凸约束优化问题的带记忆模型信赖域算法

本文我们考虑求解凸约束优化问题的信赖域方法 .与传统的方法不同 ,我们信赖域子问题的逼近模型中包括过去迭代点的信息 ,该模型使我们可以从更全局的角度来求得信赖域试探步 ,从而避免了传统信赖域方法中试探步的求取完全依赖于当前点的信息而过于局部化的困难 .全局收敛性的获得是依靠非单调技术来保证的     

一个新的既约梯度法及其收敛性

本文讨论了线性约束条件的非线性规划的既约梯度方法.文中提出了一个新的既约梯度法,并在相当弱的假设条件下证明了这个方法的收敛性.所得主要结果如下:1.设目标函数f为一阶连续可微,且约束条件满足非退化性.则从任意可行点开始,用这个方法或经有限次迭代后到达K.—T.点,或得到一点列{x^k),其任一极限点皆为K.—T.点.2.若点列{x^k}是收敛的点列,则这个方法包括的转轴运算在整个迭代过程中只有有限次.3.若目标函数f为二阶连续可微,且其Hessian矩阵为一致正定,则点列{x^k}必收敛到最优解.4.若最优解x更满足严格的互补松弛性,则{x^k}除有限个点外满足。     

图的彩虹连通数与最小度和

令G是一个阶为n且最小度为δ的连通图. 当δ很小而n很大时, 现有的依据于最小度参数的彩虹边连通数和彩虹点连通数的上界都很大, 它们是n的线性函数. 本文中, 我们用另一种参数,即k个独立点的最小度和σ_k来代替δ, 从而在很大程度上改进了彩虹边连通数和彩虹点连通数的上界. 本文证明了如果G有k个独立点, 那么rc(GG)≤3kn/(σk+k)+6k-3. 同时也证明了下面的结果, 如果σ_k≤7k或σ_k≥8k, 那么rvc(G)≤(4k+2k²)n/(σ_k+k)+5k; 如果7k<σ_k<8k, 那么rvc(G)≤(38k/9+2k²)n/(σ_k+k)+5k.文中也给出了例子说明我们的界比现有的界更好, 即我们的界为rc(G)≤9k-3和rvc(G)≤9k+2k²或rvc(G)≤83k/9+2k², 这意味着当δ很小而σ_k很大时, 我们的界是一个常数, 而现有的界却是n的线性函数.     

拟线性抛物方程组第一边界问题的有限差分法 ——Ⅲ.稳定性

讨论如下拟线性抛物组第一边值问题的显式、弱隐式和强隐式差分解u_t=(-1)^M+1A(x,t,u,…,u_x^M-1)u_x^2M+f(x,t,u,…,u_x^2M-1(x,t)∈Q_T={O<x<l,0<t≤T.}，u_x^k(0，t)=u_x^k(l，t)=0 (k=0，1，…，M -1)，0<t≤T，u(x，0)=φ(x)，0≤x≤l，其中u，φ和f是m维向量值函数，A是m×m正定矩阵，u_t=∂u/∂t，u_x^k=∂^ku/∂x^k.在以下意义下证明了该问题的一般有限差分格式的稳定性：即离散向量解在W₂^(2M，M)(Q_T)中的离散范数是连续地依赖于初始数据的H^M离散范数，以及矩阵A与自由项f的相应的离散范数.     

3维Poincaré猜想的一个证明

设M是一个连通闭3维流形而且π₁(M)=1,…,x_n;y₁,…,y_n>=1.在本文中，我们利用由π₁(M)=1得出来的条件x₁=y₁a₁y₁^-1…y_ka_ky_k^-1(其中y相似文献   

平凡主丛上提升过程的环流

紧流形M上的以向量场X为漂移项的Brown运动{x_t}_t≥0可以提升到M×T^k上的一个扩散过程{^-x_t}_t≥0(相应于一个M上的R^k值光滑微分1形式A)。研究提升过程{^-x_t}_t≥0绕环面T^k的k个圈的环流(即旋转数)。适当地选取R^k值微分1形式A。这些环流分别给出了{^-x_t}_t≥0的隐环流和{^-x_t}_t≥0绕M上某些闭圈的旋转数(这些闭圈生成M的一阶同调群H₁(M,Z))。     

结合有效集和多维滤子技术的拟Newton信赖域算法(英文)

针对界约束优化问题,提出一个修正的多维滤子信赖域算法.将滤子技术引入到拟Newton信赖域方法,在每步迭代,Cauchy点用于预测有效集,此时试探步借助于求解一个较小规模的信赖域子问题获得.在一定条件下,本文所提出的修正算法对于凸约束优化问题全局收敛.数值试验验证了新算法的实际运行结果.     

Schur函数的一个展式及其在计数几何中的应用

本文给出对称多项式的幂的Schur函数展式(x₁^k+…+x_n^k)^m=sumC_{(λ1,…,λn)}S_{(λ1,…,λn)}(x₁,…,x_n)中系数C_{(λ1,…,λn)}的计算方法,并把它和文献[1]应用于计数几何的若干问题。     

关于Grünwald插值算子及其应用

本文研究了基于Jacobi多项式J_n^(α,β)(x)(0<α,β<1)的零点{x_k}_lⁿ的Grünwald插值多项式G_n(f;x)=(?)f(x_k)l_k²(x),证明了G_n(f;x)在(-1,1)内的任一闭子区间上一致收敛于连续函数f(x);从而拓广了Grünwald所得结果。     

关于齐次加型同余式

本文研究了形如α₁λ₁^k+…+α_sλ_s^k=0的加型方程,此处诸α₁是一个次数为n的代数域K中的整数,主要结果为:若s≥(2k)ⁿ⁺¹(或当2 k时,s≥cknlog k),方程在任何-adic域中均可以非寻常求解,此处 为K中素理想。     

A trust region affine scaling method for bound constrained optimization

We study a new trust region affine scaling method for general bound constrained optimization problems. At each iteration, we compute two trial steps. We compute one along some direction obtained by solving an appropriate quadratic model in an ellipsoidal region. This region is defined by an affine scaling technique. It depends on both the distances of current iterate to boundaries and the trust region radius. For convergence and avoiding iterations trapped around nonstationary points, an auxiliary step is defined along some newly defined approximate projected gradient. By choosing the one which achieves more reduction of the quadratic model from the two above steps as the trial step to generate next iterate, we prove that the iterates generated by the new algorithm are not bounded away from stationary points. And also assuming that the second-order sufficient condition holds at some nondegenerate stationary point, we prove the Q-linear convergence of the objective function values. Preliminary numerical experience for problems with bound constraints from the CUTEr collection is also reported.     

非单调技术预条件弧线路径信赖域解无约束优化

本文提供修正近似信赖域类型路经三类预条件弧线路径方法解无约束最优化问题.使用对称矩阵的稳定Bunch-Parlett易于形成信赖域子问题的弧线路径,使用单位下三角矩阵作为最优路径和修正梯度路径的预条件因子.运用预条件因子改进Hessian矩阵特征值分布加速预条件共轭梯度路径收敛速度.基于沿着三类路径信赖域子问题产生试探步,将信赖域策略与非单调线搜索技术相结合作为新的回代步.理论分析证明在合理条件下所提供的算法是整体收敛性,并且具有局部超线性收敛速率,数值结果表明算法的有效性.     

Combining trust region and linesearch algorithm for equality constrained optimization

In this paper, a combining trust region and line search algorithm for equality constrained optimization is proposed. At each iteration, we only need to solve the trust region subproblem once, when the trust region trial step can not be accepted, we switch to line search to obtain the next iteration. Hence, the difficulty of repeated solving trust region subproblem in an iterate is avoided. In order to allow the direction of negative curvature, we add second correction step in trust region step and employ nommonotone technique in line search. The global convergence and local superlinearly rate are established under certain assumptions. Some numerical examples are given to illustrate the efficiency of the proposed algorithm.     

A subspace implementation of quasi-Newton trust region methods for unconstrained optimization

This paper studies subspace properties of trust region methods for unconstrained optimization, assuming the approximate Hessian is updated by quasi- Newton formulae and the initial Hessian approximation is appropriately chosen. It is shown that the trial step obtained by solving the trust region subproblem is in the subspace spanned by all the gradient vectors computed. Thus, the trial step can be defined by minimizing the quasi-Newton quadratic model in the subspace. Based on this observation, some subspace trust region algorithms are proposed and numerical results are also reported.     

CURVILINEAR PATHS AND TRUST REGION METHODS WITH NONMONOTONIC BACK TRACKING TECHNIQUE FOR UNCONSTRAINED OPTIMIZATION

1. Illtroductioncrust region method is a well-accepted technique in nonlinear optindzation to assure globalconvergence. One of the adVantages of the model is that it does not require the objectivefunction to be convex. Many differellt versions have been suggested in using trust regiontechnique. For each iteration, suppose a current iterate point, a local quadratic model of thefunction and a trust region with center at the point and a certain radius are given. A point thatminimizes the model f…     

无约束最优化的信赖域BB法

梯度法是求解无约束最优化的一类重要方法.步长选取的好坏与梯度法的数值表现息息相关.注意到BB步长隐含了目标函数的二阶信息,本文将BB法与信赖域方法相结合,利用BB步长的倒数去近似目标函数的Hesse矩阵,同时利用信赖域子问题更加灵活地选取梯度法的步长,给出求解无约束最优化问题的单调和非单调信赖域BB法.在适当的假设条件下,证明了算法的全局收敛性.数值试验表明,与已有的求解无约束优化问题的BB类型的方法相比,非单调信赖域BB法中e_k=‖x_k-x~*‖的下降呈现更明显的阶梯状和单调性,因此收敛速度更快.     

A Line Search and Trust Region Algorithm with Trust Region Radius Converging to Zero

In this paper, we present a new line search and trust region algorithm for unconstrained optimization problem with the trust region radius converging to zero. The new trust region algorithm performs a backtracking line search from the failed, point instead of resolving the subproblem when the trial step results in an increase in the objective function. We show that the algorithm preserves the convergence properties of the traditional trust region algorithms. Numerical results are also given.     

Two accelerated nonmonotone adaptive trust region line search methods

Hybridizing monotone and nonmonotone approaches, we employ a modified trust region ratio in which more information is provided about the agreement between the exact and the approximate models. Also, we use an adaptive trust region radius as well as two accelerated Armijo-type line search strategies to avoid resolving the trust region subproblem whenever a trial step is rejected. We show that the proposed algorithm is globally and locally superlinearly convergent. Comparative numerical experiments show practical efficiency of the proposed accelerated adaptive trust region algorithm.     

A model-hybrid approach for unconstrained optimization problems

In this paper, we propose a model-hybrid approach for nonlinear optimization that employs both trust region method and quasi-Newton method, which can avoid possibly resolve the trust region subproblem if the trial step is not acceptable. In particular, unlike the traditional trust region methods, the new approach does not use a single approximate model from beginning to the end, but instead employs quadratic model or conic model at every iteration adaptively. We show that the new algorithm preserves the strong convergence properties of trust region methods. Numerical results are also presented.