A projected gradient method with nonmonotonic backtracking technique for solving convex constrained monotone variational inequality problem

Based on a differentiable merit function proposed by Taji, et al in “Mathematical Programming, 1993, 58： 369-383”, a projected gradient trust region method for the monotone variational inequality problem with convex constraints is presented. Theoretical analysis is given which proves that the proposed algorithm is globally convergent and has a local quadratic convergence rate under some reasonable conditions. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm.     

An Affine Scaling Interior Trust Region Method via Optimal Path for Solving Monotone Variational Inequality Problem with Linear Constraints

Based on a differentiable merit function proposed by Taji et al. in ＂Math. Prog. Stud., 58, 1993, 369-383＂, the authors propose an affine scaling interior trust region strategy via optimal path to modify Newton method for the strictly monotone variational inequality problem subject to linear equality and inequality constraints. By using the eigensystem decomposition and affine scaling mapping, the authors form an affine scaling optimal curvilinear path very easily in order to approximately solve the trust region subproblem. Theoretical analysis is given which shows that the proposed algorithm is globally convergent and has a local quadratic convergence rate under some reasonable conditions.     

盒子型区域上变分不等式的一个光滑牛顿法(英文)

The box constrained variational inequality problem can be reformulated as a nonsmooth equation by using median operator.In this paper,we present a smoothing Newton method for solving the box constrained variational inequality problem based on a new smoothing approximation function.The proposed algorithm is proved to be well defined and convergent globally under weaker conditions.     

求解广义线性互补问题的光滑牛顿法(英文)

The extended linear complementarity problem(denoted by ELCP) can be reformulated as the solution of a nonsmooth system of equations. By the symmetrically perturbed CHKS smoothing function, the ELCP is approximated by a family of parameterized smooth equations. A one-step smoothing Newton method is designed for solving the ELCP. The proposed algorithm is proved to be globally convergent under suitable assumptions.     

Convergence of a non-interior continuation algorithm for the monotone SCCP

It is well known that the symmetric cone complementarity problem（SCCP） is a broad class of optimization problems which contains many optimization problems as special cases.Based on a general smoothing function,we propose in this paper a non-interior continuation algorithm for solving the monotone SCCP.The proposed algorithm solves at most one system of linear equations at each iteration.By using the theory of Euclidean Jordan algebras,we show that the algorithm is globally linearly and locally quadratically convergent under suitable assumptions.     

线性互补问题的一种新Lagrange乘子法

A new multiplier method for solving the linear complementarity problem LCP(q, M) is proposed. Based on the Lagrangian of LCP(q,M) introduced here, we construct a new differentiable merit function θ(x,λ) which containing a multiplier vector λ and satisfying θ(x,λ) ≥ 0 and θ(x,λ) = 0 if and if only x solves LCP(q,M). A simple damped Newton-type algorithm which based on the merit function θ(x,λ) is presented. The main feature of the method is that the multiplier self-adjusting step accelerates the local convergence rate without losing global convergence. When M is the P-matrix, the sequence {θ(x^k,λ^k)}where {(x^k,λ^k)} generated by the algorithm is globally linearly convergent to zero and convergent in finite number of iterations if the solution is nondegenerate. Numerical results suggest that the method is high efficient and promising.     

A GLOBAL LINEAR AND LOCAL QUADRATIC SINGLE-STEP NONINTERIOR CONTINUATION METHOD FOR MONOTONE SEMIDEFINITE COMPLEMENTARITY PROBLEMS

A noninterior continuation method is proposed for semidefinite complementarity problem (SDCP). This method improves the noninterior continuation methods recently developed for SDCP by Chen and Tseng. The main properties of our method are: (i) it is well defined for the monotones SDCP; (ii) it has to solve just one linear system of equations at each step; (iii) it is shown to be both globally linearly convergent and locally quadratically convergent under suitable assumptions.     

通过广义D-间隙函数求解变分不等式问题的全局收敛性和误差界估计

The variational inequality problem can be reformulated as an unconstrained minimization problem through the D-gap function. Recently,Peng proposed a hybrid Newton-type method for minimizing the D-gap function.In this paper,a modification with generalized D-gap function gαβ of the method proposed by Peng is presented.It is shown that the algorithm has nice global convergence.This result here have improved and generalized those in the literature.Moreover, when the parameter β is chosen in a certain interval, it is proved that the generalized D-gap function gαβ has bounded level sets for the strongly monotone VIP. An error bound estimation of the algorithm is obtained.     

A QP Free Feasible Method

In [12], a QP free feasible method was proposed for the minimization of a smooth function subject to smooth inequality constraints. This method is based on the solutions of linear systems of equations, the reformulation of the KKT optimality conditions by using the Fischer-Burmeister NCP function. This method ensures the feasibility of all iterations. In this paper, we modify the method in [12] slightly to obtain the local convergence under some weaker conditions. In particular, this method is implementable and globally convergent without assuming the linear independence of the gradients of active constrained functions and the uniformly positive definiteness of the submatrix obtained by the Newton or Quasi Newton methods. We also prove that the method has superlinear convergence rate under some mild conditions. Some preliminary numerical results indicate that this new QP free feasible method is quite promising.     

Properties of a family of merit functions and a merit function method for the NCP

A family of merit functions are proposed, which are the generalization of several existing merit functions. A number of favorable properties of the proposed merit functions are established. By using these properties, a merit function method for solving nonlinear complementarity problem is investigated, and the global convergence of the proposed algorithm is proved under some standard assumptions. Some preliminary numerical results are given.     

A TRUST REGION-TYPE METHOD FOR SOLVINGMONOTONE VARIATIONAL INEQUALITY

1.IntroductionLetSbeanonemptyclosedconvexsubsetofR"andletF:R"-R"beacontinuousmapping.ThevariatiollalillequalityproblemFindx*6Ssuchthat(F(x*),x--x*)20forallxeS(VIP)iswidelyusedtostudyvariousequilibriummodelsarisingilleconomic,operatiollsresearch,transportatiollandregionalsciellces[2'3I?where(.,.)dellotestheinnerproductinR".Manyiterativemethodsfor(VIP)havebeendeveloped,forexample,projectionmethods[7ts],thenonlinearJacobimethod[5],thesuccessiveoverrelaxation.ethod[9]andgeneralizedgradient.…     

仿射变换内点信赖域类方法解单调变分不等式问题

基于Taji引入的一类可微的简单边界约束的严格单调变分不等式问题的势函数,本文提出了仿射变换内点信赖域类修正牛顿法.进一步,作者不仅从理论上证明了该算法的整体收敛性,并且在合理的假设条件下,给出了算法具有局部二次收敛速率.     

A globally convergent Newton method for solving strongly monotone variational inequalities

Variational inequality problems have been used to formulate and study equilibrium problems, which arise in many fields including economics, operations research and regional sciences. For solving variational inequality problems, various iterative methods such as projection methods and the nonlinear Jacobi method have been developed. These methods are convergent to a solution under certain conditions, but their rates of convergence are typically linear. In this paper we propose to modify the Newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length. The purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate. It is then shown that, under the strong monotonicity assumption, the method is globally convergent and, under some additional assumptions, the rate of convergence is quadratic. Limited computational experience indicates the high efficiency of the proposed method.     

求解不可微箱约束变分不等式的下降算法

1 引 论 设X(?)Rn是非空闭集,F:Rn→Rn连续映射,变分不等式问题VI(X,F)是指:求x∈X,使 F(x)T(y-x)≥0,　 (?)y∈X,(1)记指标集N=(1,2,…,n},当 X=[a,b]≡{x∈Rn|a≤xi≤bi,i∈N},(2)其中a={a1,a2,…,an}T,b={b1,b2,…,bn}T∈Rn时,VI(X,F)化为箱约束变分不等式VI(a,b,F).若ai=0,bi=+∞,i∈N,即X=R+n≡{x∈Rn|x≥0}时,VI(a,b,F)化为非线性     

箱约束变分不等式的一种新NCP-函数及其广义牛顿法

１．引 言设，变分不等式，记为ＶＩ（Ｘ，Ｆ），是指：求ｘ＝Ｘ使记为箱式约束时，称 ＶＩ（Ｘ，Ｆ）为箱约束变分不等式，记为 ＶＩ（［ａ，ｂ］，Ｆ）．若ａｉ＝０，ｂｉ＝＋∞，　　　　　　　　　　　　　　　　　　　　　　　　　为非线性互补问题ＮＣＰ（Ｆ）：求ｘ∈Ｒ     

Quadratic Convergence of a Long-Step Interior-Point Method for Nonlinear Monotone Variational Inequality Problems

This paper offers an analysis on a standard long-step primal-dual interior-point method for nonlinear monotone variational inequality problems. The method has polynomial-time complexity and its q-order of convergence is two. The results are proved under mild assumptions. In particular, new conditions on the invariance of the rank and range space of certain matrices are employed, rather than restrictive assumptions like nondegeneracy.     

Some projection-like methods for the generalized Nash equilibria

A generalized Nash game is an m-person noncooperative game in which each player’s strategy depends on the rivals’ strategies. Based on a quasi-variational inequality formulation for the generalized Nash game, we present two projection-like methods for solving the generalized Nash equilibria in this paper. It is shown that under certain assumptions, these methods are globally convergent. Preliminary computational experience is also reported.