A class of nonlinear proximal point algorithms for variational inequality problems

This work is motivated by a recent work on an extended linear proximal point algorithm (PPA) [B.S. He, X.L. Fu, and Z.K. Jiang, Proximal-point algorithm using a linear proximal term, J. Optim. Theory Appl. 141 (2009), pp. 299–319], which aims at relaxing the requirement of the linear proximal term of classical PPA. In this paper, we make further contributions along the line. First, we generalize the linear PPA-based contraction method by using a nonlinear proximal term instead of the linear one. A notable superiority over traditional PPA-like methods is that the nonlinear proximal term of the proposed method may not necessarily be a gradient of any functions. In addition, the nonlinearity of the proximal term makes the new method more flexible. To avoid solving a variational inequality subproblem exactly, we then propose an inexact version of the developed method, which may be more computationally attractive in terms of requiring lower computational cost. Finally, we gainfully employ our new methods to solve linearly constrained convex minimization and variational inequality problems.     

A globally convergent BFGS method for pseudo-monotone variational inequality problems

In this paper, we propose a globally convergent BFGS method to solve Variational Inequality Problems (VIPs). In fact, a globalization technique on the basis of the hyperplane projection method is applied to the BFGS method. The technique, which is independent of any merit function, is applicable for pseudo-monotone problems. The proposed method applies the BFGS direction and tries to reduce the distance of iterates to the solution set. This property, called Fejer monotonicity of iterates with respect to the solution set, is the basis of the convergence analysis. The method applied to pseudo-monotone VIP is globally convergent in the sense that subproblems always have unique solutions, and the sequence of iterates converges to a solution to the problem without any regularity assumption. Finally, some numerical simulations are included to evaluate the efficiency of the proposed algorithm.     

A smoothing inexact Newton method for variational inequality problems

In this paper, we reformulate the variational inequality problem as an equivalent smooth non-linear equation system by introducing the Chen–Harker–Kanzow–Smale smoothing function. A new smoothing inexact Newton algorithm is proposed to solve the smooth equations. In each iteration, the corresponding linear system is solved approximately. We prove that the proposed algorithm converges globally and superlinearly under mild conditions. Preliminary numerical results indicate that the method is effective.     

Hybrid algorithm for generalized mixed equilibrium problems and variational inequality problems and fixed point problems

In this paper, we introduce a new iterative algorithm by hybrid method for finding a common element of the set of solutions of finite general mixed equilibrium problems and the set of solutions of a general variational inequality problem for finite inverse strongly monotone mappings and the set of common fixed points of infinite family of strictly pseudocontractive mappings in a real Hilbert space. Then we prove strong convergence of the scheme to a common element of the three above described sets. Our result improves and extends the corresponding results announced by many others.     

A globally and locally superlinearly convergent inexact Newton-GMRES method for large-scale variational inequality problem

In this paper, we propose an inexact Newton-generalized minimal residual method for solving the variational inequality problem. Based on a new smoothing function, the variational inequality problem is reformulated as a system of parameterized smooth equations. In each iteration, the corresponding linear system is solved only approximately. Under mild assumptions, it is proved that the proposed algorithm has global convergence and local superlinear convergence properties. Preliminary numerical results indicate that the method is effective for a large-scale variational inequality problem.     

Mixed type symmetric duality for multiobjective variational problems with cone constraints

In this study, a pair of multiobjective variational mixed symmetric dual programs involving cone constraints is presented and usual duality results are established using the notion of generalized ‐convexity. Self‐duality is discussed under additional condition of skew symmetry, and a time‐independence (static) version of the problems is also incorporated.     

A hybrid splitting method for variational inequality problems with separable structure

Alternating direction method (ADM), which decomposes a large-scale original variational inequality (VI) problem into a series of smaller scale subproblems, is very attractive for solving a class of VI problems with a separable structure. This type of method can greatly improve the efficiency, but cannot avoid solving VI subproblems. In this paper, we propose a hybrid splitting method with variable parameters for separable VI problems. Specifically, the proposed method solves only one strongly monotone VI subproblem and a well-posed system of nonlinear equations in each iteration. The global convergence of the new method is established under some standard assumptions as those in classical ADMs. Finally, some preliminary numerical results show that the proposed method performs favourably in practice.     

A viscosity approximation scheme for finite mixed equilibrium problems and variational inequality problems and fixed point problems

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of finite mixed equilibrium problems, the set of solutions of variational inequalities for two cocoercive mappings, the set of common fixed points of an infinite family of nonexpansive mappings and the set of common fixed points of a nonexpansive semigroup in Hilbert space. Then we prove a strong convergence theorem under some suitable conditions. The results obtained in this paper extend and improve many recent ones announced by many others.     

The quasi-newtonian method of solution of convex variational inequalities with descent decomposition

The problem of numerical solution of convex variational inequalities with nonlinear constraints is considered. An equivalent optimization problem is constructed. To solve it, a Newtonian-type method with descent decomposition with respect to direct and dual variables of a variational inequality is developed. The nonlocal convergence of the algorithm and the superlinear rate of convergence in a neighborhood of the solution are proved. Translated from Kibemetika i Sistemnyi Analiz, No. 4, pp. 90–105, July–August, 1999.     

Entanglement assisted classical capacity of a class of quantum channels with long-term memory

In this paper we evaluate the entanglement assisted classical capacity of a class of quantum channels with long-term memory, which are convex combinations of memoryless channels. The memory of such channels can be considered to be given by a Markov chain which is aperiodic but not irreducible. This class of channels was introduced by Datta and Dorlas in (J. Phys. A, Math. Theor. 40:8147–8164, 2007), where its product state capacity was evaluated.      

Strong convergence theorems based on a new modified extragradient method for variational inequality problems and fixed point problems in Banach spaces

In this paper, we introduce a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of a general variational inequality and the set of common fixed points of an infinite family of k_i-strict pseudocontractions in a Banach space. We obtain some strong convergence theorems under suitable conditions. The results obtained in this paper improve and extend the recent ones announced by many others.     

An iterative approximation scheme for solving a split generalized equilibrium,variational inequalities and fixed point problems

In this paper, we consider a common solution of three problems in Hilbert spaces including the split generalized equilibrium problem, the variational inequality problem and fixed point problem. For finding the solution, we present a new iterative method and prove the strongly convergence theorem under mild conditions. Moreover, some numerical examples are given in the last section.     

Cybersecurity investments with nonlinear budget constraints and conservation laws: variational equilibrium,marginal expected utilities,and Lagrange multipliers

In this paper, we propose a new cybersecurity investment supply chain game theory model, assuming that the demands for the product are known and fixed and, hence, the conservation law of each demand market is fulfilled. The model is a generalized Nash equilibrium model with nonlinear budget constraints for which we define the variational equilibrium, which provides us with a variational inequality formulation. We construct an equivalent formulation, enabling the analysis of the influence of the conservation laws and the importance of the associated Lagrange multipliers. We find that the marginal expected transaction utility of each retailer depends on this Lagrange multiplier and its sign. Finally, numerical examples with reported equilibrium product flows, cybersecurity investment levels, and Lagrange multipliers, along with individual firm vulnerability and network vulnerability, illustrate the obtained results.     

Short-term and long-term memory analysis of learning using 2D and 3D educational contents

ABSTRACT

The effect of 2D and 3D educational content learning on memory has been studied using electroencephalography (EEG) brain signal. A hypothesis is set that the 3D materials are better than the 2D materials for learning and memory recall. To test the hypothesis, we proposed a classification system that will predict true or false recall for short-term memory (STM) and long-term memory (LTM) after learning by either 2D or 3D educational contents. For this purpose, EEG brain signals are recorded during learning and testing; the signals are then analysed in the time domain using different types of features in various frequency bands. The features are then fed into a support vector machine (SVM)-based classifier. The experimental results indicate that the learning and memory recall using 2D and 3D contents do not have significant differences for both the STM and the LTM.     

A class of projection and contraction methods for asymmetric linear variational inequalities and their relations to Fukushima's descent method

For solving asymmetric linear variational inequalities, we present a class of projection and contraction methods under the general G-norm. The search direction of our methods is just a convex combination of two descent directions of Fukushima's merit function. However, we use the direction to reduce the distance function (1/2)u − u*²_G, where μ* is a solution point of the problem. Finally, we report some numerical results for spatial price equilibrium problems by using the presented methods.     

不确定双向联想记忆神经网络的稳定性分析

对双向联想记忆神经网络研究了平衡点的鲁棒稳定性.该网络的参数不确定,并且有时变时滞.当神经网络的激励函数满足Lipschitz连续性条件时,通过选取合适的Lyapunov-Krasovskii函数,建立了两个全局鲁棒稳定判据.由于这些判据考虑了神经元激励作用和抑制作用对网络的影响,他们和时变时滞的数值无关,并且易于使用内点算法进行检验.在注释中和已有的结果进行了对比.两个数值例子展示了所得结果的有效性.     

Global asymptotic stability of uncertain stochastic bi-directional associative memory networks with discrete and distributed delays

In this paper, the global asymptotic stability analysis problem is investigated for a class of stochastic bi-directional associative memory (BAM) networks with mixed time-delays and parameter uncertainties. The mixed time-delays consist of both the discrete and the distributed delays, the uncertainties are assumed to be norm-bounded, and the neural network are subject to stochastic disturbances described by a Brownian motion. Without assuming the monotonicity and differentiability of activation functions, we employ the Lyapunov–Krasovskii stability theory and some new developed techniques to establish sufficient conditions for the stochastic delayed BAM networks to be globally asymptotically stable in the mean square. These conditions are expressed in terms of the feasibility to a set of linear matrix inequalities (LMIs) that can be easily checked by utilizing the numerically efficient Matlab LMI toolbox. A simple example is exploited to show the usefulness of the derived LMI-based stability conditions.     

Evolutionary subgradient inclusions with nonlinear weakly continuous operators and applications

In this paper we consider the first order evolutionary inclusions with nonlinear weakly continuous operators and a multivalued term which involves the Clarke subgradient of a locally Lipschitz function. First, we provide a surjectivity result for stationary inclusion with weakly–weakly upper semicontinuous multifunction. Then, we use this result to prove the existence of solutions to the Rothe sequence and the evolutionary subgradient inclusion. Finally, we apply our results to the non-stationary Navier–Stokes equation with nonmonotone and multivalued frictional boundary conditions.     

Loop scheduling with memory access reduction subject to register constraints for DSP applications

Memory accesses introduce big‐time overhead and power consumption because of the performance gap between processors and main memory. This paper describes and evaluates a technique, loop scheduling with memory access reduction (LSMAR), that replaces hidden redundant load operations with register operations in loop kernels and performs partial scheduling for newly generated register operations subject to register constraints. By exploiting data dependence of memory access operations, the LSMAR technique can effectively reduce the number of memory accesses of loop kernels, thereby improving timing performance. The technique has been implemented into the Trimaran compiler and evaluated using a set of benchmarks from DSPstone and MiBench on the cycle‐accurate simulator of the Trimaran infrastructure. The experimental results show that when the LSMAR technique is applied, the number of memory accesses can be reduced by 18.47% on average over the benchmarks when it is not applied. The measurements also indicate that the optimizations only lead to an average 1.41% increase in code size. With such small code size expansion, the technique is more suitable for embedded systems compared with prior work.Copyright © 2013 John Wiley & Sons, Ltd.