An accurate a posteriori error estimator for the Steklov eigenvalue problem and its applications

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine mesh and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.

     

A method for generating a well-distributed Pareto set in nonlinear multiobjective optimization

A method is presented for generating a well-distributed Pareto set in nonlinear multiobjective optimization. The approach shares conceptual similarity with the Physical Programming-based method, the Normal-Boundary Intersection and the Normal Constraint methods, in its systematic approach investigating the objective space in order to obtain a well-distributed Pareto set. The proposed approach is based on the generalization of the class functions which allows the orientation of the search domain to be conducted in the objective space. It is shown that the proposed modification allows the method to generate an even representation of the entire Pareto surface. The generation is performed for both convex and nonconvex Pareto frontiers. A simple algorithm has been proposed to remove local Pareto solutions. The suggested approach has been verified by several test cases, including the generation of both convex and concave Pareto frontiers.     

Steklov eigenvalue problem and its applications An accurate a posteriori error estimator for the

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine meshes and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.     

A Robust Identification Technique for Time-Varying ARMA Processes Based on Variable Structure Systems Theory

In this paper, a novel method for extracting the values of the coefficients of time-varying ARMA processes is proposed. The approach discussed assumes solely that the orders of the numerator and the denominator polynomials are known. The algorithm is demonstrated to be stable in the sense of Lyapunov, furthermore, it is shown in the paper that the evolution in the parameter space takes place in a finite volume. The proposed method is cost effective and is based on the variable structure systems theory, which is well known with its robustness to uncertainties. In the simulation example, the coefficients of a second order ARMA process is extracted by the use of the algorithm presented. The results confirm the prominent features of the proposed technique.     

A new mechanism to maintain diversity in multi-objective metaheuristics

In this article, a new mechanism to spread the solutions generated by a multi-objective evolutionary algorithm is proposed. This approach is based on the use of stripes that are applied in objective function space and is independent of the search engine adopted. Additionally, it overcomes some of the drawbacks of other previous proposals such as the ?-dominance method. In order to validate the proposed approach, it is coupled to a multi-objective particle swarm optimizer and its performance is assessed with respect to that of state-of-the-art algorithms, using standard test problems and performance measures taken from the specialized literature. The results indicate that the proposed approach is a viable diversity maintenance mechanism that can be incorporated to any multi-objective metaheuristic used for multi-objective optimization.     

A parametric solution algorithm for a class of rank-two nonconvex programs

The aim of this paper is to propose a solution algorithm for a particular class of rank-two nonconvex programs having a polyhedral feasible region. The algorithm lies within the class of the so called “optimal level solutions” parametric methods. The subproblems obtained by means of this parametrical approach are quadratic convex ones, but not necessarily neither strictly convex nor linear. For this very reason, in order to solve in an unifying framework all of the considered rank-two nonconvex programs a new approach needs to be proposed. The efficiency of the algorithm is improved by means of the use of underestimation functions. The results of a computational test are provided and discussed.     

An Algorithm for Solving Optimization Problems with One Linear Objective Function and Finitely Many Constraints of Fuzzy Relation Inequalities

An optimization model with one linear objective function and fuzzy relation equation constraints was presented by Fang and Li (1999) as well as an efficient solution procedure was designed by them for solving such a problem. A more general case of the problem, an optimization model with one linear objective function and finitely many constraints of fuzzy relation inequalities, is investigated in this paper. A new approach for solving this problem is proposed based on a necessary condition of optimality given in the paper. Compared with the known methods, the proposed algorithm shrinks the searching region and hence obtains an optimal solution fast. For some special cases, the proposed algorithm reaches an optimal solution very fast since there is only one minimum solution in the shrunk searching region. At the end of the paper, two numerical examples are given to illustrate this difference between the proposed algorithm and the known ones.     

Genetic learning of accurate and compact fuzzy rule based systems based on the 2-tuples linguistic representation

One of the problems that focus the research in the linguistic fuzzy modeling area is the trade-off between interpretability and accuracy. To deal with this problem, different approaches can be found in the literature. Recently, a new linguistic rule representation model was presented to perform a genetic lateral tuning of membership functions. It is based on the linguistic 2-tuples representation that allows the lateral displacement of a label considering an unique parameter. This way to work involves a reduction of the search space that eases the derivation of optimal models and therefore, improves the mentioned trade-off.Based on the 2-tuples rule representation, this work proposes a new method to obtain linguistic fuzzy systems by means of an evolutionary learning of the data base a priori (number of labels and lateral displacements) and a simple rule generation method to quickly learn the associated rule base. Since this rule generation method is run from each data base definition generated by the evolutionary algorithm, its selection is an important aspect. In this work, we also propose two new ad hoc data-driven rule generation methods, analyzing the influence of them and other rule generation methods in the proposed learning approach. The developed algorithms will be tested considering two different real-world problems.     

Mobile Robot Global Localization using an Evolutionary MAP Filter

A new algorithm based on evolutionary computation concepts is presented in this paper. This algorithm is a non linear evolutive filter known as the Evolutive Localization Filter (ELF) which is able to solve the global localization problem in a robust and efficient way. The proposed algorithm searches stochastically along the state space for the best robot pose estimate. The set of pose solutions (the population) represents the most likely areas according to the perception and motion information up to date. The population evolves by using the log-likelihood of each candidate pose according to the observation and the motion error derived from the comparison between observed and predicted data obtained from the probabilistic perception and motion model. The algorithm has been tested on a mobile robot equipped with a laser range finder to demonstrate the effectiveness, robustness and computational efficiency of the proposed approach.     

Pareto Analysis vis-à-vis Balance Space Approach in Multiobjective Global Optimization

There is much controversy about the balance space approach, introduced first in Ref. 1, pp. 138–140, with the consideration of the balance number and balance vectors, and then further developed in Ref. 2, with the consideration of balance points and balance sets. There were attempts to identify the balance space approach with some other methods of multiobjective optimization, notably the method proposed in Ref. 3 and most recently Pareto analysis, as presented in Ref. 4. In this paper, we compare Pareto analysis with the balance space approach on several examples to demonstrate the interrelation and the differences of the two methods. As a byproduct, it is shown that, in some cases, the entire Pareto sets, proper and adjoint, can be determined very simply, without any special investigation of the (nonscalarized, nonconvex) multiobjective global optimization problem. The method of parameter introduction is presented in application to determining the Pareto sets and balance set. The use of computer graphics software complemented with the Gauss–Jordan matrix reduction algorithm is proposed for a class of otherwise intractable problems with nonconvex constraint sets.     

A branch and price algorithm for the minimum power multicasting problem in wireless sensor networks

The Minimum Power Multicast Problem arises in wireless sensor networks and consists in assigning a transmission power to each node of a network in such a way that the total power consumption over the network is minimized, while a source node is connected to a set of destination nodes, toward which a message has to be sent periodically. A new mixed integer programming model for the problem, based on paths, is presented. A practical exact algorithm based on column generation and branch and price is derived from this model. A comparison with state-of-the-art exact methods is presented, and it is shown that the new approach compares favorably to other algorithms when the number of destination nodes is moderate. Under this condition, the proposed method is able to solve previously unmanageable instances.     

Solving the sum-of-ratios problem by a stochastic search algorithm

In spite of the recent progress in fractional programming, the sum-of-ratios problem remains untoward. Freund and Jarre proved that this is an NP-complete problem. Most methods overcome the difficulty using the deterministic type of algorithms, particularly, the branch-and-bound method. In this paper, we propose a new approach by applying the stochastic search algorithm introduced by Birbil, Fang and Sheu to a transformed image space. The algorithm then computes and moves sample particles in the q − 1 dimensional image space according to randomly controlled interacting electromagnetic forces. Numerical experiments on problems up to sum of eight linear ratios with a thousand variables are reported. The results also show that solving the sum-of-ratios problem in the image space as proposed is, in general, preferable to solving it directly in the primal domain.     

Transgenetic algorithm for the Traveling Purchaser Problem

In this paper an evolutionary algorithm is presented for the Traveling Purchaser Problem, an important variation of the Traveling Salesman Problem. The evolutionary approach proposed in this paper is called transgenetic algorithm. It is inspired on two significant evolutionary driving forces: horizontal gene transfer and endosymbiosis. The performance of the algorithm proposed for the investigated problem is compared with other recent works presented in the literature. Computational experiments show that the proposed approach is very effective for the investigated problem with 17 and 9 new best solutions reported for capacitated and uncapacitated instances, respectively.     

A new algorithm for variance based importance analysis of models with correlated inputs

Importance analysis is aimed at finding the contributions of the inputs to the output uncertainty. For structural models involving correlated input variables, the variance contribution by an individual input variable is decomposed into correlated contribution and uncorrelated contribution in this study. Based on point estimate, this work proposes a new algorithm to conduct variance based importance analysis for correlated input variables. Transformation of the input variables from correlation space to independence space and the computation of conditional distribution in the process ensure that the correlation information is inherited correctly. Different point estimate methods can be employed in the proposed algorithm, thus the algorithm is adaptable and evolvable. Meanwhile, the proposed algorithm is also applicable to uncertainty systems with multiple modes. The proposed algorithm avoids the sampling procedure, which usually consumes a heavy computational cost. Results of several examples in this work have proven the proposed algorithm can be used as an effective tool to deal with uncertainty analysis involving correlated inputs.     

A multi-swarm PSO using charged particles in a partitioned search space for continuous optimization

Particle swarm optimization (PSO) is characterized by a fast convergence, which can lead the algorithms of this class to stagnate in local optima. In this paper, a variant of the standard PSO algorithm is presented, called PSO-2S, based on several initializations in different zones of the search space, using charged particles. This algorithm uses two kinds of swarms, a main one that gathers the best particles of auxiliary ones, initialized several times. The auxiliary swarms are initialized in different areas, then an electrostatic repulsion heuristic is applied in each area to increase its diversity. We analyse the performance of the proposed approach on a testbed made of unimodal and multimodal test functions with and without coordinate rotation and shift. The Lennard-Jones potential problem is also used. The proposed algorithm is compared to several other PSO algorithms on this benchmark. The obtained results show the efficiency of the proposed algorithm.     

Galerkin and weighted Galerkin methods for a forward-backward heat equation

Summary. Galerkin and weighted Galerkin methods are proposed for the numerical solution of parabolic partial differential equations where the diffusion coefficient takes different signs. The approach is based on a simultaneous discretization of space and time variables by using continuous finite element methods. Under some simple assumptions, error estimates and some numerical results for both Galerkin and weighted Galerkin methods are presented. Comparisons with the previous methods show that new methods not only can be used to solve a wider class of equations but also require less regularity for the solution and need fewer computations. Received March 3, 1995     

Handling equality constraints in evolutionary optimization

Over the last few decades several methods have been proposed for handling functional constraints while solving optimization problems using evolutionary algorithms (EAs). However, the presence of equality constraints makes the feasible space very small compared to the entire search space. As a consequence, the handling of equality constraints has long been a difficult issue for evolutionary optimization methods. This paper presents a Hybrid Evolutionary Algorithm (HEA) for solving optimization problems with both equality and inequality constraints. In HEA, we propose a new local search technique with special emphasis on equality constraints. The basic concept of the new technique is to reach a point on the equality constraint from the current position of an individual solution, and then explore on the constraint landscape. We believe this new concept will influence the future research direction for constrained optimization using population based algorithms. The proposed algorithm is tested on a set of standard benchmark problems. The results show that the proposed technique works very well on those benchmark problems.     

基于时域精细积分算法的瞬态传热多宗量反演

基于有限元法和精细积分算法,提出了一种求解瞬态热传导多宗量反演问题的新方法．采用有限元法和精细积分算法分别对空间、时间变量进行离散,可以得到正演问题高精度的半解析数值模型,由此建立了多宗量反演的计算模式,并给出敏度分析的计算公式．对一维和二维的热物性参数、热源项、边界条件等进行了单宗量和多宗量的反演求解,初步考虑了初值和噪音等对反演结果的影响,数值算例验证了该方法的有效性．     

Generalization errors of Laplacian regularized least squares regression

Semi-supervised learning is an emerging computational paradigm for machine learning,that aims to make better use of large amounts of inexpensive unlabeled data to improve the learning performance.While various methods have been proposed based on different intuitions,the crucial issue of generalization performance is still poorly understood.In this paper,we investigate the convergence property of the Laplacian regularized least squares regression,a semi-supervised learning algorithm based on manifold regularization.Moreover,the improvement of error bounds in terms of the number of labeled and unlabeled data is presented for the first time as far as we know.The convergence rate depends on the approximation property and the capacity of the reproducing kernel Hilbert space measured by covering numbers.Some new techniques are exploited for the analysis since an extra regularizer is introduced.     

New subgradient extragradient methods for solving monotone bilevel equilibrium problems

ABSTRACT

In this paper, we propose new subgradient extragradient methods for finding a solution of a strongly monotone equilibrium problem over the solution set of another monotone equilibrium problem which usually is called monotone bilevel equilibrium problem in Hilbert spaces. The first proposed algorithm is based on the subgradient extragradient method presented by Censor et al. [Censor Y, Gibali A, Reich S. The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl. 2011;148:318–335]. The strong convergence of the algorithm is established under monotone assumptions of the cost bifunctions with Lipschitz-type continuous conditions recently presented by Mastroeni in the auxiliary problem principle. We also present a modification of the algorithm for solving an equilibrium problem, where the constraint domain is the common solution set of another equilibrium problem and a fixed point problem. Several fundamental experiments are provided to illustrate the numerical behaviour of the algorithms and to compare with others.