一类K-单调系统产生的K-单调算子

本文研究了Ｋ－单调系统的解的渐近性质．应用Ｋ－单调算子的性质,得到了保证Ｋ－单调系统的正周期解的存在性、唯一性、全局渐近稳定性的充分条件．     

无限维Hillbert空间中单调变分不等式解的近似迭代算法

在无穷维Hillbert空间中研究了一类单调型变分不等式，把求单调型变分不等式解的问题转化为求强单调变分不等式的解，建立了一种新的迭代算法，并证明了由算法生成的迭代序列强收敛于单调变分不等式的解，从而推广了所列文献中的许多重要结果．     

一类非混合单调算子方程的耦合解定理及其非单调迭代算法

研究无穷维序Banach空间中一类非混合单调算子，它可以表示成T=T1 T2，其中T1是混合单调算子，T2是反向混合单调算子(T2≠0)，得到了其耦合解的存在性定理．当尸是正规极小锥时，通过构造一系列确界生成序列，建立了耦合解的非单调迭代算法．置后，推广了最大一最小解的概念，定义了极大一极小解并研究了其存在的条件．     

求解反应扩散对流问题的并行块单调迭代算法

考虑求解一类非线性反应扩散对流方程的块单调迭代算法,其中包括传统的块Picard,块Jacobi,以及在区域分解算法中常用的并行Schwarz算法.所讨论的算法可从问题的一个上解和下解出发,产生一个上解迭代序列和下解迭代序列并单调收敛于离散问题的解.这类算法的优点在于算法的并行结构好且可直接通过所产生的上解和下解迭代序列,得到迭代解的最大模误差界.在理论上,得到了算法的单调收敛性、线性与超线性收敛性.     

H-单调映象的广义Fuzzy隐拟变分包含

引入了一类H-单调映象的广义Fuzzy隐拟变分包含问题,利用文[1]中H-单调映象的预解算子技巧研究了这类变分包含解的迭代算法逼近,证明了其解的存在性以及由算法生成的迭代序列的收敛性。     

一类带线搜索的非单调信赖域算法

本文对于无约束最优化问题提出了一类新的非单调信赖域算法．与通常的非单调信赖域算法不同，当试探步不成功时，并不重解信赖域子问题，而采用非单调线搜索，从而减小了计算量．在适当的条件下，证明了此算法的全局收敛性．     

一类K-单调系统产生的K-单调算子

本文研究了K-单调系统的解的渐近性质.应用K-单调算子的性质,得到了保证K-单调系统的正周期解的存在性、唯一性、全局渐近稳定性的充分条件.     

非单调投影梯度方法解不等式约束优化

本文提出了投影梯度算法结合非单调信赖技术解不等式约束优化问题，获得了算法的整体收敛性的证明．     

单调全局最优化问题的凸化外逼近算法

单调优化是指目标函数与约束函数均为单调函数的全局优化问题．本文提出一种新的凸化变换方法把单调函数化为凸函数，进而把单调优化问题化为等价的凸极大或凹极小问题，然后采用Hoffman的外逼近方法来求得问题的全局最优解．我们把这种凸化方法同Tuy的Polyblock外逼近方法作了比较，通过数值比较可以看出本文提出的凸化的方法在收敛速度上明显优于Polyblock方法．     

求解单调变分不等式问题的一个连续型迭代方法

本文给出一个求解单调变分不等式问题的连续型迭代方法,对任意单调趋于零的正数序列和任意初始点,方法产生的迭代点列均收敛到所求变分不等式问题的一个解,且在适当条件下方法具有Q-超线性收敛率.数值试验结果进一步表明了所给方法的稳定性和有效性.     

一类Riccati方程组对称自反解的两种迭代算法

针对源于Markov跳变线性二次控制问题中的一类对偶代数Riccati方程组,分别采用修正共轭梯度算法和正交投影算法作为非精确Newton算法的内迭代方法,建立求其对称自反解的非精确Newton-MCG算法和非精确Newton-OGP算法.两种迭代算法仅要求Riccati方程组存在对称自反解,对系数矩阵等没有附加限定.数值算例表明,两种迭代算法是有效的.     

Necessary and sufficient condition for the existence of positive solutions of a coupled system for elastic beam equations

In this work, by using fixed point theorem and monotone iterative technique, the author establishes a necessary and sufficient condition of the existence of positive solutions for a class of nonlinear singular elastic beam differential system under reasonable conditions. Moreover, some uniqueness results of positive solution and the estimate of convergent rate of the iterative sequence of solution are obtained.     

Algorithms for Unconstrained Optimization Problems via Control Theory

Existing algorithms for solving unconstrained optimization problems are generally only optimal in the short term. It is desirable to have algorithms which are long-term optimal. To achieve this, the problem of computing the minimum point of an unconstrained function is formulated as a sequence of optimal control problems. Some qualitative results are obtained from the optimal control analysis. These qualitative results are then used to construct a theoretical iterative method and a new continuous-time method for computing the minimum point of a nonlinear unconstrained function. New iterative algorithms which approximate the theoretical iterative method and the proposed continuous-time method are then established. For convergence analysis, it is useful to note that the numerical solution of an unconstrained optimization problem is none other than an inverse Lyapunov function problem. Convergence conditions for the proposed continuous-time method and iterative algorithms are established by using the Lyapunov function theorem.     

Continuous-time method and its discretization to inverse problem of intensity-modulated radiation therapy treatment planning

We propose a novel approach for solving box-constrained inverse problems in intensity-modulated radiation therapy (IMRT) treatment planning based on the idea of continuous dynamical methods and split-feasibility algorithms. Our method can compute a feasible solution without the second derivative of an objective function, which is required for gradient-based optimization algorithms. We prove theoretically that a double Kullback–Leibler divergence can be used as the Lyapunov function for the IMRT planning system.Moreover, we propose a non-negatively constrained iterative method formulated by discretizing a differential equation in the continuous method. We give proof for the convergence of a desired solution in the discretized system, theoretically. The proposed method not only reduces computational costs but also does not produce a solution with an unphysical negative radiation beam weight in solving IMRT planning inverse problems.The convergence properties of solutions for an ill-posed case are confirmed by numerical experiments using phantom data simulating a clinical setup.     

Banach空间混合型一阶脉冲积分-微分方程初值问题的唯一解

本文在一般的序Banach空间中研究了一阶脉冲混合型积分-微分方程初值问题的唯一解。在比较广泛的上控制条件并且假定所考虑初值问题只有一个上解或下解的假设之下,我们证明了所考虑初值问题的唯一解可以由显形式表达的迭代序列的一致极限得到,并给出了逼近解的迭代序列的误差估计,本文没有使用任何紧型条件。我们的结果是最近一些结果的改进和推广。     

不连续二阶非线性微分方程的周期边值问题的唯一解

在Banach中,本文在很弱条件下,通过迭代序列得到了不连续二阶非线性微分方程的周期边值问题的唯一解存在性的一个充分条件,而且给出了迭代序列近代解的误差估计.     

Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval

By employing the monotone iterative technique, we not only establish the existence of the unique solution for a fractional integral boundary value problem on semi-infinite intervals, but also develop an explicit iterative sequence for approximating the solution and give an error estimate for the approximation, which is an important improvement of existing results.     

Sylvester矩阵方程极小范数最小二乘解的迭代解法

研究了Sylvester矩阵方程最小二乘解以及极小范数最小二乘解的迭代解法,首先利用递阶辨识原理,得到了求解矩阵方程AX+YB=C的极小范数最小二乘解的一种迭代算法,进而,将这种算法推广到一般线性矩阵方程A_iX_iB_i=C的情形,最后,数值例子验证了算法的有效性.     

Mann-Type Steepest-Descent and Modified Hybrid Steepest-Descent Methods for Variational Inequalities in Banach Spaces

In this paper, we propose three different kinds of iteration schemes to compute the approximate solutions of variational inequalities in the setting of Banach spaces. First, we suggest Mann-type steepest-descent iterative algorithm, which is based on two well-known methods: Mann iterative method and steepest-descent method. Second, we introduce modified hybrid steepest-descent iterative algorithm. Third, we propose modified hybrid steepest-descent iterative algorithm by using the resolvent operator. For the first two cases, we prove the convergence of sequences generated by the proposed algorithms to a solution of a variational inequality in the setting of Banach spaces. For the third case, we prove the convergence of the iterative sequence generated by the proposed algorithm to a zero of an operator, which is also a solution of a variational inequality.