficient global optimization method.     

基于粒子群算法的捕食者-食饵模型的参数估计

针对捕食者-食饵模型参数估计问题,基于三次Hermite插值多项式,提出了一种基于粒子群优化算法的高精度参数估计方法.数值仿真实验表明,本文提出的参数估计方法可以更精确地计算出相关参数.     

An improved particle swarm optimization algorithm combined with piecewise linear chaotic map

Particle swarm optimization (PSO) has gained increasing attention in tackling complex optimization problems. Its further superiority when hybridized with other search techniques is also shown. Chaos, with the properties of ergodicity and stochasticity, is definitely a good candidate, but currently only the well-known logistic map is prevalently used. In this paper, the performance and deficiencies of schemes coupling chaotic search into PSO are analyzed. Then, the piecewise linear chaotic map (PWLCM) is introduced to perform the chaotic search. An improved PSO algorithm combined with PWLCM (PWLCPSO) is proposed subsequently, and experimental results verify its great superiority.     

Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem

The pure azimuthal shear problem for a circular cylindrical tube of nonlinearly elastic material, both isotropic and anisotropic, is examined on the basis of a complementary energy principle. For particular choices of strain-energy function, one convex and one non-convex, closed-form solutions are obtained for this mixed boundary-value problem, for which the governing differential equation can be converted into an algebraic equation. The results for the non-convex strain energy function provide an illustration of a situation in which smooth analytic solutions of a nonlinear boundary-value problem are not global minimizers of the energy in the variational statement of the problem. Both the global minimizer and the local extrema are identified and the results are illustrated for particular values of the material parameters.      

On solving multiobjective bin packing problems using evolutionary particle swarm optimization

The bin packing problem is widely found in applications such as loading of tractor trailer trucks, cargo airplanes and ships, where a balanced load provides better fuel efficiency and safer ride. In these applications, there are often conflicting criteria to be satisfied, i.e., to minimize the bins used and to balance the load of each bin, subject to a number of practical constraints. Unlike existing studies that only consider the issue of minimum bins, a multiobjective two-dimensional mathematical model for bin packing problems with multiple constraints (MOBPP-2D) is formulated in this paper. To solve MOBPP-2D problems, a multiobjective evolutionary particle swarm optimization algorithm (MOEPSO) is proposed. Without the need of combining both objectives into a composite scalar weighting function, MOEPSO incorporates the concept of Pareto’s optimality to evolve a family of solutions along the trade-off surface. Extensive numerical investigations are performed on various test instances, and their performances are compared both quantitatively and statistically with other optimization methods to illustrate the effectiveness and efficiency of MOEPSO in solving multiobjective bin packing problems.     

An analysis of the velocity updating rule of the particle swarm optimization algorithm

The particle swarm optimization algorithm includes three vectors associated with each particle: inertia, personal, and social influence vectors. The personal and social influence vectors are typically multiplied by random diagonal matrices (often referred to as random vectors) resulting in changes in their lengths and directions. This multiplication, in turn, influences the variation of the particles in the swarm. In this paper we examine several issues associated with the multiplication of personal and social influence vectors by such random matrices, these include: (1) Uncontrollable changes in the length and direction of these vectors resulting in delay in convergence or attraction to locations far from quality solutions in some situations (2) Weak direction alternation for the vectors that are aligned closely to coordinate axes resulting in preventing the swarm from further improvement in some situations, and (3) limitation in particle movement to one orthant resulting in premature convergence in some situations. To overcome these issues, we use randomly generated rotation matrices (rather than the random diagonal matrices) in the velocity updating rule of the particle swarm optimizer. This approach makes it possible to control the impact of the random components (i.e. the random matrices) on the direction and length of personal and social influence vectors separately. As a result, all the above mentioned issues are effectively addressed. We propose to use the Euclidean rotation matrices for rotation because it preserves the length of the vectors during rotation, which makes it easier to control the effects of the randomness on the direction and length of vectors. The direction of the Euclidean matrices is generated randomly by a normal distribution. The mean and variance of the distribution are investigated in detail for different algorithms and different numbers of dimensions. Also, an adaptive approach for the variance of the normal distribution is proposed which is independent from the algorithm and the number of dimensions. The method is adjoined to several particle swarm optimization variants. It is tested on 18 standard optimization benchmark functions in 10, 30 and 60 dimensional spaces. Experimental results show that the proposed method can significantly improve the performance of several types of particle swarm optimization algorithms in terms of convergence speed and solution quality.     

An equivariant tensor product on Mackey functors

For all subgroups H of a cyclic p-group G we define norm functors that build a G-Mackey functor from an H-Mackey functor. We give an explicit construction of these functors in terms of generators and relations based solely on the intrinsic, algebraic properties of Mackey functors and Tambara functors. We use these norm functors to define a monoidal structure on the category of Mackey functors where Tambara functors are the commutative ring objects.     

Some remarks on the deterministic particle swarm optimization algorithm

In this paper, we give a simple proof for the convergence of the deterministic particle swarm optimization algorithm under the weak chaotic assumption and remark that the weak chaotic assumption does not relax the stagnation assumption in essence. Under the spectral radius assumption, we propose a convergence criterion for the deterministic particle swarm optimization algorithm in terms of the personal best and neighborhood best position of the particle that incorporates the stagnation assumption or the weak chaotic assumption as a special case.     

An efficient co-swarm particle swarm optimization for non-linear constrained optimization

This paper proposes a new co-swarm PSO (CSHPSO) for constrained optimization problems, which is obtained by hybridizing the recently proposed shrinking hypersphere PSO (SHPSO) with the differential evolution (DE) approach. The total swarm is subdivided into two sub swarms in such a way that the first sub swarms uses SHPSO and second sub swarms uses DE. Experiments are performed on a state-of-the-art problems proposed in IEEE CEC 2006. The results of the CSHPSO is compared with SHPSO and DE in a variety of fashions. A statistical approach is applied to provide the significance of the numerical experiments. In order to further test the efficacy of the proposed CSHPSO, an economic dispatch (ED) problem with valve points effects for 40 generating units is solved. The results of the problem using CSHPSO is compared with SHPSO, DE and the existing solutions in the literature. It is concluded that CSHPSO is able to give the minimal cost for the ED problem in comparison with the other algorithms considered. Hence, CSHPSO is a promising new co-swarm PSO which can be used to solve any real constrained optimization problem.     

Parameter estimation for noisy chaotic systems based on an improved particle swarm optimization algorithm

It is of great importance to estimate the unknown parameters and time delays of chaotic systems in control and synchronization. This paper is concerned with the uncertain parameters and time delays of chaotic systems corrupted with random noise. Parameters and time delays of such chaotic systems are estimated based on the improved particle swarm optimization algorithm for its global searching ability. Numerical simulations are given to show satisfactory results.     

Memetic particle swarm optimization

We propose a new Memetic Particle Swarm Optimization scheme that incorporates local search techniques in the standard Particle Swarm Optimization algorithm, resulting in an efficient and effective optimization method, which is analyzed theoretically. The proposed algorithm is applied to different unconstrained, constrained, minimax and integer programming problems and the obtained results are compared to that of the global and local variants of Particle Swarm Optimization, justifying the superiority of the memetic approach.     

Advanced particle swarm assisted genetic algorithm for constrained optimization problems

A novel hybrid evolutionary algorithm is developed based on the particle swarm optimization (PSO) and genetic algorithms (GAs). The PSO phase involves the enhancement of worst solutions by using the global-local best inertia weight and acceleration coefficients to increase the efficiency. In the genetic algorithm phase, a new rank-based multi-parent crossover is used by modifying the crossover and mutation operators which favors both the local and global exploration simultaneously. In addition, the Euclidean distance-based niching is implemented in the replacement phase of the GA to maintain the population diversity. To avoid the local optimum solutions, the stagnation check is performed and the solution is randomized when needed. The constraints are handled using an effective feasible population based approach. The parameters are self-adaptive requiring no tuning based on the type of problems. Numerical simulations are performed first to evaluate the current algorithm for a set of 24 benchmark constrained nonlinear optimization problems. The results demonstrate reasonable correlation and high quality optimum solutions with significantly less function evaluations against other state-of-the-art heuristic-based optimization algorithms. The algorithm is also applied to various nonlinear engineering optimization problems and shown to be excellent in searching for the global optimal solutions.