采用不定Dogleg路径的非单调自适应信赖域算法解无约束极小化(英文)

In this paper, we combine the nonmonotone and adaptive techniques with trust region method for unconstrained minimization problems. We set a new ratio of the actual descent and predicted descent. Then, instead of the monotone sequence, the nonmonotone sequence of function values are employed. With the adaptive technique, the radius of trust region △k can be adjusted automatically to improve the efficiency of trust region methods. By means of the Bunch-Parlett factorization, we construct a method with indefinite dogleg path for solving the trust region subproblem which can handle the indefinite approximate Hessian Bk. The convergence properties of the algorithm are established. Finally, detailed numerical results are reported to show that our algorithm is efficient.     

解约束优化问题的投影梯度型中心方法

本文提出了一种求解约束优化问题的新算法—投影梯度型中心方法．在连续可微和非退化的假设条件下，证明了其全局收敛性．本文算法计算简单且形式灵活．     

Projected Gradient Type Method of Centers for Constrained Optimization

本文提出了一种求解约束优化问题的新算法—投影梯度型中心方法．在连续可微和非退化的假设条件下，证明了其全局收敛性．本文算法计算简单且形式灵活．     

关于不等式约束的信赖域算法

对于具有不等式约束的非线性优化问题，本文给出一个依赖域算法，由于算法中依赖区域约束采用向量的∞范数约束的形式，从而使子问题变二次规划，同时使算法变得更实用。在通常假设条件下，证明了算法的整体收敛性和超线性收敛性。     

The Equality Constrained Indefinite Least Squares Problem: Theory and Algorithms

We present theory and algorithms for the equality constrained indefinite least squares problem, which requires minimization of an indefinite quadratic form subject to a linear equality constraint. A generalized hyperbolic QR factorization is introduced and used in the derivation of perturbation bounds and to construct a numerical method. An alternative method is obtained by employing a generalized QR factorization in combination with a Cholesky factorization. Rounding error analysis is given to show that both methods have satisfactory numerical stability properties and numerical experiments are given for illustration. This work builds on recent work on the unconstrained indefinite least squares problem by Chandrasekaran, Gu, and Sayed and by the present authors.     

一个等式约束问题的拟Newton—信赖域型方法及其收敛性

在[1]中,Vardi提出一个信赖域方法,而收敛性证明却是在精确λ-搜索下给出的,本文在[1]的基础上提出一个新的算法-拟Newton-信赖域型算法,并证明该算法是全局收敛的,通过利用二阶修正技术去修正该算法,我们证明了该算法是局部超线性收敛的。     

解等式约束优化的既约Hessian依赖域方法

本文提出了解等式约束优化的一个信赖域方法,该方法以既约Hessian逐步二次规划为基础,它享有信赖域方法与既约Hessian方法的优点,在通常条件下,证明了算法的全局收敛性。     

不等式约束优化的非单调可行信赖域-SQP算法

本文讨论不等式约束优化问题,给出一个信赖域方法与SQP方法相结合的新的可行算法,算法中采用了压缩技术,使得QP子问题产生的搜索方向尽可能为可行方向,并且采用了高阶校正的方法来克服算法产生的Maratos效应现象.在适当的条件下,证明了算法的全局收敛性和超线性收敛性.数值结果表明算法是有效的.     

等式约束优化的组合信赖域与拟牛顿算法

本文对等式约束问题提出了一个种组合信赖域与拟牛顿算法。该算法的特点是若Lagrangian函数的近似Ｈessian阵在等式约束Jacobi阵的零空间正定的,则选择拟牛顿算法,否则用信赖域算法,在通常信赖域算法的收敛假设下,该文证明了组合算法的全局收敛性。     

解不等式约束优化的新的序列线性方程组方法

提出一种新的序列线性方程组(SSLE)算法解非线性不等式约束优化问题.在算法的每步迭代,子问题只需解四个简化的有相同的系数矩阵的线性方程组.证明算法是可行的,并且不需假定聚点的孤立性、严格互补条件和积极约束函数的梯度的线性独立性得到算法的全局收敛性.在一定条件下,证明算法的超线性收敛率.     

一般约束优化基于识别函数的模松弛算法

借助于半罚函数和产生工作集的识别函数以及模松弛SQP算法思想, 本文建立了求解带等式及不等式约束优化的一个新算法. 每次迭代中, 算法的搜索方向由一个简化的二次规划子问题及一个简化的线性方程组产生. 算法在不包含严格互补性的温和条件下具有全局收敛性和超线性收敛性. 最后给出了算法初步的数值试验报告.     

A Globally and Superlinearly Convergent SQP Algorithm for Nonlinear Constrained Optimization

Based on a continuously differentiable exact penalty function and a regularization technique for dealing with the inconsistency of subproblems in the SQP method, we present a new SQP algorithm for nonlinear constrained optimization problems. The proposed algorithm incorporates automatic adjustment rules for the choice of the parameters and makes use of an approximate directional derivative of the merit function to avoid the need to evaluate second order derivatives of the problem functions. Under mild assumptions the algorithm is proved to be globally convergent, and in particular the superlinear convergence rate is established without assuming that the strict complementarity condition at the solution holds. Numerical results reported show that the proposed algorithm is promising.     

非线性不等式约束最优化一个超线性与二次收敛的强次可行方法

本文讨论非线性不等式约束最优化问题，借助于序列线性方程组技术和强次可行方法思想，建立了问题的一个初始点任意的快速收敛新算法．在每次迭代中，算法只需解一个结构简单的线性方程组．算法的初始迭代点不仅可以是任意的，而且不使用罚函数和罚参数，在迭代过程中，迭代点列的可行性单调不减．在相对弱的假设下，算法具有较好的收敛性和收敛速度，即具有整体与强收敛性，超线性与二次收敛性．文中最后给出一些数值试验结果．     

A Limited-Memory Multipoint Symmetric Secant Method for Bound Constrained Optimization

A new algorithm for solving smooth large-scale minimization problems with bound constraints is introduced. The way of dealing with active constraints is similar to the one used in some recently introduced quadratic solvers. A limited-memory multipoint symmetric secant method for approximating the Hessian is presented. Positive-definiteness of the Hessian approximation is not enforced. A combination of trust-region and conjugate-gradient approaches is used to explore useful information. Global convergence is proved for a general model algorithm. Results of numerical experiments are presented.     

Globally and Superlinearly Convergent QP-Free Algorithm for Nonlinear Constrained Optimization

A new, infeasible QP-free algorithm for nonlinear constrained optimization problems is proposed. The algorithm is based on a continuously differentiable exact penalty function and on active-set strategy. After a finite number of iterations, the algorithm requires only the solution of two linear systems at each iteration. We prove that the algorithm is globally convergent toward the KKT points and that, if the second-order sufficiency condition and the strict complementarity condition hold, then the rate of convergence is superlinear or even quadratic. Moreover, we incorporate two automatic adjustment rules for the choice of the penalty parameter and make use of an approximated direction as derivative of the merit function so that only first-order derivatives of the objective and constraint functions are used.     

A SQP Method for Inequality Constrained Optimization

Abstract In this paper, a new SQP method for inequality constrained optimization is proposed and the globalconvergence is obtained under very mild conditions.     

A Smoothing Newton-Type Algorithm of Stronger Convergence for the Quadratically Constrained Convex Quadratic Programming

In this paper we propose a smoothing Newton-type algorithm for the problem of minimizing a convex quadratic function subject to finitely many convex quadratic inequality constraints. The algorithm is shown to converge globally and possess stronger local superlinear convergence. Preliminary numerical results are also reported. Mathematics Subject Classification (1991): 90C33, 65K10 This author’s work was also partially supported by the Scientific Research Foundation of Tianjin University for the Returned Overseas Chinese Scholars and the Scientific Research Foundation of Liu Hui Center for Applied Mathematics, Nankai University-Tianjin University.     

一种解决不等式约束优化问题的光滑牛顿法

本通过引入松弛变量和Fischer函数把带有不等式约束优化问题的K-T条件转化为一个等价的非线性系统，并引入一参数μ，从而提出了一种新的光滑牛顿法。在适当的条件下，证明了算法的全局收敛性，并提供了数值结果。     

A Constrained Optimization Approach for LCP

In this paper, LCP is converted to an equivalent nonsmooth nonlinear equation system H(x,y) = 0 by using the famous NCP function-Fischer-Burmeister function. Note that some equations in H(x, y) = 0 are nonsmooth and nonlinear hence difficult to solve while the others are linear hence easy to solve. Then we further convert the nonlinear equation system H(x, y) = 0 to an optimization problem with linear equality constraints. After that we study the conditions under which the K-T points of the optimization problem are the solutions of the original LCP and propose a method to solve the optimization problem. In this algorithm, the search direction is obtained by solving a strict convex programming at each iterative point, However, our algorithm is essentially different from traditional SQP method. The global convergence of the method is proved under mild conditions. In addition, we can prove that the algorithm is convergent superlinearly under the conditions: M is P0 matrix and the limit point is a strict complementarity solution of LCP. Preliminary numerical experiments are reported with this method.