A RELAXED INERTIAL FACTOR OF THE MODIFIED SUBGRADIENT EXTRAGRADIENT METHOD FOR SOLVING PSEUDO MONOTONE VARIATIONAL INEQUALITIES IN HILBERT SPACES

In this paper,we investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces.For solving this problem,we propose a new method that combines the advantages of the subgradient extragradient method and the projection contraction method.Some very recent papers have considered different inertial algorithms which allowed the inertial factor is chosen in[0;1].The purpose of this work is to continue working in this direction,we propose another inertial subgradient extragradient method that the inertial factor can be chosen in a special case to be 1.Under suitable mild conditions,we establish the weak convergence of the proposed algorithm.Moreover,linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions.Finally,some numerical illustrations are given to confirm the theoretical analysis.

Bi-extrapolated subgradient projection algorithm for solving multiple-sets split feasibility problem

This paper deals with a bi-extrapolated subgradient projection algorithm by intro- ducing two extrapolated factors in the iterative step to solve the multiple-sets split feasibility problem. The strategy is intend to improve the convergence. And its convergence is proved un- der some suitable conditions. Numerical results illustrate that the bi-extrapolated subgradient projection algorithm converges more quickly than the existing algorithms.

An Extrapolated Parallel Subgradient Projection Algorithm with Centering Technique for the Convex Feasibility Problem

In this paper,we present an extrapolated parallel subgradient projection method with the centering technique for the convex feasibility problem,the algorithm improves the convergence by reason of using centering techniques which reduce the oscillation of the corresponding sequence.To prove the convergence in a simply way,we transmit the parallel algorithm in the original space to a sequential one in a newly constructed product space.Thus,the convergence of the parallel algorithm is derived with the help of the sequential one under some suitable conditions.Numerical results show that the new algorithm has better convergence than the existing algorithms.

NONDESCENT SUBGRADIENT METHOD FOR NONSMOOTH CONSTRAINED MINIMIZATION

A kind of nondecreasing subgradient algorithm with appropriate stopping rule has been proposed for nonsmooth constrained minimization problem. The dual theory is invoked in dealing with the stopping rule and general global minimiizing algorithm is employed as a subroutine of the algorithm. The method is expected to tackle a large class of nonsmooth constrained minimization problem.

CURRENTS CARRIED BY THE SUBGRADIENT GRAPHS OF SEMI-CONVEX FUNCTIONS AND APPLICATIONS TO HESSIAN MEASURES

In this paper we study integer multiplicity rectifiable currents carried by the subgradient （subdifferential） graphs of semi-convex functions on an n-dimensional convex domain, and show a weak continuity theorem with respect to pointwise convergence for such currents. As an application, the structure theorem of the Lagrangian currents for semi-convex functions is given and the k-Hessian measures are calculated by a different method in terms of currents.

Projected subgradient method for non-Lipschitz set-valued mixed variational inequalities

A projected subgradient method for solving a class of set-valued mixed variational inequalities （SMVIs） is proposed when the mapping is not necessarily Lipschitz. Under some suitable conditions, it can be proven that the sequence generated by the method can strongly converge to the unique solution to the problem in the Hilbert spaces.

Modified Subgradient Extragradient Method for Variational Inequality Problems and Fixed Point Problems

Many approaches inquiring into variational inequality problems have been put forward,among which subgradient extragradient method is of great significance.A novel algorithm is presented in this article for resolving quasi-nonexpansive fixed point problem and pseudomonotone variational inequality problem in a real Hilbert interspace.In order to decrease the execution time and quicken the velocity of convergence,the proposed algorithm adopts an inertial technology.Moreover,the algorithm is by virtue of a non-monotonic step size rule to acquire strong convergence theorem without estimating the value of Lipschitz constant.Finally,numerical results on some problems authenticate that the algorithm has preferable efficiency than other algorithms.

Subgradient Extragradient Methods for Equilibrium Problems and Fixed Point Problems in Hilbert Space

Inspired by inertial methods and extragradient algorithms,two algorithms were proposed to investigate fixed point problem of quasinonexpansive mapping and pseudomonotone equilibrium problem in this study.In order to enhance the speed of the convergence and reduce computational cost,the algorithms used a new step size and a cutting hyperplane.The first algorithm was proved to be weak convergence,while the second algorithm used a modified version of Halpern iteration to obtain strong convergence.Finally,numerical experiments on several specific problems and comparisons with other algorithms verified the superiority of the proposed algorithms.

Modified Subgradient Extragradient Method for Pseudomonotone Variational Inequalities

Many approaches have been put forward to resolve the variational inequality problem. The subgradient extragradient method is one of the most effective. This paper proposes a modified subgradient extragradient method about classical variational inequality in a real Hilbert interspace. By analyzing the operator’s partial message, the proposed method designs a non-monotonic step length strategy which requires no line search and is independent of the value of Lipschitz constant, and is extended to solve the problem of pseudomonotone variational inequality. Meanwhile, the method requires merely one map value and a projective transformation to the practicable set at every iteration. In addition, without knowing the Lipschitz constant for interrelated mapping, weak convergence is given and R-linear convergence rate is established concerning algorithm. Several numerical results further illustrate that the method is superior to other algorithms.

THE WEAK BENSON PROPER EFFICIENT SUBGRADIENT AND THE OPTIMALITY CONDITIONS OF SET-VALUED OPTIMIZATION

The subgradient, under the weak Benson proper efficiency, of a set-valued mapping in ordered Banach space is developed, and the weak Benson proper efficient Hahn-Banach theorem of a set-valued mapping is established, with which the existence of the subgradient is proved and the characterizations of weak Benson proper efficient elements of constrained(unconstrained) vector set-valued optimization problems are presented.

Subgradient-based feedback neural networks for non-differentiable convex optimization problems

This paper developed the dynamic feedback neural network model to solve the convex nonlinear programming problem proposed by Leung et al. and introduced subgradient-based dynamic feedback neural networks to solve non-differentiable convex optimization problems. For unconstrained non-differentiable convex optimization problem, on the assumption that the objective function is convex coercive, we proved that with arbitrarily given initial value, the trajectory of the feedback neural network constructed by a projection subgradient converges to an asymptotically stable equilibrium point which is also an optimal solution of the primal unconstrained problem. For constrained non-differentiable convex optimization problem, on the assumption that the objective function is convex coercive and the constraint functions are convex also, the energy functions sequence and corresponding dynamic feedback subneural network models based on a projection subgradient are successively constructed respectively, the convergence theorem is then obtained and the stopping condition is given. Furthermore, the effective algorithms are designed and some simulation experiments are illustrated.

Distributed projection subgradient algorithm for two-network zero-sum game with random sleep scheme

In this paper,a zero-sum game Nash equilibrium computation problem with a common constraint set is investigated under two time-varying multi-agent subnetworks,where the two subnetworks have opposite payoff function.A novel distributed projection subgradient algorithm with random sleep scheme is developed to reduce the calculation amount of agents in the process of computing Nash equilibrium.In our algorithm,each agent is determined by an independent identically distributed Bernoulli decision to compute the subgradient and perform the projection operation or to keep the previous consensus estimate,it effectively reduces the amount of computation and calculation time.Moreover,the traditional assumption of stepsize adopted in the existing methods is removed,and the stepsizes in our algorithm are randomized diminishing.Besides,we prove that all agents converge to Nash equilibrium with probability 1 by our algorithm.Finally,a simulation example verifies the validity of our algorithm.

Non-monotonous Sequential Subgradient Projection Algorithm for Convex Feasibility Problem

The existing methods of projection for solving convex feasibility problem may lead to slow conver- gence when the sequences enter some narrow＂corridor＂ between two or more convex sets. In this paper, we apply a technique that may interrupt the monotonity of the constructed sequence to the sequential subgradient pro- jection algorithm to construct a nommonotonous sequential subgradient projection algorithm for solving convex feasibility problem, which can leave such corridor by taking a big step at different steps during the iteration. Under some suitable conditions, the convergence is proved.We also compare the numerical performance of the proposed algorithm with that of the monotonous algorithm by numerical experiments.

TIME PERIODIC SOLUTIONS TO THE EVOLUTIONARY OSEEN MODEL FOR A GENERALIZED NEWTONIAN INCOMPRESSIBLE FLUID

In this paper we study a nonstationary Oseen model for a generalized Newtonian incompressible fluid with a time periodic condition and a multivalued,nonmonotone friction law.First,a variational formulation of the model is obtained;that is a nonlinear boundary hemivariational inequality of parabolic type for the velocity field.Then,an abstract first-order evolutionary hemivariational inequality in the framework of an evolution triple of spaces is investigated.Under mild assumptions,the nonemptiness and weak compactness of the set of periodic solutions to the abstract inequality are proven.Furthermore,a uniqueness theorem for the abstract inequality is established by using a monotonicity argument.Finally,we employ the theoretical results to examine the nonstationary Oseen model.

ROBUST ACOUSTIC SOURCE LOCALIZATION FOR DIGITAL HEARING AIDS IN NOISE AND REVERBERANT ENVIRONMENT

A new method in digital hearing aids to adaptively localize the speech source in noise and reverberant environment is proposed. Based on the room reverberant model and the multichannel adaptive eigenvalue decomposition （MCAED） algorithm, the proposed method can iteratively estimate impulse response coefficients between the speech source and microphones by the adaptive subgradient projection method. Then, it acquires the time delays of microphone pairs, and calculates the source position by the geometric method. Compared with the traditional normal least mean square （NLMS） algorithm, the adaptive subgradient projection method achieves faster and more accurate convergence in a low signal-to-noise ratio （SNR） environment. Simulations for glasses digital hearing aids with four-component square array demonstrate the robust performance of the proposed method.

凸规划对偶的统一性

根据凸函数的性质 ,讨论一般凸规划的 L agrange对偶、Subgradient对偶和 Wolfe对偶等三种不同对偶形式之间的关系 ,给出它们之间等价性成立的条件。

A novel Lagrangian relaxation level approach for scheduling steelmaking-refining-continuous casting production

A Lagrangian relaxation(LR) approach was presented which is with machine capacity relaxation and operation precedence relaxation for solving a flexible job shop(FJS) scheduling problem from the steelmaking-refining-continuous casting process. Unlike the full optimization of LR problems in traditional LR approaches, the machine capacity relaxation is optimized asymptotically, while the precedence relaxation is optimized approximately due to the NP-hard nature of its LR problem. Because the standard subgradient algorithm(SSA) cannot solve the Lagrangian dual(LD) problem within the partial optimization of LR problem, an effective deflected-conditional approximate subgradient level algorithm(DCASLA) was developed, named as Lagrangian relaxation level approach. The efficiency of the DCASLA is enhanced by a deflected-conditional epsilon-subgradient to weaken the possible zigzagging phenomena. Computational results and comparisons show that the proposed methods improve significantly the efficiency of the LR approach and the DCASLA adopting capacity relaxation strategy performs best among eight methods in terms of solution quality and running time.

Optimal congestion control algorithm for ad hoc networks: Penalty function-based approach

In this paper, based on the inherent characteristic of the contention relation between flows in ad hoc networks, we introduce the notion of the link＇s interference set, extend the utility maximization problem representing congestion control in wireline networks to ad hoc networks, apply the penalty function approach and the subgradient method to solve this problem, and propose the congestion control algorithm Penalty function-based Optical Congestion Control （POCC） which is implemented in NS2- simulator. Specifically, each link transmits periodically the information on its congestion state to its interference set; the set ; the sermon at each source adjusts the transmission rate based on the optimal tradeoffbetween the utility value and the congestion level which the interference set of the links that this session goes though suffers from. MATLAB-based simulation results showed that POCC can approach the globally optimal solution. The NS2-based simulation results showed that POCC outperforms default TCP and ATCP to achieve efficient and fair resource allocation in ad hoc networks.

An efficient algorithm for multi-dimensional nonlinear knapsack problems

Multi-dimensional nonlinear knapsack problem is a bounded nonlinear integer programming problem that maximizes a separable nondecreasing function subject to multiple separable nondecreasing constraints. This problem is often encountered in resource allocation, industrial planning and computer network. In this paper, a new convergent Lagrangian dual method was proposed for solving this problem. Cutting plane method was used to solve the dual problem and to compute the Lagrangian bounds of the primal problem. In order to eliminate the duality gap and thus to guarantee the convergence of the algorithm, domain cut technique was employed to remove certain integer boxes and partition the revised domain to a union of integer boxes. Extensive computational results show that the proposed method is efficient for solving large-scale multi-dimensional nonlinear knapsack problems. Our numerical results also indicate that the cutting plane method significantly outperforms the subgradient method as a dual search procedure.

EXISTENCE AND CONVERGENCE RESULTS FOR AN ELASTIC FRICTIONAL CONTACT PROBLEM WITH NONMONOTONE SUBDIFFERENTIAL BOUNDARY CONDITIONS

The goal of this paper is to study a mathematical model of a nonlinear static frictional contact problem in elasticity with the mixed boundary conditions described by a combination of the Signorini unilateral frictionless contact condition,and nonmonotone multivalued contact,and friction laws of subdifferential form.First,under suitable assumptions,we deliver the weak formulation of the contact model,which is an elliptic system with Lagrange multipliers,and which consists of a hemivariational inequality and a variational inequality.Then,we prove the solvability of the contact problem.Finally,employing the notion of H-convergence of nonlinear elasticity tensors,we provide a result on the convergence of solutions under perturbations which appear in the elasticity operator,body forces,and surface tractions.