变分不等式的几类求解方法

本文转为系统地分析和概述了变分不等式问题中几类占有重要地位的求解方法,包括方法产生的背景,主要结果及应用等,这几类算法分别为连续算法,（拟）牛顿型算法,一般迭代模型,投影算法,投影收缩算法等。     

Variable metric methods for minimizing a class of nondifferentiable functions

We develop a class of methods for minimizing a nondifferentiable function which is the maximum of a finite number of smooth functions. The methods proceed by solving iteratively quadratic programming problems to generate search directions. For efficiency the matrices in the quadratic programming problems are suggested to be updated in a variable metric way. By doing so, the methods possess many attractive features of variable metric methods and can be viewed as their natural extension to the nondifferentiable case. To avoid the difficulties of an exact line search, a practical stepsize procedure is also introduced. Under mild assumptions the resulting method converge globally.Research supported by National Science Foundation under grant number ENG 7903881.     

Path-following barrier and penalty methods for linearly constrained problems

In the present paper some barrier and penalty methods (e.g. logarithmic barriers, SUMT, exponential penalties), which define a continuously differentiable primal and dual path, applied to linearly constrained convex problems are studied, in particular, the radius of convergence of Newton’s method depending on the barrier and penalty para-meter is estimated, Unlike using self-concordance properties the convergence bounds are derived by direct estimations of the solutions of the Newton equations. The obtained results establish parameter selection rules which guarantee the overall convergence of the considered barrier and penalty techniques with only a finite number of Newton steps at each parameter level. Moreover, the obtained estimates support scaling method which uses approximate dual multipliers as available in barrier and penalty methods     

Adaptive methods for solvings minimax problems ∗

We consider finite-dimensional minimax problems for two traditional models: firstly,with box constraints at variables and,secondly,taking into account a finite number of tinear inequalities. We present finite exact primal and dual methods. These methods are adapted to a great extent to the specific structure of the cost function which is formed by a finite number of linear functions. During the iterations of the primal method we make use of the information from the dual problem, thereby increasing effectiveness. To improve the dual method we use the “long dual step” rule (the principle of ullrelaxation).The results are illustrated by numerical experiments.     

Generalized semi-infinite programming: numerical aspects

Generalized semi-infinite optimization problems (GSIP) are considered. It is investigated how the numerical methods for standard semi-infinite programming (SIP) can be extended to GSIP. Newton methods can be extended immediately. For discretization methods the situation is more complicated. These difficulties are discussed and convergence results for a discretization- and an exchange method are derived under fairly general assumptions on GSIP. The question is answered under which conditions GSIP represents a convex problem.     

A weighted gram-schmidt method for convex quadratic programming

Range-space methods for convex quadratic programming improve in efficiency as the number of constraints active at the solution decreases. In this paper we describe a range-space method based upon updating a weighted Gram-Schmidt factorization of the constraints in the active set. The updating methods described are applicable to both primal and dual quadratic programming algorithms that use an active-set strategy. Many quadratic programming problems include simple bounds on all the variables as well as general linear constraints. A feature of the proposed method is that it is able to exploit the structure of simple bound constraints. This allows the method to retain efficiency when the number ofgeneral constraints active at the solution is small. Furthermore, the efficiency of the method improves as the number of active bound constraints increases. This research was supported by the U.S. Department of Energy Contract DE-AC03-76SF00326, PA No. DE-AT03-76ER72018; National Science Foundation Grants MCS-7926009 and ECS-8012974; the Office of Naval Research Contract N00014-75-C-0267; and the U.S. Army Research Office Contract DAAG29-79-C-0110. The work of Nicholas Gould was supported by the Science and Engineering Research Council of Great Britain.     

ASYMPTOTICALLY OPTIMAL SUCCESSIVE OVERRELAXATION METHODS FOR SYSTEMS OF LINEAR EQUATIONS

We present a class of asymptotically optimal successive overrelaxation methods for solving the large sparse system of linear equations. Numerical computations show that these new methods are more efficient and robust than the classical successive overrelaxation method.     

信赖域方法最优曲线性质分析

本文描述了信赖域方法最优曲线在二维子空间内投影的几个性质,分析了几种信赖域折线法与该投影的关系,为推导理邹的求解信赖域子问题的折线近似提供理论依据。     

Iterative schemes for quasimonotone mixed variational inequalities

We consider some new iterative methods for solving quasimonotone mixed variational inequalities by updating the solution. These algorithms are based on combining extrapolation and splitting techniques. The convergence analysis of these new methods is considered. These new methods are versatile and are easy to implement. Our method of proof of convergence is very simple and uses either monotonicity or quasimonotonicity of the operator.     

Optimal approximation of sparse hessians and its equivalence to a graph coloring problem

We consider the problem of approximating the Hessian matrix of a smooth non-linear function using a minimum number of gradient evaluations, particularly in the case that the Hessian has a known, fixed sparsity pattern. We study the class of Direct Methods for this problem, and propose two new ways of classifying Direct Methods. Examples are given that show the relationships among optimal methods from each class. The problem of finding a non-overlapping direct cover is shown to be equivalent to a generalized graph coloring problem—the distance-2 graph coloring problem. A theorem is proved showing that the general distance-k graph coloring problem is NP-Complete for all fixedk≥2, and hence that the optimal non-overlapping direct cover problem is also NP-Complete. Some worst-case bounds on the performance of a simple coloring heuristic are given. An appendix proves a well-known folklore result, which gives lower bounds on the number of gradient evaluations needed in any possible approximation method. This research was partially supported by the Department of Energy Contract AM03-76SF00326. PA#DE-AT03-76ER72018; Army Research Office Contract DAA29-79-C-0110; Office of Naval Research Contract N00014-74-C-0267; National Science Foundation Grants MCS76-81259, MCS-79260099 and ECS-8012974.     

Globally convergent conjugate gradient algorithms

The paper examines a method of implementing an angle test for determining when to restart conjugate gradient methods in a steepest descent direction. The test is based on guaranteeing that the cosine of the angle between the search direction and the negative gradient is within a constant multiple of the cosine of the angle between the Fletcher-Reeves search direction and the negative gradient. This guarantees convergence, for the Fletcher-Reeves method is known to converge. Numerical results indicate little, if anything, is lost in efficiency, and indicate gains may well be possible for large problems.     

A class of methods for linear programming

A class of methods is presented for solving standard linear programming problems. Like the simplex method, these methods move from one feasible solution to another at each iteration, improving the objective function as they go. Each such feasible solution is also associated with a basis. However, this feasible solution need not be an extreme point and the basic solution corresponding to the associated basis need not be feasible. Nevertheless, an optimal solution, if one exists, is found in a finite number of iterations (under nondegeneracy). An important example of a method in the class is the reduced gradient method with a slight modification regarding selection of the entering variable.     

On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem

We present practical conditions under which the existence and uniqueness of a finite solution to a given equality quadratic program may be examined. Different sets of conditions allow this examination to take place when null-space, range-space or Lagrangian methods are used to find stationary points for the quadratic program.This research supported in part by the Natural Sciences and Engineering Research Council, Canada.     

Computational experiments with scaled initial hessian approximation for the broyden family methods ∗

In this paper we consider two alternative choices for the factor used to scale the initial Hessian approximation, before updating by a member of the Broyden family of updates for quasi-Newton optimization methods. By extensive computational experiments carried out on a set of standard test problems from the CUTE collection, using efficient implemen-tations of the quasi-Newton method, we show that the proposed new scaling factors are better, in terms of efficiency achieved (number of iterations, number of function and gradient evaluations), than the standard choice proposed in the literature     

A quasi-Newton method can be obtained from a method of conjugate directions

In a recent paper McCormick and Ritter consider two classes of algorithms, namely methods of conjugate directions and quasi-Newton methods, for the problem of minimizing a function ofn variablesF(x). They show that the former methods possess ann-step superlinear rate of convergence while the latter are every step superlinear and therefore inherently superior. In this paper a simple and computationally inexpensive modification of a method of conjugate directions is presented. It is shown that the modified method is a quasi-Newton method and is thus every step superlinearly convergent. It is also shown that under certain assumptions on the second derivatives ofF the rate of convergence of the modified method isn-step quadratic.This work was supported by the National Research Council of Canada under Research Grant A8189.     

Experiments with primal - dual decomposition and subgradient methods for the uncapacitatied facility location problem

For optimization problems that are structured both with respect to the constraints and with respect to the variables, it is possible to use primal–dual solution approaches, based on decomposition principles. One can construct a primal subproblem, by fixing some variables, and a dual subproblem, by relaxing some constraints and king their Lagrange multipliers, so that both these problems are much easier to solve than the original problem. We study methods based on these subproblems, that do not include the difficult Benders or Dantzig-Wolfe master problems, namely primal–dual subgradient optimization methods, mean value cross decomposition, and several comtbinations of the different techniques. In this paper, these solution approaches are applied to the well-known uncapacitated facility location problem. Computational tests show that some combination methods yield near-optimal solutions quicker than the classical dual ascent method of Erlenkotter     

Some examples of cycling in variable metric methods for constrained minimization

Although variable metric methods for constrained minimization generally give good numerical results, many of their convergence properties are still open. In this note two examples are presented to show that variable metric methods may cycle between two points instead of converging to the required solution.     

Problem structuring methods ‘in the Dock’: Arguing the case for Soft OR

Problem structuring methods (‘soft’ OR) have been around for approximately 40 years and yet these methods are still very much overlooked in the OR world. Whilst there is almost certainly a number of explanations for this, two key stumbling blocks are: (1) the subjective nature of the modelling yielding insights rather than testable results, and (2) the demand on users to both manage content (through modelling) and processes (work with rather than ‘on behalf’ of groups). However, as evidenced from practice there are also a number of significant benefits. This paper therefore aims to examine the case of Soft OR through examining the case for and against problem structuring methods.     

基于模糊理论的软件寿命周期费用评价模型

软件寿命周期费用评价模型涉及到软件开发、使用和维护过程中各种资源最有效利用的权衡分析。由于软件开发不是一门严谨的精确科学,往往存在大量具有不确定性的需求以及许多未知和不确定因素,所有这些都给软件寿命周期费用评价带来模糊效用。本文将模糊理论应用于软件寿命周期费用的评价,给出了从评价属性模糊值的确定、模糊评价模型的建立,到模型求解和最优方案选择的模糊评价方法,并通过对一个算例的分析,证明了该模型的可行性。     

Iterative methods for variational and complementarity problems

In this paper, we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems. Included among such methods are the Newton and several successive overrelaxation algorithms. For the most part, the study is concerned with the family of linear approximation methods. These are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems. Convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches. The paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria. Finally, several convergence results are obtained for some nonlinear approximation methods.This research was based on work supported by the National Science Foundation under grant ECS-7926320.