求解变量带简单界约束的非线性规划问题的信赖域方法

1．引言。本文考虑下述变量带简单界约束的非线性规划问题：问题（1．1）不仅是实际应用中出现的简单的约束最优化问题，而且相当一部分最优化问题可以把变量限制在有意义的区间内181．因此，无论在理论方面还是在实际应用方面，都有必要研究此种问题．给出简便而且有效的算法．有些文章提出了一些特殊的方法．如011和［2］．14］及16］提出了一类信赖域方法，它们都借助于某种辅助点，证明了算法的全局收敛性．在收敛速度的分析方面，除要求在＊－T点满足严格互补松弛外，它们还要求另一个条件，即在每次迭代中，辅助点的有效约束必须在尝…     

On finite convergence of proximal point algorithms for variational inequalities

In this paper, we first characterize finite convergence of an arbitrary iterative algorithm for solving the variational inequality problem (VIP), where the finite convergence means that the algorithm can find an exact solution of the problem in a finite number of iterations. By using this result, we obtain that the well-known proximal point algorithm possesses finite convergence if the solution set of VIP is weakly sharp. As an extension, we show finite convergence of the inertial proximal method for solving the general variational inequality problem under the condition of weak g-sharpness.     

Convergence Properties of Projection and Contraction Methods for Variational Inequality Problems

In this paper we develop the convergence theory of a general class of projection and contraction algorithms (PC method), where an extended stepsize rule is used, for solving variational inequality (VI) problems. It is shown that, by defining a scaled projection residue, the PC method forces the sequence of the residues to zero. It is also shown that, by defining a projected function, the PC method forces the sequence of projected functions to zero. A consequence of this result is that if the PC method converges to a nondegenerate solution of the VI problem, then after a finite number of iterations, the optimal face is identified. Finally, we study local convergence behavior of the extragradient algorithm for solving the KKT system of the inequality constrained VI problem. \keywords{Variational inequality, Projection and contraction method, Predictor-corrector stepsize, Convergence property.} \amsclass{90C30, 90C33, 65K05.} Accepted 5 September 2000. Online publication 16 January 2001.     

A note on conjugate gradient convergence – Part III

Summary. In this paper we again consider the rate of convergence of the conjugate gradient method. We start with a general analysis of the conjugate gradient method for uniformly bounded solutions vectors and matrices whose eigenvalues are uniformly bounded and positive. We show that in such cases a fixed finite number of iterations of the method gives some fixed amount of improvement as the the size of the matrix tends to infinity. Then we specialize to the finite element (or finite difference) scheme for the problem . We show that for some classes of function we see this same effect. For other functions we show that the gain made by performing a fixed number of iterations of the method tends to zero as the size of the matrix tends to infinity. Received July 9, 1998 / Published online March 16, 2000     

On Finite Convergence of Iterative Methods for Variational Inequalities in Hilbert Spaces

In a Hilbert space, we study the finite termination of iterative methods for solving a monotone variational inequality under a weak sharpness assumption. Most results to date require that the sequence generated by the method converges strongly to a solution. In this paper, we show that the proximal point algorithm for solving the variational inequality terminates at a solution in a finite number of iterations if the solution set is weakly sharp. Consequently, we derive finite convergence results for the gradient projection and extragradient methods. Our results show that the assumption of strong convergence of sequences can be removed in the Hilbert space case.     

Multigrid methods with web-splines

Summary. We describe and analyze a multigrid algorithm for finite element approximations of second order elliptic boundary value problems with weightedextended b-splines (web-splines). This new technique provides high accuracy with relatively low-dimensional subspaces, does not require any grid generation, and is ideally suited for hierarchical solution techniques. In particular, we show that the standard W-cycle yields uniform convergence, i.e., the required number of iterations is bounded independent of the grid width. Received August 17, 2000 / Published online August 17, 2001     

On the finite termination of the Douglas-Rachford method for the convex feasibility problem

In this paper we apply the Douglas–Rachford (DR) method to solve the problem of finding a point in the intersection of the interior of a closed convex cone and a closed convex set in an infinite-dimensional Hilbert space. For this purpose, we propose two variants of the DR method which can find a point in the intersection in a finite number of iterations. In order to analyse the finite termination of the methods, we use some properties of the metric projection and a result regarding the rate of convergence of fixed point iterations. As applications of the results, we propose the methods for solving the conic and semidefinite feasibility problems, which terminate at a solution in a finite number of iterations.     

解带有二次约束非凸二次规划问题的一个分枝缩减方法

在这篇论文里,有机地把外逼近方法与分枝定界技术结合起来,提出了解带有二次约束非凸二次规划问题的一个分枝缩减方法;给出了原问题的一个新的线性规划松弛,以便确定它在超矩形上全局最优值的一个下界;利用超矩形的一个深度二级剖分方法,以及超矩形的缩减和删除技术,提高算法的收敛速度;证明了在知道原问题可行点的条件下,该算法在有限步里就可以获得原问题的一个全局最优化解,并且用一个例子说明了该算法是有效的.     

Nonmonotone adaptive trust region method

In this paper, we propose a nonmonotone adaptive trust region method for unconstrained optimization problems. This method can produce an adaptive trust region radius automatically at each iteration and allow the functional value of iterates to increase within finite iterations and finally decrease after such finite iterations. This nonmonotone approach and adaptive trust region radius can reduce the number of solving trust region subproblems when reaching the same precision. The global convergence and convergence rate of this method are analyzed under some mild conditions. Numerical results show that the proposed method is effective in practical computation.     

等式与界约束非线性优化的信赖域增广Lagrangian算法

１．引 言本文讨论如下非线性约束优化问题：其中； 是Ｒｎ→Ｒ的可微函数，　　　　　　．记 问题（１．１）是非线性约束优化问题中的一类重要类型，事实上任一个非线性等式与不等式约束优化均可引入松驰变量转化为（１．１）的形式．因此（１．１）的求解是人们讨论的热点问     

Subgradient Method for Convex Feasibility on Riemannian Manifolds

In this paper, a subgradient type algorithm for solving convex feasibility problem on Riemannian manifold is proposed and analysed. The sequence generated by the algorithm converges to a solution of the problem, provided the sectional curvature of the manifold is non-negative. Moreover, assuming a Slater type qualification condition, we analyse a variant of the first algorithm, which generates a sequence with finite convergence property, i.e., a feasible point is obtained after a finite number of iterations. Some examples motivating the application of the algorithm for feasibility problems, nonconvex in the usual sense, are considered.     

Bias optimality for multichain continuous-time Markov decision processes

This paper deals with the bias optimality of multichain models for finite continuous-time Markov decision processes. Based on new performance difference formulas developed here, we prove the convergence of a so-called bias-optimal policy iteration algorithm, which can be used to obtain bias-optimal policies in a finite number of iterations.     

A Proximal Point-Type Method for Multicriteria Optimization

In this paper, we present a proximal point algorithm for multicriteria optimization, by assuming an iterative process which uses a variable scalarization function. With respect to the convergence analysis, firstly we show that, for any sequence generated from our algorithm, each accumulation point is a Pareto critical point for the multiobjective function. A more significant novelty here is that our paper gets full convergence for quasi-convex functions. In the convex or pseudo-convex cases, we prove convergence to a weak Pareto optimal point. Another contribution is to consider a variant of our algorithm, obtaining the iterative step through an unconstrained subproblem. Then, we show that any sequence generated by this new algorithm attains a Pareto optimal point after a finite number of iterations under the assumption that the weak Pareto optimal set is weak sharp for the multiobjective problem.     

A new finitely convergent algorithm for systems of nonlinear inequalities

In this work, combining the generalized projection techniques with the idea of a strongly sub-feasible direction method, a new algorithm for solving systems of nonlinear inequalities is presented. At each iteration of the proposed algorithm, the search direction is yielded by just one new explicit formula. The proposed algorithm is proved not only to possess global and strong convergence but also to be able to produce a solution in a finite number of iterations. Finally, some interesting numerical results are reported.     

Preconditioners Based on Fundamental Solutions

We consider a new preconditioning technique for the iterative solution of linear systems of equations that arise when discretizing partial differential equations. The method is applied to finite difference discretizations, but the ideas apply to other discretizations too. If E is a fundamental solution of a differential operator P, we have E*(Pu) = u. Inspired by this, we choose the preconditioner to be a discretization of an approximate inverse K, given by a convolution-like operator with E as a kernel. We present analysis showing that if P is a first order differential operator, KP is bounded, and numerical results show grid independent convergence for first order partial differential equations, using fixed point iterations. For the second order convection-diffusion equation convergence is no longer grid independent when using fixed point iterations, a result that is consistent with our theory. However, if the grid is chosen to give a fixed number of grid points within boundary layers, the number of iterations is independent of the physical viscosity parameter. AMS subject classification (2000) 65F10, 65N22     

A New Superlinearly Convergent Strongly Subfeasible Sequential Quadratic Programming Algorithm for Inequality-Constrained Optimization

Combining the ideas of generalized projection and the strongly subfeasible sequential quadratic programming (SQP) method, we present a new strongly subfeasible SQP algorithm for nonlinearly inequality-constrained optimization problems. The algorithm, in which a new unified step-length search of Armijo type is introduced, starting from an arbitrary initial point, produces a feasible point after a finite number of iterations and from then on becomes a feasible descent SQP algorithm. At each iteration, only one quadratic program needs to be solved, and two correctional directions are obtained simply by explicit formulas that contain the same inverse matrix. Furthermore, the global and superlinear convergence results are proved under mild assumptions without strict complementarity conditions. Finally, some preliminary numerical results show that the proposed algorithm is stable and promising.     

Accelerated reflection projection algorithm and its application to the LMI problem

We discuss accelerated version of the alternating projection method which can be applied to solve the linear matrix inequality (LMI) problem. The alternating projection method is a well-known algorithm for the convex feasibility problem, and has many generalizations and extensions. Bauschke and Kruk proposed a reflection projection algorithm for computing a point in the intersection of an obtuse cone and a closed convex set. We carry on this research in two directions. First, we present an accelerated version of the reflection projection algorithm, and prove its weak convergence in a Hilbert space; second, we prove the finite termination of an algorithm which is based on the proposed algorithm and provide an explicit upper bound for the required number of iterations under certain assumptions. Numerical experiments for the LMI problem are provided to demonstrate the effectiveness and merits of the proposed algorithms.     

An improved working set selection for SMO-type decomposition method

The sequential minimization optimization (SMO) is a simple and efficient decomposition algorithm for solving support vector machines (SVMs). In this paper, an improved working set selection and a simplified minimization step are proposed for the SMO-type decomposition method that reduces the learning time for SVM and increases the efficiency of SMO. Since the working set is selected directly according to the Karush–Kuhn–Tucker (KKT) conditions, the minimization step of subproblem is simplified, accordingly the learning time for SVM is reduced and the convergence is accelerated. Following Keerthi’s method, the convergence of the proposed algorithm is analyzed. It is proven that within a finite number of iterations, solution that is based on satisfaction of the KKT conditions will be obtained by using the improved algorithm.     

Preconditioned conjugate gradient algorithms for nonconvex problems with box constraints

The paper describes new conjugate gradient algorithms for large scale nonconvex problems with box constraints. In order to speed up convergence the algorithms employ scaling matrices which transform the space of original variables into the space in which Hessian matrices of the problem’s functionals have more clustered eigenvalues. This is done by applying limited memory BFGS updating matrices. Once the scaling matrix is calculated, the next few conjugate gradient iterations are performed in the transformed space. The box constraints are treated efficiently by the projection. We also present a limited memory quasi-Newton method which is a special version of our general algorithm. The presented algorithms have strong global convergence properties, in particular they identify constraints active at a solution in a finite number of iterations. We believe that they are competitive to the L-BFGS-B method and present some numerical results which support our claim.     

The preconditioned GMRES method for systems of coupled FEM-BEM equations

We analyze the generalized minimal residual method (GMRES) as a solver for coupled finite element and boundary element equations. To accelerate the convergence of GMRES we apply a hierarchical basis block preconditioner for piecewise linear finite elements and piecewise constant boundary elements. It is shown that the number of iterations which is necessary to reach a given accuracy grows only poly-logarithmically with the number of unknowns. This revised version was published online in June 2006 with corrections to the Cover Date.