可交易瓶颈许可证的交通网络对偶均衡模型

为了能实际求解可交易瓶颈许可证(TBP)均衡系统,利用网络对偶均衡理论建立了TBP的交通网络对偶均衡模型。以路段和节点的均衡条件代替路径均衡条件,并放松了节点流量的平衡条件和分时OD流量与总需求的平衡条件,从而得到了TBP的互补问题方程。利用互补问题与变分不等式基本等价定理,得到动态交通网络变分不等式模型。建构等价的非线性优化问题,说明新模型的有效性,并给出了基于时空网络的模型解的存在性判别方法。算例的求解验证了模型的可行性。     

A preconditioner for solving large-scale variational inequality problems by a semismooth inexact approach

The numerical solution of a large-scale variational inequality problem can be obtained using the generalization of an inexact Newton method applied to a semismooth nonlinear system. This approach requires a sparse and large linear system to be solved at each step. In this work we obtain an approximate solution of this system by the LSQR algorithm of Paige and Saunders combined with a convenient preconditioner that is a variant of the incomplete LU–factorization. Since the computation of the factorization of the preconditioning matrix can be very expensive and memory consuming, we propose a preconditioner that admits block-factorization. Thus the direct factorization is only applied to submatrices of smaller sizes. Numerical experiments on a set of test-problems arising from the literature show the effectiveness of this approach.     

Generalized projection method for a variational inequality system with different mapping in Banach spaces

The approximate solution of a system for variational inequality with different mapping in Hilbert spaces has been studied, based on the convergence of projection methods. However, little research has been done in Banach space. The primary reason is that projection mapping lacks preferable properties in Banach space. In this paper, we introduce generalized projection methods. By using these methods, the results presented in this paper have extended the main results of Chang et al.’s and Huang and Noor’s recent works [S.S. Chang, H.W. Joseph Lee, C.K. Chan, Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces, Appl. Math. Lett. 20 (2007) 329–334; Zhenyu Huang, Muhammad Aslam Noor, An explicit projection method for a system of nonlinear variational inequalities with different (γ,r)-cocoercive mappings, Appl. Math. Comput. 190 (2007) 356–361] from Hilbert spaces to Banach space.     

A continuous method model for solving general variational inequality

Based on the projection operator, this paper presents a continuous method model for solving general variational inequality problems (VIPs) with bound constraints. A main feature of the proposed model is that it does not involve any form of matrix information in analysing its convergence properties. Under some reasonable assumptions, the convergence results of the proposed method model are established. Numerical results on some problems show that the proposed approach is efficient and can be applied to solve large scale VIPs with bound constraints.     

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.     

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.     

Two new self-adaptive projection methods for variational inequality problems

In this paper, we propose two new projection methods for solving variational inequality problems (VI). The method is simple; it uses only function evaluation and projection onto the feasible set. Under the conditions that the underlying function is continuous and satisfies some generalized monotonicity assumption, the methods are proven to converge to a solution of variational inequality globally. Some preliminary computational results are reported to illustrate the efficiency of the methods.     

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 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.     

Penalty method for variational inequalities with multivalued mappings. Part I

The regularization method of variational inequalities is generalized by means of penalty operators to a class of variational inequalities with multivalued mappings. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 57–69, July–August, 2000.     

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.     

New cooperative projection neural network for nonlinearly constrained variational inequality

This paper proposes a new cooperative projection neural network (CPNN), which combines automatically three individual neural network models with a common projection term. As a special case, the proposed CPNN can include three recent recurrent neural networks for solving monotone variational inequality problems with limit or linear constraints, respectively. Under the monotonicity condition of the corresponding Lagrangian mapping, the proposed CPNN is theoretically guaranteed to solve monotone variational in...     

A linear matrix inequality approach to H∞ control

The continuous- and discrete-time H_∞ control problems are solved via elementary manipulations on linear matrix inequalities (LMI). Two interesting new features emerge through this approach: solvability conditions valid for both regular and singular problems, and an LMI-based parametrization of all H_∞-suboptimal controllers, including reduced-order controllers. The solvability conditions involve Riccati inequalities rather than the usual indefinite Riccati equations. Alternatively, these conditions can be expressed as a system of three LMIs. Efficient convex optimization techniques are available to solve this system. Moreover, its solutions parametrize the set of H_∞ controllers and bear important connections with the controller order and the closed-loop Lyapunov functions. Thanks to such connections, the LMI-based characterization of H_∞ controllers opens new perspectives for the refinement of H_∞ design. Applications to cancellation-free design and controller order reduction are discussed and illustrated by examples.     

A variational approach to multi-sensor fusion of images

Past research into multi-modality sensor data fusion has given rise to approaches that are generally heuristic and ad hoc. In this paper we utilize the calculus of variations as the underlying framework for fusing registered images of different modalities when models relating these modalities are available. The result is a mathematically rigorous method for improving the accuracy with which parameters can be estimated. Using both dense and sparse simulated range and intensity data, the proposed approach is demonstrated on the problem of estimating the surface representing the three dimensional structure of a scene. The results indicate that a four to five-fold increase in surface estimation accuracy with respect to the original input data can be realized. Furthermore, an 8%–250% increase in accuracy over surface estimation from each sensing modality alone (i.e., via shape from shading or surface reconstruction) can be realized.H. Pien is supported by Draper Laboratory under IR&D No. 451; J. Gauch is partially supported by the National Science Foundation under Grant IRI-9109431.     

A variational inequality approach to the inelastic stress-unilateral analysis of cable-structures

The present paper deals with the inelastic, stress-unilateral analysis of cable-structures undergoing large displacements. The response of the cable-structures to load- and initial strain-increments is described by a set of equations and inequalities, which are formulated compactly as variational inequalities. Two dual extremum principles—generalizations of the minimum of the potential and complementary energy—with inequalities as subsidiary conditions are derived. The numerical calculation is preformed iteratively by using the decomposition techniques of the multilevel optimization and applying in the resulting substructures the algorithms of non linear optimization. As an example the elastic, elastic-perfectly plastic and workhardening behaviour of a cable-structure is examined and some useful results from the comparison of the solutions are obtained.     

Evolutionary variational inequality with long-term memory and applications to economic networks

The aim of this paper is to present the relation between an evolutionary variational inequality with long-term memory and Lagrange multipliers. More precisely, we study the oligopolistic market equilibrium problem in which the profit function depends also on previous events of the market by means of a long-term memory which takes into account the previous states of the equilibrium. Moreover, thanks to the variational formulation, we are able to show existence and regularity results for equilibrium solutions. Then, we apply the infinite dimensional duality theory through which we obtain the existence of Lagrange multipliers which are great utility in order to understand the behaviour of the market. Finally, an example is provided, which allows to analyse the influence of the long-term memory on the equilibrium solution.     

A variational inequality for measurement feedback almost-dissipative control

In this paper, an optimal measurement feedback control problem that yields an almost- (or practically) dissipative closed loop system is considered. That is, the aim is to consider optimal control problems which, when solved, yield a closed loop system which almost satisfies a dissipation inequality. The main idea is that by weakening the required dissipation inequality, a broader class of open loop systems and controllers are admissible, leading to broader application. In obtaining the main results of this paper, dynamic programming is applied to the optimal control problem of interest to derive a variational inequality that generalizes the information state based partial differential equation associated with measurement feedback nonlinear dissipative control. This variational inequality can in principle be used to derive the optimal controller. In the special case of certainty equivalence, an explicit solution of the variational inequality exists and is a functional of the solution of the corresponding optimal state feedback almost-dissipative control problem.     

Convergence analysis and error estimates of the element-free Galerkin method for a class of parabolic evolutionary variational inequalities

This paper is presented for the convergence analysis of the element-free Galerkin method for a class of parabolic evolutionary variational inequalities arising from the heat-servo control problem. The error estimates illustrate that the convergence order depends not only on the number of basis functions in the moving least-squares approximation but also the relationship with the time step and the spatial step. Numerical examples verify the convergence analysis and the error estimates.