Safe bounds for the solutions of nonlinear problems using a parallel multisplitting method

For some systems of nonlinear equationsF(x)=0 we derive an algorithm which iteratively constructs tight lower and upper bounds for the zeros ofF. The algorithm is based on a multisplitting of certain matrices thus showing a natural parallelism. We prove criteria for the convergence of the bounds towards the zeros and we investigate the speed of convergence.     

Nonlinear continuous multi-response problems: a novel two-phase hybrid genetic based metaheuristic

Generally the most real world production systems are tackling several different responses and the problem is optimizing these responses concurrently. This study strives to present a new two-phase hybrid genetic based metaheuristic for optimizing nonlinear continuous multi-response problems. Premature convergence and getting stuck in local optima, which makes the algorithm time consuming, are common problems dealing with genetic algorithms (GAs). So we hybridize GA with a clustering approach and particle swarm optimization algorithm (PSO) to make a balanced relationship between time consuming and premature termination. The proposed algorithm also tries to find Ideal Points (IPs) for response functions. IPs are considered as improvement measures that determine when PSO should start. PSO based local search exploit Pareto archive solutions to enhance performance of the algorithm by expanding the search space. Since there is no standard benchmark in this field, we use two case studies from distinguished paper in multi-response optimization and compare the results with some of the mentioned algorithms in the literature. Results show the outperformance of the proposed algorithm than all of them.     

Adaptive recursive algorithm with logarithmic transformation for nonlinear system identification in α-stable noise

Although the least mean pth power (LMP) and normalized LMP (NLMP) algorithms of adaptive Volterra filters outperform the conventional least mean square (LMS) algorithm in the presence of α-stable noise, they still exhibit slow convergence and high steady-state kernel error in nonlinear system identification. To overcome these limitations, an enhanced recursive least mean pth power algorithm with logarithmic transformation (RLogLMP) is proposed in this paper. The proposed algorithm is adjusted to minimize the new cost function with the p-norm logarithmic transformation of the error signal. The logarithmic transformation, which can diminish the significance of outliers under α-stable noise environment, increases the robustness of the proposed algorithm and reduces the steady-state kernel error. Moreover, the proposed method improves the convergence rate by the enhanced recursive scheme. Finally, simulation results demonstrate that the proposed algorithm is superior to the LMP, NLMP, normalized least mean absolute deviation (NLMAD), recursive least squares (RLS) and nonlinear iteratively reweighted least squares (NIRLS) algorithms in terms of convergence rate and steady-state kernel error.     

<Emphasis Type="Italic">hp</Emphasis>-Adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

This work is concerned with the derivation of an a posteriori error estimator for Galerkin approximations to nonlinear initial value problems with an emphasis on finite-time existence in the context of blow-up. The structure of the derived estimator leads naturally to the development of both h and hp versions of an adaptive algorithm designed to approximate the blow-up time. The adaptive algorithms are then applied in a series of numerical experiments, and the rate of convergence to the blow-up time is investigated.     

A new class of hybrid global optimization algorithms for peptide structure prediction: integrated hybrids

A novel class of hybrid global optimization methods for application to the structure prediction in protein-folding problem is introduced. These optimization methods take the form of a hybrid between a deterministic global optimization algorithm, the αBB, and a stochastically based method, conformational space annealing (CSA), and attempt to combine the beneficial features of these two algorithms. The αBB method as previously extant exhibits consistency, as it guarantees convergence to the global minimum for twice-continuously differentiable constrained nonlinear programming problems, but can benefit from improvements in the computational front. Computational studies for met-enkephalin demonstrate the promise for the proposed hybrid global optimization method.     

An algorithm for solving nonlinear least-squares problems with a new curvilinear search

We propose a modification of an algorithm introduced by Martínez (1987) for solving nonlinear least-squares problems. Like in the previous algorithm, after the calculation of an approximated Gauss-Newton directiond, we obtain the next iterate on a two-dimensional subspace which includesd. However, we simplify the process of searching the new point, and we define the plane using a scaled gradient direction, instead of the original gradient. We prove that the new algorithm has global convergence properties. We present some numerical experiments.     

Type-2 fuzzy sliding mode control without reaching phase for nonlinear system

A new sliding mode control (SMC) algorithm for the nth order nonlinear system suffering from parameters uncertainty and subjected to an external perturbation is proposed. The algorithm employs a time-varying switching plane. At the initial time t=t₀, the plane passes through the point determined by the system initial conditions in the error state space. Afterwards, the plane moves to the origin of the state space. Since the nonlinear system is sensible to the perturbations and uncertainties during the reaching phase, the elimination of such phase yields in a considerable amelioration of system robustness. Switching plane is chosen such that: (1) the reaching phase is eliminated, (2) the nonlinear system is insensitive to the external disturbance and the model uncertainty from the initial time (3) the convergence of the tracking error to zero. Furthermore, a Type-2 fuzzy system is used to approximate system dynamics (assumed to be unknown) and to construct the equivalent controller such that: (1) all signals of closed-loop system are uniformly ultimately bounded, (2) the problems related to adaptive fuzzy controllers like singularity and constraints on the control gain are resolved. To ensure the robustness of the overall closed-loop system, analytical demonstration using Lyapunov theorem is considered. Finally, a robot manipulator is considered as a real time system in order to confirm the efficiency of the proposed approach. The experimentation is done for both regulation and tracking control problems.     

On a nonlinear iterative method in applied mechanics-part 1

A particular iterative method that has proven effective for finite difference solution of nonlinear membrane and plate problems is studied. The iteration is shown to belong to a general class of iterations termed SOR-Newton m_k step iteration and corresponds to the choice m_k = 1. As a result of this characterization we proceed to give the theoretical basis for studying convergence of the iteration. From this standpoint one is better able to evaluate the utility and limitations of the iterative scheme and compare it with alternative competitive schemes for various classes of problems and nonlinear systems in applied mechanics.     

Analysis of the nonlinear Uzawa algorithm for symmetric saddle point problems

In this paper we discuss the convergence behaviour of the nonlinear Uzawa algorithm for solving saddle point problems presented in a recent paper of Cao [Z.H. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math. 46 (2003), pp. 157–171]. For a general case, the results on the convergence of the algorithm are given.     

A New Functional Iterative Algorithm for the Regularized Long-Wave Equation Using an Integral Equation Formalism

A fundamental question in computational nonlinear partial differential equations is raised to discover if one could construct a functional iterative algorithm for the regularized long-wave (RLW) equation (or the Benjamin–Bona–Mahony equation) based on an integral equation formalism? Here, the RLW equation is a third-order nonlinear partial differential equation, describing physically nonlinear dispersive waves in shallow water. For the question, the concept of pseudo-parameter, suggested by Jang (Commun Nonlinear Sci Numer Simul 43:118–138, 2017), is introduced and incorporated into the RLW equation. Thereby, dual nonlinear integral equations of second kind involving the parameter are formulated. The application of the fixed point theorem to the integral equations results in a new (semi-analytic and derivative-free) functional iteration algorithm (as required). The new algorithm allows the exploration of new regimes of pseudo-parameters, so that it can be valid for a much wider range (in the complex plane) of pseudo-parameter values than that of Jang (2017). Being fairly simple (or straightforward), the iteration algorithm is found to be not only stable but accurate. Specifically, a numerical experiment on a solitary wave is performed on the convergence and accuracy of the iteration for various complex values of the pseudo-parameters, further providing the regions of convergence subject to some constraints in the complex plane. Moreover, the algorithm yields a particularly relevant physical investigation of the nonlinear behavior near the front of a slowly varying wave train, in which, indeed, interesting nonlinear wave features are demonstrated. As a consequence, the preceding question may be answered.     

基于改进粒子群算法的BP神经网络及其应用

目前BP神经网络是一种有效的预测方法,但在实际应用当中存在着一些自身的缺点,为此提出了一种基于改进粒子群算法的BP神经网络。通过动态调整粒子群算法中的惯性因子ω,有效地增强了算法对非线性问题的处理能力,同时提高了算法的收敛速度和搜索全局最优值的能力。建立改进后的BP网络模型,通过该模型和逐步回归方法对某市降水量进行实例分析。分析结果表明,改进后的BP网络模型具有较高的准备预报能力和稳定性。     

Orthogonal PSO algorithm for economic dispatch of thermal generating units under various power constraints in smart power grid

Particle swarm optimization (PSO) algorithm has been successfully applied to solve various optimization problems in science and engineering. One such popular one is called global PSO (GPSO) algorithm. One of major drawback of GPSO algorithm is the phenomenon of “zigzagging”, that leads to premature convergence by falling into local minima. In addition, the performance of GPSO algorithm deteriorates for high-dimensional problems, especially in presence of nonlinear constraints. In this paper we propose a novel algorithm called, orthogonal PSO (OPSO) that alleviates the shortcomings of the GPSO algorithm. In OPSO algorithm, the m particles of the swarm are divided into two groups: active group and passive group. The d particles of the active group undergo an orthogonal diagonalization process and are updated in such way that their position vectors become orthogonally diagonalized. In the OPSO algorithm, the particles are updated using only one guide, thus avoiding the conflict between the two guides that occurs in the GPSO algorithm. We applied the OPSO algorithm for solving economic dispatch (ED) problem by taking three power systems under several power constraints imposed by thermal generating units (TGUs) and smart power grid (SPG), for example, ramp rate limits, and prohibited operating zones. In addition, the OPSO algorithm is also applied for ten selected shifted and rotated CEC 2015 benchmark functions. With extensive simulation studies, we have shown superior performance of OPSO algorithm over GPSO algorithm and several existing evolutional computation techniques in terms of several performance measures, e.g., minimum cost, convergence rate, consistency, and stability. In addition, using unpaired t-Test, we have shown the statistical significance of the OPSO algorithm against several contending algorithms including top-ranked CEC 2015 algorithms.     

A Quasi-Newton method with modification of one column per iteration

In this paper we introduce a new Quasi-Newton method for solving nonlinear simultaneous equations. At each iteration only one column ofB _k is changed to obtainB _k+1 . This permits to use the well-known techniques of Linear Programming for modifying the factorization ofB _k . We present a local convergence theorem for a restarted version of the method. The new algorithm is compared numerically with some other methods which were introduced for solving the same kind of problems.     

Nonlinear system fault diagnosis based on adaptive estimation

An approach to fault diagnosis for a class of nonlinear systems is proposed in this paper. It is based on a new adaptive estimation algorithm for recursive estimation of the parameters related to faults. This algorithm is designed in a constructive manner through a nontrivial combination of a high gain observer and a recently developed linear adaptive observer, without resort to any linearization. Its global exponential convergence is ensured by an easy-to-check persistent excitation condition. A numerical example is presented for illustration.     

A Non-monotone Line Search Algorithm for?Unconstrained Optimization

The monotone line search schemes have been extensively used in the iterative methods for solving various optimization problems. It is well known that the non-monotone line search technique can improve the likelihood of finding a global optimal solution and the numerical performance of the methods, especially for some difficult nonlinear problems. The traditional non-monotone line search approach requires that a maximum of recent function values decreases. In this paper, we propose a new line search scheme which requires that a convex combination of recent function values decreases. We apply the new line search technique to solve unconstrained optimization problems, and show the proposed algorithm possesses global convergence and R-linear convergence under suitable assumptions. We also report the numerical results of the proposed algorithm for solving almost all the unconstrained testing problems given in CUTEr, and give numerical comparisons of the proposed algorithm with two famous non-monotone methods.     

An efficient artificial bee colony algorithm with application to nonlinear predictive control

In this paper a constrained nonlinear predictive control algorithm, that uses the artificial bee colony (ABC) algorithm to solve the optimization problem, is proposed. The main objective is to derive a simple and efficient control algorithm that can solve the nonlinear constrained optimization problem with minimal computational time. Indeed, a modified version, enhancing the exploring and the exploitation capabilities, of the ABC algorithm is proposed and used to design a nonlinear constrained predictive controller. This version allows addressing the premature and the slow convergence drawbacks of the standard ABC algorithm, using a modified search equation, a well-known organized distribution mechanism for the initial population and a new equation for the limit parameter. A convergence statistical analysis of the proposed algorithm, using some well-known benchmark functions is presented and compared with several other variants of the ABC algorithm. To demonstrate the efficiency of the proposed algorithm in solving engineering problems, the constrained nonlinear predictive control of the model of a Multi-Input Multi-Output industrial boiler is considered. The control performances of the proposed ABC algorithm-based controller are also compared to those obtained using some variants of the ABC algorithms.     

A many-objective evolutionary algorithm based on a projection-assisted intra-family election

In recent years, many researchers have put emphasis on the study of how to keep a good balance between convergence and diversity in many-objective optimization. This paper proposes a new many-objective evolutionary algorithm based on a projection-assisted intra-family election. In the proposed algorithm, basic evolution directions are adaptively generated according to the current population and potential evolution directions are excavated in each individual's family. Based on these evolution directions, a strategy of intra-family election is performed in every family and elite individuals are elected as representatives of the specific family to join the next stage, which can enhance the convergence of the algorithm. Moreover, a selection procedure based on angles is used to maintain the diversity. The performance of the proposed algorithm is verified and compared with several state-of-the-art many-objective evolutionary algorithms on a variety of well-known benchmark problems ranging from 5 to 20 objectives. Empirical results demonstrate that the proposed algorithm outperforms other peer algorithms in terms of both the diversity and the convergence of the final solutions set on most of the test instances. In particular, our proposed algorithm shows obvious superiority when handling the problems with larger number of objectives.     

On the computation of solution spaces in high dimensions

A stochastic algorithm that computes box-shaped solution spaces for nonlinear, high-dimensional and noisy problems with uncertain input parameters has been proposed in Zimmermann and von Hoessle (Int J Numer Methods Eng 94(3):290–307, 2013). This paper studies in detail the quality of the results and the efficiency of the algorithm. Appropriate benchmark problems are specified and compared with exact solutions that were derived analytically. The speed of convergence decreases as the number of dimensions increases. Relevant mechanisms are identified that explain how the number of dimensions affects the performance. The optimal number of sample points per iteration is determined in dependence of the preference for fast convergence or a large volume.     

Accelerating the quadratic lower-bound algorithm via optimizing the shrinkage parameter

When the Newton-Raphson algorithm or the Fisher scoring algorithm does not work and the EM-type algorithms are not available, the quadratic lower-bound (QLB) algorithm may be a useful optimization tool. However, like all EM-type algorithms, the QLB algorithm may also suffer from slow convergence which can be viewed as the cost for having the ascent property. This paper proposes a novel ‘shrinkage parameter’ approach to accelerate the QLB algorithm while maintaining its simplicity and stability (i.e., monotonic increase in log-likelihood). The strategy is first to construct a class of quadratic surrogate functions Q_r(θ|θ^(t)) that induces a class of QLB algorithms indexed by a ‘shrinkage parameter’ r (r∈R) and then to optimize r over R under some criterion of convergence. For three commonly used criteria (i.e., the smallest eigenvalue, the trace and the determinant), we derive a uniformly optimal shrinkage parameter and find an optimal QLB algorithm. Some theoretical justifications are also presented. Next, we generalize the optimal QLB algorithm to problems with penalizing function and then investigate the associated properties of convergence. The optimal QLB algorithm is applied to fit a logistic regression model and a Cox proportional hazards model. Two real datasets are analyzed to illustrate the proposed methods.     

Convergence analysis of a self-adaptive multi-objective evolutionary algorithm based on grids

Evolutionary algorithms have been successfully applied to various multi-objective optimization problems. However, theoretical studies on multi-objective evolutionary algorithms, especially with self-adaption, are relatively scarce. This paper analyzes the convergence properties of a self-adaptive (μ+1)-algorithm. The convergence of the algorithm is defined, and general convergence conditions are studied. Under these conditions, it is proven that the proposed self-adaptive (μ+1)-algorithm converges in probability or almost surely to the Pareto-optimal front.