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Journal of Systems Science & Complexity 2008年第1期21卷 85-113页

作者： Peihua GUO Detong ZHU Department of Mathematics Shanghai Normal University Shanghai 200234 China. Detong ZHU Business College Shanghai Normal University Shanghai 200234 China.

The authors propose an affine scaling modified gradient path method in association with reduced projective Hessian and nonmonotonic interior backtracking line search techniques for solving the linear equality constrained optimization subject to bounds on variables. By employing the QR decomposition of the constraint matrix and the eigensystem decomposition of reduced projective Hes- sian matrix in the subproblem, the authors form affine scaling modified gradient curvilinear path very easily. By using interior backtracking line search technique, each iterate switches to trial step of strict interior feasibility. The global convergence and fast local superlinear/quadratical convergence rates of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm.

关键词： affine scaling convergence interior point modified gradient path nonmonotonic technique QR decomposition.

AN affine scaling DERIVATIVE-FREE TRUST REGION METHOD WITH INTERIOR BACKTRACKING TECHNIQUE FOR BOUNDED-CONSTRAINED NONLINEAR PROGRAMMING

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Journal of Systems Science & Complexity 2014年第3期27卷 537-564页

作者： GAO Jing ZHU Detong Department of Mathematics Shanghai Normal University

This paper proposes an arlene scaling derivative-free trust region method with interior backtracking technique for bounded-constrained nonlinear programming. This method is designed to get a stationary point for such a problem with polynomial interpolation models instead of the objective function in trust region subproblem. Combined with both trust region strategy and line search technique, at each iteration, the affine scaling derivative-free trust region subproblem generates a backtracking direction in order to obtain a new accepted interior feasible step. Global convergence and fast local convergence properties are established under some reasonable conditions. Some numerical results are also given to show the effectiveness of the proposed algorithm.

关键词： affine scaling backtracking technique box constrains derivative-free optimization non-linear programming trust region method.

An affine scaling Interior Trust Region Method via Optimal Path for Solving Monotone Variational Inequality Problem with Linear Constraints

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Chinese Annals of Mathematics,Series B 2008年第3期29卷 273-290页

作者： Yunjuan WANG Detong ZHU Mathematics and Science College Shanghai Normal University Shanghai 200234 ChinaArts and Science School Shanghai Dianji University Shanghai 200240 China Business College Shanghai Normal University Shanghai 200234 China

Based on a differentiable merit function proposed by Taji et al. in ＂Math. Prog. Stud., 58, 1993, 369-383＂, the authors propose an affine scaling interior trust region strategy via optimal path to modify Newton method for the strictly monotone variational inequality problem subject to linear equality and inequality constraints. By using the eigensystem decomposition and affine scaling mapping, the authors form an affine scaling optimal curvilinear path very easily in order to approximately solve the trust region subproblem. Theoretical analysis is given which shows that the proposed algorithm is globally convergent and has a local quadratic convergence rate under some reasonable conditions.

关键词： Trust region affine scaling Interior point Optimal path Variational inequality problem

Superlinear Convergence of affine scaling Interior Point Newton Method for Linear Inequality Constrained Minimization without Strict Complementarity

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Acta Mathematicae Applicatae Sinica 2009年第2期25卷 183-194页

作者： De-tong Zhu Business College Shanghai Normal University Shanghai 200234 China

In this paper we extend and improve the classical affine scaling interior-point Newton method for solving nonlinear optimization subject to linear inequality constraints in the absence of the strict complementarity assumption. Introducing a computationally efficient technique and employing an identification function for the definition of the new affine scaling matrix, we propose and analyze a new affine scaling interior-point Newton method which improves the Coleman and Li affine sealing matrix in [2] for solving the linear inequlity constrained optimization. Local superlinear and quadratical convergence of the proposed algorithm is established under the strong second order sufficiency condition without assuming strict complementarity of the solution.

关键词： Interior method affine scaling strict complementarity

Global and Local Convergence of a New affine scaling Trust Region Algorithm for Linearly Constrained Optimization

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Acta Mathematica Sinica,English Series 2016年第10期32卷 1203-1213页

作者： Chao GU De Tong ZHU School of Mathematics and Information Shanghai LiXin University of Commerce Department of Mathematics Shanghai Normal University

Chen and Zhang [Sci. China, Set. A, 45, 1390-1397 （2002）] introduced an affine scaling trust region algorithm for linearly constrained optimization and analyzed its global convergence. In this paper, we derive a new affine scaling trust region algorithm with dwindling filter for linearly constrained optimization. Different from Chen and Zhang＇s work, the trial points generated by the new algorithm axe accepted if they improve the objective function or improve the first order necessary optimality conditions. Under mild conditions, we discuss both the global and local convergence of the new algorithm. Preliminary numerical results are reported.

关键词： Linearly constrained optimization affine scaling trust region dwindling filter~ conver-gence

Superlinearly Convergent affine scaling Interior Trust-Region Method for Linear Constrained LC^1 Minimization

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Acta Mathematica Sinica,English Series 2008年第12期24卷 2081-2100页

作者： De Tong ZHU Business College Shanghai Normal University Shanghai 200234 P. R. China

We extend the classical affine scaling interior trust region algorithm for the linear constrained smooth minimization problem to the nonsmooth case where the gradient of objective function is only locally Lipschitzian. We propose and analyze a new affine scaling trust-region method in association with nonmonotonic interior backtracking line search technique for solving the linear constrained LC1 optimization where the second-order derivative of the objective function is explicitly required to be locally Lipschitzian. The general trust region subproblem in the proposed algorithm is defined by minimizing an augmented affine scaling quadratic model which requires both first and second order information of the objective function subject only to an affine scaling ellipsoidal constraint in a null subspace of the augmented equality constraints. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required. Applications of the algorithm to some nonsmooth optimization problems are discussed.

关键词： trust region method backtracking nonmonotonic technique interior point LC^1 minimization affine scaling

A Trust Region affine scaling Method for Bound Constrained Optimization

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Acta Mathematica Sinica,English Series 2013年第1期29卷 159-182页

作者： Xiao WANG School of Mathematical Sciences University of Chinese Academy of Sciences

We study a new trust region affine scaling method for general bound constrained optimiza- tion 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.

关键词： Bound constrained optimization affine scaling trust region approximate projected gradient

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有界约束非线性方程组的仿射共轭梯度路径法

高等学校计算数学学报 2011年第1期33卷 67-80页

作者：张勇朱德通上海师范大学数理学院上海师范大学商学院

1引言本文研究约束非线性方程组的问题F(x)=0,x∈Ω.(1.1)这里F:X→

关键词： affine scaling conjugate gradient nonlinear systems

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线性约束优化的仿射内点修正梯度路径方法

高等学校计算数学学报 2005年第S1期 93-99页

作者：王云娟朱德通上海电机学院文理系上海师范大学数学系

1 引言本文主要考虑带有线性约束的非线性优化问题: 针对问题(1．1)中具有不等式约束这一特性,本文技巧性的引用仿射变换构造信赖域子问题;同时采用修正梯度路径搜索与线搜索相结合的技术,通过修正梯度路径搜索求得模型的迭代方向,然后... 详细信息

1 引言本文主要考虑带有线性约束的非线性优化问题: 针对问题(1．1)中具有不等式约束这一特性,本文技巧性的引用仿射变换构造信赖域子问题;同时采用修正梯度路径搜索与线搜索相结合的技术,通过修正梯度路径搜索求得模型的迭代方向,然后沿此方向通过非单调线搜索获得步长因子,既能使迭代点严格

关键词： modified gradient path linear inequality constraint interior point method affine scaling nonmonotonic technique.

A FAMILY OF THE LOCAL CONVERGENCE OF THE IMPROVED SECANT METHODS FOR NONLINEAR EQUALITY CONSTRAINED OPTIMIZATION SUBJECT TO BOUNDS ON VARIABLES

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Journal of Systems Science & Complexity 2014年第2期27卷 307-326页

作者： ZHANG Yong ZHU Detong School of Mathematics and Physics Changzhou University Mathematics and Science College Shanghai Normal University

This paper studies a family of the local convergence of the improved secant methods for solving the nonlinear equality constrained optimization subject to bounds on variables. The Hessian of the Lagrangian is approximated using the DFP or the BFGS secant updates. The improved secant methods are used to generate a search direction. Combining with a suitable step size, each iterate switches to trial step of strict interior feasibility. When the Hessian is only positive definite in an affine null subspace, one shows that the algorithms generate the sequences converging q-linearly and two-step q-superlinearly. Yhrthermore, under some suitable assumptions, some sequences generated by the algorithms converge locally one-step q-superlinearly. Finally, some numerical results are presented to illustrate the effectiveness of the proposed algorithms.

关键词： affine scaling local convergence secant methods second order correction.