非线性最优化的抽象算法模型及非闭性收敛条件的研究(下)

本文叙述了具有单调性的最优化算法的若干重要的收敛性条件,包括这方面最近的新成果,并且证明了新的收敛性条件比文献中已有的条件要严格地弱;其次讨论了常见的可行点算法类的一致可行性收敛条件,证明了本文介绍的新的收敛性条件比一致可行性收敛条件要弱。 1.单调最优化算法的全局收敛性大多数具体的最优化算法是单调算法,即对应于迭代点列{x_i}的某一函数f(目标函数或特定的另一函数)的值{f(x_i)}是单调数列,所以文献中对于单调的抽象算法模型的全局收敛性研究很多。Zangwill提出的第一个抽象算法和相应的收敛性条件就是关于单调算法的。对于这类算法,函数值{f(x_i)}的单调性与算法的全局收敛性有密切关系。一般而言,单调算法的收敛性条件比较简单些,见文献以[1～6],[8～15]。在文献[12]中,     

伪单调广义混合变分不等式的改进k步预解算法

在算子的分裂技巧基础上介绍了求解伪单调广义混合变分不等式的改进五步预解算法，算法的收敛性只要求算子的g-伪单调和g—Lipschitz连续性，算子的伪单调比单调更弱．本文的新算法推广了文献中某些已有的结果．     

非线性最优化的抽象算法模型及非闭性收敛条件的研究(上)

本文叙述了若干种具有重要意义的抽象算法模型的结构,以及相应的收敛性条件。其次叙述了在去掉闭性和严格单调性的限制下目前得到的几组新的收敛性条件,并且对一些重要的收敛条件的关系进行了讨论。最后考虑了一种广义单调算法,讨论了相应的收敛性结果。 1.抽象算法模型非线性最优化算法在六十年代有了迅速的发展,例如无约束最优化问题的变尺度算法     

Elman网络梯度学习法的收敛性

考虑有限样本集上Elman网络梯度学习法的确定性收敛性．证明了误差函数的单调递减性．给出了一个弱收敛性结果和一个强收敛结果,表明误差函数的梯度收敛于0,权值序列收敛于固定点．通过数值例子验证了理论结果的正确性．     

矩阵分裂的单调收敛性

本文在非负矩阵分裂条件下证明了迭代算法(3)的单调收敛性,它不仅推广了[1]～[5]中的相应结果,而且在比[7]中定理较弱的条件下,得到了广义AOR迭代法的单调收敛性。本文最后还给出了一个数值例子。     

求解二阶锥绝对值方程组的非单调光滑牛顿算法

本文提出了求解二阶锥绝对值方程组(SOCAVE)的非单调光滑牛顿算法.在适当的条件下分析了算法的全局收敛性和局部二次收敛性.数值结果表明用非单调光滑牛顿算法求解SOCAVE是可行且高效的.     

一类非线性Signorini问题的有限元逼近

本文讨论了一类与非线性单调算子相联系的变分不等式问题——一类非线性Signorini问题。证明了解的存在性和唯一性,给出解的一个表征性质。随后,构造了问题的一个有限元逼近格式;得到了有限元近似解的收敛性结果和误差估计,关键词:非线性单调算子,变分不等式,Signorini问题,有限元逼近,收敛性,误差估计。     

新的非单调线搜索规则BFGS算法的全局收敛性

本文在Zhang H.C.的非单调线搜索规则的基础上,设计了求解无约束最优化问题的新的非单调线搜索BFGS算法,在一定 的条件下证明了算法的线性收敛性和超线性收敛性分析.数值例子表明算法是有效的.     

一类非线性不连续集值发展型方程的广义单调迭代法

本文利用广义单调迭代法研究了一类非线性不连续集值发展型方程的数值解法,利用序理论给出其迭代格式,得到了迭代解的收敛性结果.在一种较弱的条件下,给出了离散解集收敛性的若干结论.     

一类锥模型非单调信赖域算法及收敛性分析

本文对无约束优化问题提出了一类基于锥模型的非单调信赖域算法.二次模型非单调信赖域算法是新算法的特例.在适当的条件下,证明了算法的全局收敛性及Q-二次收敛性.     

Tight global linear convergence rate bounds for Douglas–Rachford splitting

Recently, several authors have shown local and global convergence rate results for Douglas–Rachford splitting under strong monotonicity, Lipschitz continuity, and cocoercivity assumptions. Most of these focus on the convex optimization setting. In the more general monotone inclusion setting, Lions and Mercier showed a linear convergence rate bound under the assumption that one of the two operators is strongly monotone and Lipschitz continuous. We show that this bound is not tight, meaning that no problem from the considered class converges exactly with that rate. In this paper, we present tight global linear convergence rate bounds for that class of problems. We also provide tight linear convergence rate bounds under the assumptions that one of the operators is strongly monotone and cocoercive, and that one of the operators is strongly monotone and the other is cocoercive. All our linear convergence results are obtained by proving the stronger property that the Douglas–Rachford operator is contractive.     

弱相对非扩张映像不动点单调CQ算法与应用

Kamimura和Takahashi$^{[7]}$证明了相对非扩张映像CQ迭代算法的强收敛定理.该文构造了单调CQ算法, 用来逼近弱相对非扩张映像不动点, 证明了强收敛定理. 并将结果应用于逼近Banach空间极大单调算子的零点. 单调CQ算法比目前的CQ算法收敛速度快. 另外, 为证明弱相对非扩张映像不动点强收敛定理,该文运用了新的Cauchy列证明方法, 而不用Kadec-Klee性质, 该文结果改进了S.Matsushita 和 W.Takahashi及其它人的结果.     

Monotone convergence of finite element approximations of obstacle problems

The purpose of the work is to study the monotone convergence of numerical solutions of obstacle problems under mesh refinement when the obstacle is convex. We prove monotone convergence of piecewise linear finite element approximations for one-dimensional obstacle problems. We demonstrate by giving a example that such monotone convergence will not hold in the two-dimensional case.     

单调集函数的连续性与可测函数序列的收敛

引了单调集函数的几种连续性并且讨论了它们与可测函数依测度收敛之间的关系，给出可加测度论中的Lesbegue定理在单调测度空间上的4种推广形式。讨论单调集函数的连续性和模糊积分与Choquet积分的单调收敛定理之间的等价性。证明Choquet积分的控制收敛定理。     

Convergence of Finite Volume Schemes for Hamilton-Jacobi Equations with Dirichlet Boundary Conditions

We study numerical methods for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. We first propose a new class of abstract monotone approximation schemes and get a convergence rate of 1/2 . Then, according to the abstract convergence results, by newly constructing monotone finite volume approximations on interior and boundary points, we obtain convergent finite volume schemes for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. Finally give some numerical results.     

Uniform quadratic convergence of monotone iterates for nonlinear singularly perturbed parabolic problems

This paper deals with a monotone iterative method for solving nonlinear singularly perturbed parabolic problems. Monotone sequences, based on the method of upper and lower solutions, are constructed for a nonlinear difference scheme which approximates the nonlinear parabolic problem. This monotone convergence leads to the existence-uniqueness theorem. The monotone sequences possess quadratic convergence rate. An analysis of uniform convergence of the monotone iterative method to the solutions of the nonlinear difference scheme and to the continuous problem is given. Numerical experiments are presented.     

Graphical convergence of sums of monotone mappings

This paper gives sufficient conditions for graphical convergence of sums of maximal monotone mappings. The main result concerns finite-dimensional spaces and it generalizes known convergence results for sums. The proof is based on a duality argument and a new boundedness result for sequences of monotone mappings which is of interest on its own. An application to the epi-convergence theory of convex functions is given. Counterexamples are used to show that the results cannot be directly extended to infinite dimensions.

     

Global convergence analysis of a class of epidemic models

This paper addresses the global convergence of the epidemic models whose infected subsystems are monotone in the sense of Hirsch (1984). By invoking results from monotone system theory and nonlinear control theory, a simple method is proposed for determining the global asymptotic stability of a disease free equilibrium (DFE) and the global convergence to an endemic equilibrium (EE). Typical epidemic models are studied to illustrate the applicability of the proposed methodology.     

On the convergence of a linear recursive sequence with nonnegative coefficients

A simple proof to some known results on the convergence of linear recursive sequences with nonnegative coefficients is given, using the technique of monotone convergence.     

A First-Order Stochastic Primal-Dual Algorithm with Correction Step

In this article, we investigate the convergence properties of a stochastic primal-dual splitting algorithm for solving structured monotone inclusions involving the sum of a cocoercive operator and a composite monotone operator. The proposed method is the stochastic extension to monotone inclusions of a proximal method studied in the literature for saddle point problems. It consists in a forward step determined by the stochastic evaluation of the cocoercive operator, a backward step in the dual variables involving the resolvent of the monotone operator, and an additional forward step using the stochastic evaluation of the cocoercive operator introduced in the first step. We prove weak almost sure convergence of the iterates by showing that the primal-dual sequence generated by the method is stochastic quasi-Fejér-monotone with respect to the set of zeros of the considered primal and dual inclusions. Additional results on ergodic convergence in expectation are considered for the special case of saddle point models.