线性不等式约束优化的弧线路径信赖域算法

提供了弧线路径结合仿射内点信赖域策略的非单调回代算法解线性不等式约束的优化问题.基于仿射投影的信赖域子问题获得新的搜索方向,采用弧线路径的近似信赖域和线搜索结合技术得到回代步,获得新的步长.通过证明所提供的弧线路径具有一系列良好性质,从而在合理的条件下,证明所提供的算法不仅具有整体收敛性,而且保持算法的局部超线性收敛速率.数值测试表明了算法的有效性与可靠性.     

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.     

仿射内点既约投影Hessian算法解非线性约束优化

本文结合非单调内点回代技术,提供了新的仿射信赖域方法解含有非负变量约束和非线性等式约束的优化问题.为求解大规模问题,采用等式约束的Jacobian矩阵的QR分解和两块校正的双边既约Hessian矩阵投影,将问题分解成零空间和值空间两个信赖域子问题.零空间的子问题为通常二次目标函数只带椭球约束的信赖域子问题,而值空间的子问题使用满足信赖域约束参数的值空间投影向量方向.通过引入Fletcher罚函数作为势函数,将由两个子问题结合信赖域策略构成的合成方向,并使用非单调线搜索技术回代于可接受的非负约束内点步长.在合理的条件下,算法具有整体收敛性且两块校正的双边既约Hessian投影法将保持超线性收敛速率.非单调技术将克服高度非线性情况,加快收敛进展.     

仿射内点既约投影Hessian算法解非线性约束优化

本文结合非单调内点回代技术,提供了新的仿射信赖域方法解含有非负变量约束和非线性等式约束的优化问题.为求解大规模问题,采用等式约束的Jacobian矩阵的QR分解和两块校正的双边既约Hessian矩阵投影,将问题分解成零空间和值空间两个信赖域子问题.零空间的子问题为通常二次目标函数只带椭球约束的信赖域子问题,而值空间的子问题使用满足信赖域约束参数的值空间投影向量方向.通过引入Fletcher罚函数作为势函数,将由两个子问题结合信赖域策略构成的合成方向,并使用非单调线搜索技术回代于可接受的非负约束内点步长.在合理的条件下,算法具有整体收敛性且两块校正的双边既约Hessian投影法将保持超线性收敛速率.非单调技术将克服高度非线性情况,加快收敛进展.     

线性不等式约束的广义非线性互补问题的仿射内点信赖域方法

提供了一种新的非单调内点回代线搜索技术的仿射内点信赖域方法解线性不等式约束的广义非线性互补问题(GCP).基于广义互补问题构成的半光滑方程组的广义Jacobian矩阵,算法使用l_2范数作为半光滑方程组的势函数,形成的信赖域子问题为一个带椭球约束的线性化的二次模型.利用广义牛顿方程计算试探迭代步,通过内点映射回代技术确保迭代点是严格内点,保证了算法的整体收敛性.在合理的条件下,证明了信赖域算法在接近最优点时可转化为广义拟牛顿步,进而具有局部超线性收敛速率.非单调技术将克服高度非线性情况加速收敛进展.最后,数值结果表明了算法的有效性.     

线性不等式约束的广义非线性互补问题的仿射内点信赖域方法

提供了一种新的非单调内点回代线搜索技术的仿射内点信赖域方法解线性不等式约束的广义非线性互补问题(GCP).基于广义互补问题构成的半光滑方程组的广义Jacobian矩阵,算法使用l2范数作为半光滑方程组的势函数,形成的信赖域子问题为一个带椭球约束的线性化的二次模型.利用广义牛顿方程计算试探迭代步,通过内点映射回代技术确保迭代点是严格内点,保证了算法的整体收敛性.在合理的条件下,证明了信赖域算法在接近最优点时可转化为广义拟牛顿步,进而具有局部超线性收敛速率.非单调技术将克服高度非线性情况加速收敛进展.最后,数值结果表明了算法的有效性.     

一类等式约束非线性优化问题的信赖域新算法

提出一类信赖域新算法用于求解等式约束的非线性优化问题,在构造增广拉格朗日函数的基础上,提出了信赖域子问题的求解公式,研究了拉格朗日乘子和罚因子的修正公式,并使用滤子技巧,放松了接受尝试步的条件,证明了算法的收敛性.最后进行了数值试验.     

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.     

非线性等式约束优化问题的仿射信赖域方法

提出非线性等式和有界约束优化问题的结合非单调技术的仿射信赖域方法. 结合信赖域方法和内点回代线搜索技术, 每一步迭代转到由一般信赖域子问题产生的回代步中且满足严格内点可行条件. 在合理的假设条件下, 证明了算法的整体收敛性和局部超线性收敛速率. 最后, 数值结果表明了所提供的算法具有有效性.     

信赖域方法在Hölderian局部误差界下的收敛性质

信赖域方法是求解非线性方程组的一种重要方法.本文研究了求解非线性方程组的信赖域半径趋于零的信赖域算法在Jacobi矩阵Hölderian连续条件下的全局收敛性质,以及其在Hölderian局部误差界和Jacobi矩阵Hölderian连续条件下的收敛速度.     

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…     

NONMONOTONE PRECONDITIONAL CURVILINEAR PATH ALGORITHMS FOR UNCONSTRAINED OPTIMIZATION

This paper presents nonmonotonic quasi-Newton algorithms via two pre-conditional curvilinear paths, the preconditional modified gradient path and the precon-ditional optimal path, for unconstrained optimization problem. We employ the stable Bunch-Parlett factorization method to form two curvilinear paths very easily. The nonmonotone criterion is used to speed up the convergence progress in the contours of objective function with large curvature. Theoretical analyses are given which prove that the proposed algorithms are globally convergent and have a local superlinear convergence rate under some reasonable conditions. The results of numerical experiments are reported to show the effectiveness of the proposed algorithms.     

AN ADAPTIVE NONMONOTONIC TRUST REGION METHOD WITH CURVILINEAR SEARCHES

In this paper,an algorithm for unconstrained optimization that employs both trustregion techniques and curvilinear searches is proposed.At every iteration,we solve thetrust region subproblem whose radius is generated adaptively only once.Nonmonotonicbacktracking curvilinear searches are performed when the solution of the subproblem isunacceptable.The global convergence and fast local convergence rate of the proposedalgorithms are established under some reasonable conditions.The results of numericalexperiments are reported to show the effectiveness of the proposed algorithms.     

线性等式约束优化的既约预条件共轭梯度路径法

采用既约预条件共轭梯度路径结合非单调技术解线性等式约束的非线性优化问题.基于广义消去法将原问题转化为等式约束矩阵的零空间中的一个无约束优化问题,通过一个增广系统获得既约预条件方程,并构造共轭梯度路径解二次模型,从而获得搜索方向和迭代步长.基于共轭梯度路径的良好性质,在合理的假设条件下,证明了算法不仅具有整体收敛性,而且保持快速的超线性收敛速率.进一步,数值计算表明了算法的可行性和有效性.     

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.     

On the convergence of curvilinear search algorithms in unconstrained optimization

The purpose of this paper is to unify the conditions under which curvilinear algorithms for unconstrained optimization converge. Particularly, two gradient path approximation algorithms and a trust region curvilinear algorithm are examined in this context.     

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.     

求解一般界约束优化问题的积极集信赖域方法

本文提出了一种求解一般界约束优化问题的新方法. 每步迭代分为两个阶段. 在第一阶段, 从 当前迭代点x_k 出发, 沿着经过仿射变换后的梯度步, 得到试探点x_k¹, 记录下它的积极集. 这里用到的仿射变换矩阵不仅依赖于变量到边界的距离, 还依赖于当前迭代点的梯度以及该步迭代中的信赖域半 径. 在第二阶段, 从x_k¹ 出发, 通过在积极约束的零空间里面求解一个信赖域子问题得到新的试探点. 然后判断是否接受这个试探点作为下一个迭代点. 文中证明了算法的全局收敛性, 并且迭代点列的每 个聚点都是一阶稳定点. 文中还对国际著名的CUTEr 算例库中所有的界约束优化问题进行了测试. 数值结果表明我们的方法是有效的, 并且可以与L-BFGS-B 方法相媲美.     

A Projected Indefinite Dogleg Path Method for Equality Constrained Optimization

In this paper, we propose a 2-step trust region indefinite dogleg path method for the solution of nonlinear equality constrained optimization problems. The method is a globally convergent modification of the locally convergent Fontecilla method and an indefinite dogleg path method is proposed to get approximate solutions of quadratic programming subproblems even if the Hessian in the model is indefinite. The dogleg paths lie in the null space of the Jacobian matrix of the constraints. An ₁ exact penalty function is used in the method to determine if a trial point is accepted. The global convergence and the local two-step superlinear convergence rate are proved. Some numerical results are presented.