We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit funct...
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We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original variational inequality problem is reformulated as simple bounded minimization. Under certain assumptions, we show that any stationary point of the optimization problem is a solution of the problem considered. Finally, we propose a descent method for the variational inequality problem and prove its global convergence.
作者：
Peng, JMAssistant Professor
State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific Engineering Computing Academia Sinica Beijing China
The implicit Lagrangian has attracted much attention recently because of its utility in reformulating complementarity and variational inequality problems as unconstrained minimization problems, II was first proposed b...
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The implicit Lagrangian has attracted much attention recently because of its utility in reformulating complementarity and variational inequality problems as unconstrained minimization problems, II was first proposed by Mangasarian and Solodov as a merit function for the nonlinear complementarity problem (Ref. 1). Three open problems were also raised in the same paper, This paper addresses, among other issues, one of these problems by giving the properties of the implicit Lagrangian and establishing its convexity under appropriate assumptions.
Many stochastic models in queueing, inventory, communications, and dam theories, etc., result in the problem of numerically determining the minimal nonnegative solutions for a class of nonlinear matrix equations. Vari...
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Many stochastic models in queueing, inventory, communications, and dam theories, etc., result in the problem of numerically determining the minimal nonnegative solutions for a class of nonlinear matrix equations. Various iterative methods have been proposed to determine the matrices of interest. We propose a new, efficient successive-substitution Moser method and a Newton-Moser method which use the Moser formula (which, originally, is just the Schulz method). These new methods avoid the inverses of the matrices, and thus considerable savings on the computational workloads may be achieved. Moreover, they are much more suitable for implementing on parallel multiprocessor systems. Under certain conditions, we establish monotone convergence of these new methods, and prove local linear convergence for the substitution Moser method and superlinear convergence for the Newton-Moser method. (C) 1997 Elsevier Science Inc.
For the large-scale system of linear equations with symmetric positive definite block coefficient matrix resulting from the discretization of a self-adjoint elliptic boundary-value problem, by making use of the blocke...
For the large-scale system of linear equations with symmetric positive definite block coefficient matrix resulting from the discretization of a self-adjoint elliptic boundary-value problem, by making use of the blocked multilevel iteration idea, we construct preconditioning matrices for the coefficient matrix and set up a class of parallel hybrid algebraic multilevel iterative methods for solving this kind of system of linear equations. Theoretical analysis shows that not only do these new methods lend themselves to parallel computation, but also their convergence rates are independent of both the sizes and the level numbers of the grids, and their computational work loads are also hounded by linear functions of the step sizes of the finest grids. (C) 1997 Elsevier Science Inc.
A class of parallel hybrid iteration method and its accelerated overrelaxation variant are established for solving the large sparse block bordered system of linear equations, and their convergence are proved when the ...
A class of parallel hybrid iteration method and its accelerated overrelaxation variant are established for solving the large sparse block bordered system of linear equations, and their convergence are proved when the coefficient matrix of the linear system is an M-matrix, an II-matrix, and a symmetric positive definite matrix, respectively. (C) Elsevier Science Inc., 1997.
The hydrogen bonded ammonia chain model is studied by means of the standard self-consistent phonon approach and two modified versions. The effective crystal constant, force constant, free energy, and ratio of the firs...
The hydrogen bonded ammonia chain model is studied by means of the standard self-consistent phonon approach and two modified versions. The effective crystal constant, force constant, free energy, and ratio of the first-order free energy to the zero order as a function of temperature are numerically obtained with the three approaches, and then compared. The standard approach gives the lowest free energy. Violation of the convexity is found in one of the modified approaches near the temperature which is regarded as the melting temperature.
We set up a class of multi-parameter relaxed parallel matrix multisplitting methods for solving the linear complementarity problems on the SIMD multiprocessor systems. This class of methods can not only includes all t...
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We set up a class of multi-parameter relaxed parallel matrix multisplitting methods for solving the linear complementarity problems on the SIMD multiprocessor systems. This class of methods can not only includes all the existing relaxed methods for the linear complementarity problems, but also can yields a lot of novel ones in the sense of multisplitting. Thus, it is reasonably general. We set up the convergence theory of these relaxed methods under the condition that the system matrix is an H-matrix with positive diagonal elements.
This paper investigates the global convergence properties of the Fletcher-Reeves (FR) method for unconstrained optimization. In a simple way, we prove that a kind of inexact line search condition can ensure the conver...
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This paper investigates the global convergence properties of the Fletcher-Reeves (FR) method for unconstrained optimization. In a simple way, we prove that a kind of inexact line search condition can ensure the convergence of the FR method. Several examples are constructed to show that, if the search conditions are relaxed, the FR method may produce an ascent search direction, which implies that our result cannot be improved.
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