Search

In this paper, we present several remarks on the paper by Yao et al. (citeyearcite.nine). The results presented in the present paper are interesting improvements on the main results of Yao et al. [ABSTRACT FROM AUTHOR]

The controlled autoregressive autoregressive moving average (CARARMA) models are of popularity to describe the evolution characteristics of dynamical systems. To overcome the identification obstacle resulting from colored noises, this paper studies the identification of the CARARMA models by forming an intermediate correlated noise model. In order to realize the real-time prediction function of the models, the on-line identification scheme is developed by constructing the dynamical objective functions based on the real-time sampled observations. Firstly, a rolling optimization cost function is built based on the observation at a single sampling instant to catch the modal information at a single time point and a generalized extended stochastic gradient (GESG) algorithm is proposed through the stochastic gradient optimization. Secondly, a rolling window cost function is built in accordance with the dynamical batch observations within data window by extending the proposed GESG algorithm and the multi-innovation generalized extended stochastic gradient algorithm is derived. Thirdly, from the perspective of theoretical analysis, the convergence proof of the proposed algorithm is provided based on the stochastic martingale convergence theory. Finally, the simulation analysis and comparison studies are provided to show the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

We revisit two old and apparently little known papers by Basuev (Teoret Mat Fiz 37(1):130-134, , Teoret Mat Fiz 39(1):94-105, ) and show that the results contained there yield strong improvements on current lower bounds of the convergence radius of the Mayer series for continuous particle systems interacting via a very large class of stable and tempered potentials, which includes the Lennard-Jones type potentials. In particular we analyze the case of the classical Lennard-Jones gas under the light of the Basuev scheme and, using also some new results (Yuhjtman in J Stat Phys 160(6): 1684-1695, ) on this model recently obtained by one of us, we provide a new lower bound for the Mayer series convergence radius of the classical Lennard-Jones gas, which improves by a factor of the order 10 on the current best lower bound recently obtained in de Lima and Procacci (J Stat Phys 157(3):422-435, ). [ABSTRACT FROM AUTHOR]

In this paper, we study the almost sure boundedness and the convergence of the stochastic approximation (SA) algorithm. At present, most available convergence proofs are based on the ODE method, and the almost sure boundedness of the iterations is an assumption and not a conclusion. In Borkar and Meyn (SIAM J Control Optim 38:447–469, 2000), it is shown that if the ODE has only one globally attractive equilibrium, then under additional assumptions, the iterations are bounded almost surely, and the SA algorithm converges to the desired solution. Our objective in the present paper is to provide an alternate proof of the above, based on martingale methods, which are simpler and less technical than those based on the ODE method. As a prelude, we prove a new sufficient condition for the global asymptotic stability of an ODE. Next we prove a "converse" Lyapunov theorem on the existence of a suitable Lyapunov function with a globally bounded Hessian, for a globally exponentially stable system. Both theorems are of independent interest to researchers in stability theory. Then, using these results, we provide sufficient conditions for the almost sure boundedness and the convergence of the SA algorithm. We show through examples that our theory covers some situations that are not covered by currently known results, specifically Borkar and Meyn (2000). [ABSTRACT FROM AUTHOR]

Stochastic alternating algorithms for bi-objective optimization are considered when optimizing two conflicting functions for which optimization steps have to be applied separately for each function. Such algorithms consist of applying a certain number of steps of gradient or subgradient descent on each single objective at each iteration. In this paper, we show that stochastic alternating algorithms achieve a sublinear convergence rate of O (1 / T) , under strong convexity, for the determination of a minimizer of a weighted-sum of the two functions, parameterized by the number of steps applied on each of them. An extension to the convex case is presented for which the rate weakens to O (1 / T) . These rates are valid also in the non-smooth case. Importantly, by varying the proportion of steps applied to each function, one can determine an approximation to the Pareto front. [ABSTRACT FROM AUTHOR]

This paper is concerned with convergence of stochastic gradient algorithms with momentum terms in the nonconvex setting. A class of stochastic momentum methods, including stochastic gradient descent, heavy ball and Nesterov's accelerated gradient, is analyzed in a general framework under mild assumptions. Based on the convergence result of expected gradients, the authors prove the almost sure convergence by a detailed discussion of the effects of momentum and the number of upcrossings. It is worth noting that there are not additional restrictions imposed on the objective function and stepsize. Another improvement over previous results is that the existing Lipschitz condition of the gradient is relaxed into the condition of Hölder continuity. As a byproduct, the authors apply a localization procedure to extend the results to stochastic stepsizes. [ABSTRACT FROM AUTHOR]

This paper aims to examine convergence of income inequality in 21 OECD countries using several empirical techniques. In particular, we have used a new panel stationarity test, which allows for structural changes and cross-sectional dependence to examine the stochastic convergence of income inequality. We also employed a time series approach, residual augmented least squares-Lagrange multiplier unit root test. The empirical results show evidence for absolute, conditional, and sigma convergence. The conditional convergence test results suggest that countries are converging, but conditional on the two structural factors-economic and population growth. The stochastic convergence test results indicate the existence of convergence at the country-specific level. The results further confirm the existence of convergent clubs among OECD countries. [ABSTRACT FROM AUTHOR]

In this paper, we propose a high-order nonlinear algorithm based on a finite difference method modification to the regularized long wave equation and the Benjamin–Bona–Mahony–Burgers equation subject to the homogeneous boundary. The consequence system of nonlinear equations typically trades with high computation burden. This dilemma can be overcome by establishing a fast numerical algorithm procedure without a reduction of numerical accuracy. The proposed algorithm forms a linear system with constant coefficient matrix at each time step and produces numerical solutions, which remarkably gains many computational advantages. In terms of analysis, a priori estimation for the numerical solution is derived to obtain the convergence and stability analysis. Additionally, the algorithm is globally mass preserving to avoid nonphysical behavior. Two benchmarks, including a single solitary wave to both equations, are given to validate the applicability and accuracy of the proposed method. Numerical results are obtained and compared to other approaches available in the literature. From the comparisons it is clear that the proposed approach produces accurate and precise results at low computational cost. Besides, the proposed scheme is applied to study the effect of the viscous term on a single solitary wave. It is shown that the viscous term results in the amplitude and width of the initial condition but not in its velocities in the case of a single solitary wave. As a consequence, theoretical and numerical findings provide a new area to investigate and expand the high-order algorithm for the family of wave equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider stochastic composite convex optimization problems with the objective function satisfying a stochastic bounded gradient condition, with or without a quadratic functional growth property. These models include the most well-known classes of objective functions analyzed in the literature: nonsmooth Lipschitz functions and composition of a (potentially) nonsmooth function and a smooth function, with or without strong convexity. Based on the flexibility offered by our optimization model, we consider several variants of stochastic first-order methods, such as the stochastic proximal gradient and the stochastic proximal point algorithms. Usually, the convergence theory for these methods has been derived for simple stochastic optimization models satisfying restrictive assumptions, and the rates are in general sublinear and hold only for specific decreasing stepsizes. Hence, we analyze the convergence rates of stochastic first-order methods with constant or variable stepsize under general assumptions covering a large class of objective functions. For constant stepsize, we show that these methods can achieve linear convergence rate up to a constant proportional to the stepsize and under some strong stochastic bounded gradient condition even pure linear convergence. Moreover, when a variable stepsize is chosen we derive sublinear convergence rates for these stochastic first-order methods. Finally, the stochastic gradient mapping and the Moreau smoothing mapping introduced in the present paper lead to simple and intuitive proofs. [ABSTRACT FROM AUTHOR]

Traditional classification algorithms work well on general small-scale microarray datasets, but for large-scale scenarios, general machines are not capable of supporting the operation of these algorithms anymore for the memory and time costs. In this paper, we design a new application framework to perform the computation of at the fastest speed. First, the synthetic minority over-sampling technique is used to sample a few classes of sample for obtaining the balanced data. Then, a large-scale algorithm for L 2 -SVM based on the stochastic gradient descent method is proposed and used for microarray classification. Also, We give a simple proof of the convergence of stochastic gradient descent algorithm. Next, various large-scale algorithms for support vector machines are performed on the microarray datasets to identify the most appropriate algorithm. Finally, a comparative analysis of loss functions is done to clearly understand the differences. The experimental results show that the stochastic gradient descent algorithm and the squared hinge loss is an attractive choice, which can achieve high accuracy in seconds. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical approach for solving composite primal-dual monotone inclusions with a priori information. The underlying a priori information set is represented by the intersection of fixed point sets of a finite number of operators, and we propose an algorithm that activates the corresponding set by following a finite-valued random variable at each iteration. Our formulation is flexible and includes, for instance, deterministic and Bernoulli activations over cyclic schemes, and Kaczmarz-type random activations. The almost sure convergence of the algorithm is obtained by means of properties of stochastic Quasi-Fejér sequences. We also recover several primal-dual algorithms for monotone inclusions without a priori information and classical algorithms for solving convex feasibility problems and linear systems. In the context of convex optimization with inequality constraints, any selection of the constraints defines the a priori information set, in which case the operators involved are simply projections onto half spaces. By incorporating random projections onto a selection of the constraints to classical primal-dual schemes, we obtain faster algorithms as we illustrate by means of a numerical application to a stochastic arc capacity expansion problem in a transport network. [ABSTRACT FROM AUTHOR]

In the paper, an averaging principle problem of stochastic McKean-Vlasov equations with slow and fast time-scale is considered. Firstly, existence and uniqueness of the strong solutions of stochastic McKean-Vlasov equations with two time-scale is proved by using the Picard iteration. Secondly, we show that there exists an exponential convergence to the invariant measure for solutions of the fast equation of stochastic McKean-Vlasov equations with slow and fast time-scale. Finally, strong averaging principle for two-time-scale stochastic McKean-Vlasov equations is investigated. [ABSTRACT FROM AUTHOR]

The stochastic Frank–Wolfe method has recently attracted much general interest in the context of optimization for statistical and machine learning due to its ability to work with a more general feasible region. However, there has been a complexity gap in the dependence on the optimality tolerance ε in the guaranteed convergence rate for stochastic Frank–Wolfe compared to its deterministic counterpart. In this work, we present a new generalized stochastic Frank–Wolfe method which closes this gap for the class of structured optimization problems encountered in statistical and machine learning characterized by empirical loss minimization with a certain type of "linear prediction" property (formally defined in the paper), which is typically present in loss minimization problems in practice. Our method also introduces the notion of a "substitute gradient" that is a not-necessarily-unbiased estimate of the gradient. We show that our new method is equivalent to a particular randomized coordinate mirror descent algorithm applied to the dual problem, which in turn provides a new interpretation of randomized dual coordinate descent in the primal space. Also, in the special case of a strongly convex regularizer our generalized stochastic Frank–Wolfe method (as well as the randomized dual coordinate descent method) exhibits linear convergence. Furthermore, we present computational experiments that indicate that our method outperforms other stochastic Frank–Wolfe methods for a sufficiently small optimality tolerance, consistent with the theory developed herein. [ABSTRACT FROM AUTHOR]

Stochastic variational inequalities (SVI) provide a unified framework for the study of a general class of nonlinear optimization and Nash-type equilibrium problems with uncertain model data. Often the true solution to an SVI cannot be found directly and must be approximated. This paper considers the use of a sample average approximation (SAA), and proposes a new method to compute confidence intervals for individual components of the true SVI solution based on the asymptotic distribution of SAA solutions. We estimate the asymptotic distribution based on one SAA solution instead of generating multiple SAA solutions, and can handle inequality constraints without requiring the strict complementarity condition in the standard nonlinear programming setting. The method in this paper uses the confidence regions to guide the selection of a single piece of a piecewise linear function that governs the asymptotic distribution of SAA solutions, and does not rely on convergence rates of the SAA solutions in probability. It also provides options to control the computation procedure and investigate effects of certain key estimates on the intervals. [ABSTRACT FROM AUTHOR]

We propose in this paper a unifying scheme for several algorithms from the literature dedicated to the solving of monotone inclusion problems involving compositions with linear continuous operators in infinite dimensional Hilbert spaces. We show that a number of primal-dual algorithms for monotone inclusions and also the classical ADMM numerical scheme for convex optimization problems, along with some of its variants, can be embedded in this unifying scheme. While in the first part of the paper, convergence results for the iterates are reported, the second part is devoted to the derivation of convergence rates obtained by combining variable metric techniques with strategies based on suitable choice of dynamical step sizes. The numerical performances, which can be obtained for different dynamical step size strategies, are compared in the context of solving an image denoising problem. [ABSTRACT FROM AUTHOR]

In this paper, we trace back the genesis of Aitken's Δ2 process and Shanks' sequence transformation. These methods, which are extrapolation methods, are used for accelerating the convergence of sequences of scalars, vectors, matrices, and tensors. They had, and still have, many important applications in numerical analysis and in applied mathematics. They are related to continued fractions and Padé approximants. We go back to the roots of these methods and analyze the original contributions. New and detailed explanations on the building and properties of Shanks' transformation and its kernel are provided. We then review their historical algebraic and algorithmic developments. We also analyze how they were involved in the solution of systems of linear and nonlinear equations, in particular in the methods of Steffensen, Pulay, and Anderson. Testimonies by various actors of the domain are given. The paper can also serve as an introduction to this domain of numerical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we establish the convergence of the proximal alternating direction method of multipliers (ADMM) and block coordinate descent (BCD) method for nonseparable minimization models with quadratic coupling terms. The novel convergence results presented in this paper answer several open questions that have been the subject of considerable discussion. We firstly extend the 2-block proximal ADMM to linearly constrained convex optimization with a coupled quadratic objective function, an area where theoretical understanding is currently lacking, and prove that the sequence generated by the proximal ADMM converges in point-wise manner to a primal-dual solution pair. Moreover, we apply randomly permuted ADMM (RPADMM) to nonseparable multi-block convex optimization, and prove its expected convergence for a class of nonseparable quadratic programming problems. When the linear constraint vanishes, the 2-block proximal ADMM and RPADMM reduce to the 2-block cyclic proximal BCD method and randomly permuted BCD (RPBCD). Our study provides the first iterate convergence result for 2-block cyclic proximal BCD without assuming the boundedness of the iterates. We also theoretically establish the expected iterate convergence result concerning multi-block RPBCD for convex quadratic optimization. In addition, we demonstrate that RPBCD may have a worse convergence rate than cyclic proximal BCD for 2-block convex quadratic minimization problems. Although the results on RPADMM and RPBCD are restricted to quadratic minimization models, they provide some interesting insights: (1) random permutation makes ADMM and BCD more robust for multi-block convex minimization problems; (2) cyclic BCD may outperform RPBCD for "nice" problems, and RPBCD should be applied with caution when solving general convex optimization problems especially with a few blocks. [ABSTRACT FROM AUTHOR]

Abstract: Since the invention by Peaceman and Rachford, more than 60 years ago, of the well celebrated ADI finite difference scheme for parabolic initial-boundary problems on rectangular regions, many papers have been concerned with prescribing the boundary values for the intermediate approximations at half time levels in the case of nonzero Dirichlet boundary conditions. In the present paper, for variable coefficient parabolic problems and time-stepsize sufficiently small, we prove second order accuracy in the discrete norm of the ADI finite difference scheme in which the intermediate approximations do not involve the so called "perturbation term". As a byproduct of our stability analysis we also show that, for variable coefficients and time-stepsize sufficiently small, the ADI scheme with the perturbation term converges with order two in the discrete norm. Our convergence results generalize previous results obtained for the heat equation. [ABSTRACT FROM AUTHOR]

Genetic algorithm (GA) is a random search algorithm, which has been commonly used to solve optimization problems. A new encoding method of GA, adaptive degressive ary number encoding, is proposed in this paper. This paper firstly introduces the N-ary encoding genetic algorithm and then defines the degressive ary number encoded genetic algorithm. Based on degressive ary number encoded genetic algorithm, this paper proposes a feasible adaptive change rule of the ary number encoding, which can change ary number with the fitness function’s value. All the parameters are selected according to the parameters’ experiments. The proposed algorithm is used to solve the function optimization problems to test the performance. The performances of the proposed algorithm are compared with GA, some classic algorithms and some latest algorithms. The experiments show that the improved adaptive degressive ary number encoded genetic algorithm has a better searching ability and a faster rate of convergence. [ABSTRACT FROM AUTHOR]

In this paper we discuss the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations driven by space-time white noise on T 2 . First we prove that the convergence rate for stochastic 2D heat equation is of order α — δ in Besov space C − α for α ∈ (0,1) and δ > 0 arbitrarily small. Then we obtain the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations of order α — δ in C − α for α ∈ (0,2/9) and δ > 0 arbitrarily small. [ABSTRACT FROM AUTHOR]

The Alternating Direction Multipliers Method (ADMM) is a very popular algorithm for computing the solution of convex constrained minimization problems. Such problems are important from the application point of view, since they occur in many fields of science and engineering. ADMM is a powerful numerical tool, but unfortunately its main drawback is that it can exhibit slow convergence. Several approaches for its acceleration have been proposed in the literature and in this paper we present a new general framework devoted to this aim. In particular, we describe an algorithmic framework that makes possible the application of any acceleration step while still having the guarantee of convergence. This result is achieved thanks to a guard condition that ensures the monotonic decrease of the combined residual. The proposed strategy is applied to image deblurring problems. Several acceleration techniques are compared; to the best of our knowledge, some of them are investigated for the first time in connection with ADMM. Numerical results show that the proposed framework leads to a faster convergence with respect to other acceleration strategies recently introduced for ADMM. [ABSTRACT FROM AUTHOR]

Grey wolf optimizer (GWO) is one of recent metaheuristics swarm intelligence methods. It has been widely tailored for a wide variety of optimization problems due to its impressive characteristics over other swarm intelligence methods: it has very few parameters, and no derivation information is required in the initial search. Also it is simple, easy to use, flexible, scalable, and has a special capability to strike the right balance between the exploration and exploitation during the search which leads to favourable convergence. Therefore, the GWO has recently gained a very big research interest with tremendous audiences from several domains in a very short time. Thus, in this review paper, several research publications using GWO have been overviewed and summarized. Initially, an introductory information about GWO is provided which illustrates the natural foundation context and its related optimization conceptual framework. The main operations of GWO are procedurally discussed, and the theoretical foundation is described. Furthermore, the recent versions of GWO are discussed in detail which are categorized into modified, hybridized and paralleled versions. The main applications of GWO are also thoroughly described. The applications belong to the domains of global optimization, power engineering, bioinformatics, environmental applications, machine learning, networking and image processing, etc. The open source software of GWO is also provided. The review paper is ended by providing a summary conclusion of the main foundation of GWO and suggests several possible future directions that can be further investigated. [ABSTRACT FROM AUTHOR]

In this paper, we propose an iterative algorithm to find the common element of set of solutions of a split equilibrium problem and set of fixed points of an asymptotically nonexpansive mapping in Hilbert spaces. The new method is used to prove the strong convergence for the result of this paper. The result extends the corresponding one in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we prove that in macroscopic times of order one, the solutions to the truncated BBGKY hierarchy (to second order) converge in the weak coupling limit to the solution of the nonlinear spatially homogeneous Landau equation. The truncated problem describes the formal leading order behavior of the underlying particle dynamics, and can be reformulated as a non-Markovian hyperbolic equation that converges to the Markovian evolution described by the parabolic Landau equation. The analysis in this paper is motivated by Bogolyubov’s derivation of the kinetic equation by means of a multiple time scale analysis of the BBGKY hierarchy. [ABSTRACT FROM AUTHOR]

In this paper we present two strategies to enable “parallelization across the method” for spectral deferred corrections (SDC). Using standard low-order time-stepping methods in an iterative fashion, SDC can be seen as preconditioned Picard iteration for the collocation problem. Typically, a serial Gauß-Seidel-like preconditioner is used, computing updates for each collocation node one by one. The goal of this paper is to show how this process can be parallelized, so that all collocation nodes are updated simultaneously. The first strategy aims at finding parallel preconditioners for the Picard iteration and we test three choices using four different test problems. For the second strategy we diagonalize the quadrature matrix of the collocation problem directly. In order to integrate non-linear problems we employ simplified and inexact Newton methods. Here, we estimate the speed of convergence depending on the time-step size and verify our results using a non-linear diffusion problem. [ABSTRACT FROM AUTHOR]

In this paper, we propose a two-parameter preconditioned variant of the deteriorated PSS iteration method (J. Comput. Appl. Math., 273, 41-60 (2015)) for solving singular saddle point problems. Semi-convergence analysis shows that the new iteration method is convergent unconditionally. The new iteration method can also be regarded as a preconditioner to accelerate the convergence of Krylov subspace methods. Eigenvalue distribution of the corresponding preconditioned matrix is presented, which is instructive for the Krylov subspace acceleration. Note that, when the leading block of the saddle point matrix is symmetric, the new iteration method will reduce to the preconditioned accelerated HSS iteration method (Numer. Algor., 63 (3), 521-535 2013), the semi-convergence conditions of which can be simplified by the results in this paper. To further improve the effectiveness of the new iteration method, a relaxed variant is given, which has much better convergence and spectral properties. Numerical experiments are presented to investigate the performance of the new iteration methods for solving singular saddle point problems. [ABSTRACT FROM AUTHOR]

This paper investigates the asymptotic behaviour of solutions to certain infinite systems of coupled recurrence relations. In particular, we obtain a characterisation of those initial values which lead to a convergent solution, and for initial values satisfying a slightly stronger condition we obtain an optimal estimate on the rate of convergence. By establishing a connection with a related problem in continuous time, we are able to use this optimal estimate to improve the rate of convergence in the continuous setting obtained by the authors in a previous paper. We illustrate the power of the general approach by using it to study several concrete examples, both in continuous and in discrete time. [ABSTRACT FROM AUTHOR]

The paper introduces and analyzes the convergence of a new iterative algorithm for approximating solutions of equilibrium problems involving strongly pseudomonotone and Lipschitz-type bifunctions in Hilbert spaces. The algorithm uses a stepsize sequence which is non-increasing, diminishing, and non-summable. This leads to the main advantage of the algorithm, namely that the construction of solution approximations and the proof of its convergence are done without the prior knowledge of the modulus of strong pseudomonotonicity and Lipschitz-type constants of bifunctions. The strongly convergent theorem is established under suitable assumptions. The paper also discusses the assumptions used in the formulation of the convergent theorem. Several numerical results are reported to illustrate the behavior of the algorithm with different sequences of stepsizes and also to compare it with others. [ABSTRACT FROM AUTHOR]

In this paper, we suggest and analyze a Krasnoselski-Mann type iterative method to approximate a common element of solution sets of a hierarchical fixed point problem for nonexpansive mappings and a split mixed equilibrium problem. We prove that sequences generated by the proposed iterative method converge weakly to a common element of solution sets of these problems. Further, we derive some consequences from our main result. Furthermore, we extend the considered iterative method to a split monotone variational inclusion problem and deduce some consequences. Finally, we give a numerical example to justify the main result. The method and results presented in this paper generalize and unify the corresponding known results in this area. [ABSTRACT FROM AUTHOR]

Based on multiquadric trigonometric spline quasi-interpolation, the paper proposes a scheme for numerical differentiation of noisy data, which is a well-known ill-posed problem in practical applications. In addition, in the perspective of kernel regression, the paper studies its large sample properties including optimal bandwidth selection, convergence rate, almost sure convergence, and uniformly asymptotic normality. Simulations are provided at the end of the paper to demonstrate features of the scheme. Both theoretical results and simulations show that the scheme is simple, easy to compute, and efficient for numerical differentiation of noisy data. [ABSTRACT FROM AUTHOR]

In this paper, the complete convergence and the complete moment convergence for extended negatively dependent (END, in short) random variables without identical distribution are investigated. Under some suitable conditions, the equivalence between the moment of random variables and the complete convergence is established. In addition, the equivalence between the moment of random variables and the complete moment convergence is also proved. As applications, the Marcinkiewicz-Zygmund-type strong law of large numbers and the Baum-Katz-type result for END random variables are established. The results obtained in this paper extend the corresponding ones for independent random variables and some dependent random variables. [ABSTRACT FROM AUTHOR]

Both barycentric Lagrange interpolation and barycentric rational interpolation are thought to be stable and effective methods for approximating a given function on some special point sets. A direct evaluation of these interpolants due to N interpolation points at M sampling points requires $\mathcal {O}(NM)$ arithmetic operations. In this paper, we introduce two fast multipole methods to reduce the complexity to $\mathcal {O}(\max \left \{N,M\right \})$ . The convergence analysis is also presented in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new algorithm for solving a bilevel equilibrium problem in a real Hilbert space. In contrast to most other projection-type algorithms, which require to solve subproblems at each iteration, the subgradient method proposed in this paper requires only to calculate, at each iteration, two subgradients of convex functions and one projection onto a convex set. Hence, our algorithm has a low computational cost. We prove a strong convergence theorem for the proposed algorithm and apply it for solving the equilibrium problem over the fixed point set of a nonexpansive mapping. Some numerical experiments and comparisons are given to illustrate our results. Also, an application to Nash-Cournot equilibrium models of a semioligopolistic market is presented. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate the strong convergence and exponential stability in mean square of the exponential Euler method to semi-linear stochastic delay differential equations (SLSDDEs). It is proved that the exponential Euler approximation solution converges to the analytic solution with the strong order $\frac{1}{2}$ to SLSDDEs. On the one hand, the classical stability theorem to SLSDDEs is given by the Lyapunov functions. However, in this paper we study the exponential stability in mean square of the exact solution to SLSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to SLSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to SLSDDEs is proved to share the same stability for any step size by the property of logarithmic norm. [ABSTRACT FROM AUTHOR]

In this paper, multiobjective transformer design (TD) optimization is carried out using multiobjective evolutionary algorithm (MOEA) based on decomposition with dynamical resource allocation (MOEA/D-DRA) for four sets of conflicting TD bi-objectives such as (i) purchase cost and total loss, (ii) purchase cost and total lifetime cost (TLTC), (iii) total mass and total loss and (iv) total mass and TLTC, subjected to 14 various practical constraints. Significant decision variables with enlarged search space are employed for obtaining reliable and efficient TD with minimum losses and TLTC. TD is accompanied by 3D-finite element method assessment to validate the designed no-load loss calculated from analytical equations. To improve the searching ability of MOEA/D-DRA (MDRA) in solving this complex multimodal TD optimization problem (TDOP), this paper proposes integration of chaos with MDRA, enabling chaotic variation in the crossover rate and mutation scaling factor. To prove the effectiveness of chaos-assisted MOEA, logistic chaotic map-assisted MDRA, and iterative chaotic map with infinite collapses- (ICMIC) assisted MDRA (ICMDRA) have been successfully applied to multiobjective TDOP and their TD results are compared with those of MDRA, knee point-driven evolutionary multiobjective optimization algorithm (KnEA), and non-dominated sorting genetic algorithm (NSGA) II. This paper identifies which chaotic map can assist MDRA and solve TDOP by comparative analysis of performance of logistic and ICMIC chaotic maps. Efficient TD results and two MOEA performance indicators confirm the superiority of ICMDRA over all the other MOEAs in terms of diversity and convergence in solving TDOP. [ABSTRACT FROM AUTHOR]

A new fast direct linear equation solver for the boundary element method (BEM) is presented in this paper. The idea of the new fast direct solver stems from the concept of the hierarchical off-diagonal low-rank matrix. The hierarchical off-diagonal low-rank matrix can be decomposed into the multiplication of several diagonal block matrices. The inverse of the hierarchical off-diagonal low-rank matrix can be calculated efficiently with the Sherman-Morrison-Woodbury formula. In this paper, a more general and efficient approach to approximate the coefficient matrix of the BEM with the hierarchical off-diagonal low-rank matrix is proposed. Compared to the current fast direct solver based on the hierarchical off-diagonal low-rank matrix, the proposed method is suitable for solving general 3-D boundary element models. Several numerical examples of 3-D potential problems with the total number of unknowns up to above 200,000 are presented. The results show that the new fast direct solver can be applied to solve large 3-D BEM models accurately and with better efficiency compared with the conventional BEM. [ABSTRACT FROM AUTHOR]

The fundamental phenomenon that has been used to enhance the convergence speed of learning automata (LA) is that of incorporating the running maximum likelihood (ML) estimates of the action reward probabilities into the probability updating rules for selecting the actions. The frontiers of this field have been recently expanded by replacing the ML estimates with their corresponding Bayesian counterparts that incorporate the properties of the conjugate priors. These constitute the Bayesian pursuit algorithm (BPA), and the discretized Bayesian pursuit algorithm. Although these algorithms have been designed and efficiently implemented, and are, arguably, the fastest and most accurate LA reported in the literature, the proofs of their $$\epsilon$$ -optimal convergence has been unsolved. This is precisely the intent of this paper. In this paper, we present a single unifying analysis by which the proofs of both the continuous and discretized schemes are proven. We emphasize that unlike the ML-based pursuit schemes, the Bayesian schemes have to not only consider the estimates themselves but also the distributional forms of their conjugate posteriors and their higher order moments-all of which render the proofs to be particularly challenging. As far as we know, apart from the results themselves, the methodologies of this proof have been unreported in the literature-they are both pioneering and novel. [ABSTRACT FROM AUTHOR]

In this paper, we propose an $$l_0$$ -norm penalized shrinkage linear affine projection ( $$l_0$$ -SL-AP) algorithm and an $$l_0$$ -norm penalized shrinkage widely linear affine projection ( $$l_0$$ -SWL-AP) algorithm. The proposed algorithms provide variable step-size by minimizing the noise-free a posteriori error at each iteration and introduce an $$l_0$$ -norm constraint to the cost function. The $$l_0$$ -SWL-AP algorithm also exploits noncircular properties of the input signal. In contrast with conventional AP algorithms, the proposed algorithms increase the estimation accuracy for time-varying sparse system identification. A quantitative analysis of the convergence behavior for the $$l_0$$ -SWL-AP algorithm verifies the capabilities of the proposed algorithms. To reduce the complexity, we also introduce dichotomous coordinate descent (DCD) iterations to the proposed algorithms ( $$l_0$$ -SL-DCD-AP and $$l_0$$ -SWL-DCD-AP) in this paper. Simulations indicate that the $$l_0$$ -SL-AP and $$l_0$$ -SWL-AP algorithms provide faster convergence speed and lower steady-state misalignment than the previous APA-type algorithms. The $$l_0$$ -SL-DCD-AP and $$l_0$$ -SWL-DCD-AP algorithms perform similarly to their counterparts but with reduced complexity. [ABSTRACT FROM AUTHOR]

In the papers, dealing with derivation and applications of operational matrices of Bernstein polynomials, a basis transformation, commonly a transformation to power basis, is used. The main disadvantage of this method is that the transformation may be ill-conditioned. Moreover, when applied to the numerical simulation of a functional differential equation, it leads to dense operational matrices and so a dense coefficient matrix is obtained. In this paper, we present a new property for Bernstein polynomials. Using this property, we build exact banded operational matrices for derivatives of Bernstein polynomials. Next, as an application, we propose a new numerical method based on a Petrov-Galerkin variational formulation and the new operational matrices utilizing the dual Bernstein basis for the time-fractional advection-dispersion equation. We show that the proposed method leads to a narrow-banded linear system and so less computational effort is required to obtain the desired accuracy for the approximate solution. We also obtain the error estimation for the method. Some numerical examples are provided to demonstrate the efficiency of the method and to support the theoretical claims. [ABSTRACT FROM AUTHOR]

This paper concerns characterizations of approximation classes associated with adaptive finite element methods with isotropic h-refinements. It is known from the seminal work of Binev, Dahmen, DeVore and Petrushev that such classes are related to Besov spaces. The range of parameters for which the inverse embedding results hold is rather limited, and recently, Gaspoz and Morin have shown, among other things, that this limitation disappears if we replace Besov spaces by suitable approximation spaces associated with finite element approximation from uniformly refined triangulations. We call the latter spaces multievel approximation spaces and argue that these spaces are placed naturally halfway between adaptive approximation classes and Besov spaces, in the sense that it is more natural to relate multilevel approximation spaces with either Besov spaces or adaptive approximation classes, than to go directly from adaptive approximation classes to Besov spaces. In particular, we prove embeddings of multilevel approximation spaces into adaptive approximation classes, complementing the inverse embedding theorems of Gaspoz and Morin. Furthermore, in the present paper, we initiate a theoretical study of adaptive approximation classes that are defined using a modified notion of error, the so-called total error, which is the energy error plus an oscillation term. Such approximation classes have recently been shown to arise naturally in the analysis of adaptive algorithms. We first develop a sufficiently general approximation theory framework to handle such modifications, and then apply the abstract theory to second-order elliptic problems discretized by Lagrange finite elements, resulting in characterizations of modified approximation classes in terms of memberships of the problem solution and data into certain approximation spaces, which are in turn related to Besov spaces. Finally, it should be noted that throughout the paper we paid equal attention to both conforming and non-conforming triangulations. [ABSTRACT FROM AUTHOR]

A new method for deriving estimates of the rate of convergence of optimal methods for solving problems of smooth (strongly) convex stochastic optimization is described. The method is based on the results of stochastic optimization derived from results on the convergence of optimal methods under the conditions of inexact gradients with small noises of nonrandom nature. In contrast to earlier results, all estimates in the present paper are obtained in model generality. [ABSTRACT FROM AUTHOR]

In the present paper, we construct a new class of positive linear λ-Bernstein operators based on (p, q)-integers. We obtain a Korovkin type approximation theorem, study the rate of convergence of these operators by using the conception of K-functional and moduli of continuity, and also give a convergence theorem for the Lipschitz continuous functions. [ABSTRACT FROM AUTHOR]

This paper stems from the idea of adopting a new approach to solve some classical optimal packing problems for balls. In fact, we attack this kind of problems (which are of discrete nature) by means of shape optimization techniques, applied to suitable Γ -converging sequences of energies associated to Cheeger type problems. More precisely, in a first step we prove that different optimal packing problems are limits of sequences of optimal clusters associated to the minimization of energies involving suitable (generalized) Cheeger constants. In a second step, we propose an efficient phase field approach based on a multiphase Γ -convergence result of Modica–Mortola type, in order to compute those generalized Cheeger constants, their optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions. Our continuous shape optimization approach to solve discrete packing problems circumvents the NP-hard character of these ones, and efficiently leads to configurations close to the global minima. [ABSTRACT FROM AUTHOR]

This paper investigates stochastic convergence in income inequality across Australian states and territories since the end of World War II by utilising the LM and RALS-LM unit root tests that allow for endogenously determined structural breaks. We find that income inequality for Australia's capital city—the Australain Capital Territoy—converges to a stable steady-state when we account for endogenously determined trend-breaks. The null hypothesis of a unit root is rejected when utilising the two-break unit root test for New South Wales, Northern Territory, Queensland, South Australia, Tasmania and Western Australia indicating that income inequalities in these states converge to a stable steady-state when accounting for multiple structural breaks. In contrast, the state of Victoria consistently shows evidence of divergence regardless of the number of endogenously determined structural breaks. Structural changes in income inequality across states and territories may be linked to the mining boom, the transition from manufacturing to a service-based economy, changes in government welfare, and favourable changes in tax policies and superannuation. [ABSTRACT FROM AUTHOR]

This paper proposes a new nonlinear stochastic SIVS epidemic model with double epidemic hypothesis and Lévy jumps. The main purpose of this paper is to investigate the threshold dynamics of the stochastic SIVS epidemic model. By using the technique of a series of stochastic inequalities, we obtain sufficient conditions for the persistence in mean and extinction of the stochastic system and the threshold which governs the extinction and the spread of the epidemic diseases. Finally, this paper describes the results of numerical simulations investigating the dynamical effects of stochastic disturbance. Our results significantly improve and generalize the corresponding results in recent literatures. The developed theoretical methods and stochastic inequalities technique can be used to investigate the high-dimensional nonlinear stochastic differential systems. [ABSTRACT FROM AUTHOR]

One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the weak convergence of the gradients, fluxes and energy densities of the first-order correctors (under blow-down) which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument, completing the program initiated in Armstrong et al. (Commun. Math. Phys. 347(2):315-361, 2016): using the regularity theory recently developed for stochastic homogenization, we reduce the error in additivity as we pass to larger and larger length scales. In the second part of the paper, we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables. In particular, we prove that the first-order correctors converge, in the large-scale limit, to a variant of the Gaussian free field. [ABSTRACT FROM AUTHOR]

This is the second one of two companion papers. We have focused on the spectral entity determination in the first paper where we have considered the Hilbert-Schmidt and Pincherle-Goursat kernels. The basic idea has been the development of a perturbation expansion around the zero interval limit therein. We have emphasized on the case of most dominant eigenvalue and corresponding eigenfunction by taking the half-interval length as the perturbation parameter after universalizing the given (finite) interval of the integral operator. The basic issues in the formulation of the perturbation expansion and certain technicalities were kept as the main theme of the paper in the first companion paper. This second companion paper, however, has been designed to focus on the convergence discussions and confirmative implementations. It also presents a numerical comparison between proposed method and various well known approximation methods residing in scientific literature. [ABSTRACT FROM AUTHOR]

The uncertainties related to long-term forecasts of oil prices impose significant financial risk on ventures of oil production. To minimize risk, oil companies are inclined to maximize profit over short-term horizons ranging from months to a few years. In contrast, conventional production optimization maximizes long-term profits over horizons that span more than a decade. To address this challenge, the oil literature has introduced short-term versus long-term optimization. Ideally, this problem is solved by a posteriori multi-objective optimization methods that generate an approximation to the Pareto front of optimal short-term and long-term trade-offs. However, such methods rely on a large number of reservoir simulations and scale poorly with the number of objectives subject to optimization. Consequently, the large-scale nature of production optimization severely limits applications to real-life scenarios. More practical alternatives include ad hoc hierarchical switching schemes. As a drawback, such methods lack robustness due to unclear convergence properties and do not naturally generalize to cases of more than two objectives. Also, as this paper shows, the hierarchical formulation may skew the balance between the objectives, leaving an unfulfilled potential to increase profits. To promote efficient and reliable short-term versus long-term optimization, this paper introduces a natural way to characterize desirable Pareto points and proposes a novel least squares (LS) method. Unlike hierarchical approaches, the method is guaranteed to converge to a Pareto optimal point. Also, the LS method is designed to properly balance multiple objectives, independently of Pareto front's shape. As such, the method poses a practical alternative to a posteriori methods in situations where the frontier is intractable to generate. [ABSTRACT FROM AUTHOR]

In this paper, we propose two multi-objective memetic algorithms (MOMAs) using two different adaptive mechanisms to address combinatorial optimization problems (COPs). One mechanism adaptively selects solutions for local search based on the solutions' convergence toward the Pareto front. The second adaptive mechanism uses the convergence and diversity information of an external set (dominance archive), to guide the selection of promising solutions for local search. In addition, simulated annealing is integrated in this framework as the local refinement process. The multi-objective memetic algorithms with the two adaptive schemes (called uMOMA-SA and aMOMA-SA) are tested on two COPs and compared with some well-known multi-objective evolutionary algorithms. Experimental results suggest that uMOMA-SA and aMOMA-SA outperform the other algorithms with which they are compared. The effects of the two adaptive mechanisms are also investigated in the paper. In addition, uMOMA-SA and aMOMA-SA are compared with three single-objective and three multi-objective optimization approaches on software next release problems using real instances mined from bug repositories (Xuan et al. IEEE Trans Softw Eng 38(5):1195-1212, 2012). The results show that these multi-objective optimization approaches perform better than these single-objective ones, in general, and that aMOMA-SA has the best performance among all the approaches compared. [ABSTRACT FROM AUTHOR]

Our contribution in this paper is to propose an iterative algorithm which does not require prior knowledge of operator norm and prove a strong convergence theorem for approximating a solution of split equality fixed point problem for quasi-nonexpansive mappings in a real Hilbert space. So many have used algorithms involving the operator norm for solving split equality fixed point problem, but as widely known the computation of these algorithms may be difficult and for this reason, some researchers have recently started constructing iterative algorithms with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm. To the best of our knowledge most of the works in literature that do not involve the calculation or estimation of the operator norm only obtained weak convergence results. In this paper, by appropriately modifying the simultaneous iterative algorithm introduced by Zhao, we state and prove a strong convergence result for solving split equality problem. We present some applications of our result and then give some numerical example to study its efficiency and implementation at the end of the paper. [ABSTRACT FROM AUTHOR]