一类新的曲线搜索下的多步下降算法

提出一类新的曲线搜索下的多步下降算法,在较弱条件下证明了算法具有全局收敛性和线性收敛速率.算法利用前面多步迭代点的信息和曲线搜索技巧产生新的迭代点,收敛稳定,不用计算和存储矩阵,适于求解大规模优化问题.数值试验表明算法是有效的.     

一类新的多步曲线搜索下的超记忆梯度法

研究一类新的求解无约束优化问题的超记忆梯度法,分析了算法的全局收敛性和线性收敛速率.算法利用一种多步曲线搜索准则产生新的迭代点,在每步迭代时同时确定下降方向和步长,并且不用计算和存储矩阵,适于求解大规模优化问题.数值试验表明算法是有效的.     

等式约束优化非单调信赖域算法

设计了一个新的求解等式约束优化问题的非单调信赖域算法.该算法不需要罚函数也无需滤子.在每次迭代过程中只需求解满足下降条件的拟法向步及切向步.新算法产生的迭代步比滤子方法更易接受,计算量比单调算法小.在一般条件下,算法具有全局收敛性.     

非单调多步曲线搜索方法的收敛性

提出一种求解无约束优化问题的非单调多步曲线搜索方法.此方法具有如下特点:(1)算法在产生下一个迭代点时不仅利用了当前迭代点的信息,而且还可能利用前m个迭代点的信息.这就是多步法;(2)下降方向和步长同时确定,而不是先找到方向,再由线性搜索寻找步长.这就是曲线搜索技术;(3)采用非单调搜索技巧.在较弱的条件下,我们证明了此方法的收敛性.     

基于(A,η)-极大单调算子的强收敛定理

采用三步迭代算法研究(A,η)-极大单调算子的不动点问题及用预解算子研究包含问题的解.同时给出了在相关条件下,由三步迭代算法迭代产生出的数列的强收敛性.文中的算法是在Noor,Huang[1]的两步算法产生的弱收敛定理及Ram U Verma[2]的关于解决包含问题解的方法的启发下得到.     

非光滑凸规划不可行拟牛顿束方法的收敛性分析

利用改进函数将非光滑凸约束优化问题转化成无约束优化问题,构造了一个具有迫近形式的不可行拟牛顿束算法.值得注意的是,随着每次迭代的进行,该算法的无约束优化子问题的目标函数可能发生改变(取零步目标函数不改变,取下降步则更新目标函数),为此必须做必要的调整以保证算法的收敛性.本文主要采用了Sagastizabal和So1odov的不可行束方法的思想,在每个迭代点不一定是原始可行的情况下,得出了算法产生序列的每一个聚点是原问题最优解的收敛性结果.进一步,本文针对目标函数强凸情况下的BFGS拟牛顿算法,得到了全局收敛结果中保证拟牛顿矩阵有界的条件以及迭代序列的R-线性收敛结果.     

一类带线搜索的自适应信赖域算法

本文对无约束优化问题提出了一类带线搜索的自适应信赖域算法,新算法在试验步失败时不重解子问题,而是采用线搜索,从而减少了计算量,不同于一般的带线搜索的信赖域算法,新算法根据实际下降量与预估下降量的比值按照变化的速率对信赖域半径进行调整.文中在一定的条件下证明了算法的收敛性,并且给出了相应的数值实验结果.     

大规模多设施Weber问题的改进Cooper算法

 多设施Weber问题（multi-source Weber problem,MWP）是设施选址中的重要模型之一,而Cooper算法是求解MWP最为常用的数值方法.Cooper算法包含选址步和分配步,两步交替进行直至达到局部最优解.本文对Cooper算法的选址步和分配步分别引入改进策略,提出改进Cooper算法：选址步中将Weiszfeld算法和adaptive Barzilai-Borwein （ABB）算法结合,提出收敛速度更快的ABB-Weiszfeld算法求解选址子问题;分配步中提出贪婪簇分割策略来处理退化设施,由此进一步提出具有更好性质的贪婪混合策略.数值实验表明本文提出的改进策略有效地提高了Cooper算法的计算效率,改进算法有着更好的数值表现.     

非凸优化带松弛步的对称ADMM局部线性收敛率分析

基于对称交替方向乘子法(ADMM),结合松弛步技巧,该文提出一种带松弛步的对称ADMM用于求解两分块线性约束非凸优化问题.同时,新算法乘子更新步采用不同的松弛因子.常规假设下,给出新算法子序列的收敛性证明.误差界条件下,分析并获得由新算法产生的迭代点列以线性收敛的速率局部趋于问题稳定点,相应增广拉格朗日函数序列亦线性收敛.最后,初步试验结果表明新算法是有效的.     

用MATHEMATICA求解商人渡河问题

提出了一种求解商人渡河问题的算法,并给出了用数学软件MATHEMATICA实现该算法的源代码,列出了部分计算结果供进一步研究.该算法可以应用于求解更一般的多步决策问题.     

一类新的记忆梯度法及其全局收敛性

研究了求解无约束优化问题的记忆梯度法,利用当前和前面迭代点的信息产生下降方向,得到了一类新的无约束优化算法,在Wolfe线性搜索下证明了其全局收敛性.新算法结构简单,不用计算和存储矩阵,适于求解大型优化问题.数值试验表明算法有效.     

A Non-Interior Continuation Algorithm for the P0 or P* LCP with Strong Global and Local Convergence Properties

We propose a non-interior continuation algorithm for the solution of the linear complementarity problem (LCP) with a P₀ matrix. The proposed algorithm differentiates itself from the current continuation algorithms by combining good global convergence properties with good local convergence properties under unified conditions. Specifically, it is shown that the proposed algorithm is globally convergent under an assumption which may be satisfied even if the solution set of the LCP is unbounded. Moreover, the algorithm is globally linearly and locally superlinearly convergent under a nonsingularity assumption. If the matrix in the LCP is a P_* matrix, then the above results can be strengthened to include global linear and local quadratic convergence under a strict complementary condition without the nonsingularity assumption.     

Solving monotone inclusions with linear multi-step methods

 In this paper a new class of proximal-like algorithms for solving monotone inclusions of the form T(x)∋0 is derived. It is obtained by applying linear multi-step methods (LMM) of numerical integration in order to solve the differential inclusion , which can be viewed as a generalization of the steepest decent method for a convex function. It is proved that under suitable conditions on the parameters of the LMM, the generated sequence converges weakly to a point in the solution set T ⁻¹ (0). The LMM is very similar to the classical proximal point algorithm in that both are based on approximately evaluating the resolvants of T. Consequently, LMM can be used to derive multi-step versions of many of the optimization methods based on the classical proximal point algorithm. The convergence analysis allows errors in the computation of the iterates, and two different error criteria are analyzed, namely, the classical scheme with summable errors, and a recently proposed more constructive criterion. Received: April 2001 / Accepted: November 2002 Published online: February 14, 2003 Key Words. proximal point algorithm – monotone operator – numerical integration – strong stability – relative error criterion Mathematics Subject Classification (1991): 20E28, 20G40, 20C20     

An SQP-type algorithm for nonlinear second-order cone programs

We propose an SQP-type algorithm for solving nonlinear second-order cone programming (NSOCP) problems. At every iteration, the algorithm solves a convex SOCP subproblem in which the constraints involve linear approximations of the constraint functions in the original problem and the objective function is a convex quadratic function. Those subproblems can be transformed into linear SOCP problems, for which efficient interior point solvers are available. We establish global convergence and local quadratic convergence of the algorithm under appropriate assumptions. We report numerical results to examine the effectiveness of the algorithm. This work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science.     

具简单界约束变分不等式的拟牛顿算法的收敛性分析

１．引言 牛顿型方法是解变分不等式的一类重要数值迭代算法．其局部收敛性质的研究也取得了很好的成果（见［５］等）．近几年来,此类算法的全局收敛性研究也得到了许多进展．如阻尼牛顿法的局部超线性乃至二阶收敛性质的研究（见［４,６,９; １１, １２, １４; １６］等）．然而,对于计算上更为实用的拟牛顿法的研究还不多见．文［１８］基于祁力群等在［１４］中给出的逐次逼近牛顿型法,建立了一种解非线性互补问题的拟牛顿法,并得到了类Ｂｒｏｙｄｅｎ算法的全局收敛性．但是,该方法有以下两个缺陷：１．线搜索可能不能实现…     

On quadratic andO\left( {\sqrt {nL} } \right) convergence of a predictor—corrector algorithm for LCP

Recently several new results have been developed for the asymptotic (local) convergence of polynomial-time interior-point algorithms. It has been shown that the predictor—corrector algorithm for linear programming (LP) exhibits asymptotic quadratic convergence of the primal—dual gap to zero, without any assumptions concerning nondegeneracy, or the convergence of the iteration sequence. In this paper we prove a similar result for the monotone linear complementarity problem (LCP), assuming only that a strictly complementary solution exists. We also show by example that the existence of a strictly complementarity solution appears to be necessary to achieve superlinear convergence for the algorithm.Research supported in part by NSF Grants DDM-8922636 and DDM-9207347, and an Interdisciplinary Research Grant of the University of Iowa, Iowa Center for Advanced Studies.     

An Algorithm for Continuous Type Optimal Location Problem

In this paper, we extend the ordinary discrete type facility location problems to continuous type ones. Unlike the discrete type facility location problem in which the objective function isn't everywhere differentiable, the objective function in the continuous type facility location problem is strictly convex and continuously differentiable. An algorithm without line search for solving the continuous type facility location problems is proposed and its global convergence, linear convergence rate is proved. Numerical experiments illustrate that the algorithm suggested in this paper have smaller amount of computation, quicker convergence rate than the gradient method and conjugate direction method in some sense.     

Superlinear Convergence of a Newton-Type Algorithm for Monotone Equations

We consider the problem of finding solutions of systems of monotone equations. The Newton-type algorithm proposed in Ref. 1 has a very nice global convergence property in that the whole sequence of iterates generated by this algorithm converges to a solution, if it exists. Superlinear convergence of this algorithm is obtained under a standard nonsingularity assumption. The nonsingularity condition implies that the problem has a unique solution; thus, for a problem with more than one solution, such a nonsingularity condition cannot hold. In this paper, we show that the superlinear convergence of this algorithm still holds under a local error-bound assumption that is weaker than the standard nonsingularity condition. The local error-bound condition may hold even for problems with nonunique solutions. As an application, we obtain a Newton algorithm with very nice global and superlinear convergence for the minimum norm solution of linear programs.This research was supported by the Singapore-MIT Alliance and the Australian Research Council.