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nonmonotonic REDUCED PROJECTED HESSIAN METHOD VIA AN AFFINE SCALING INTERIOR MODIFIED GRADIENT PATH FOR BOUNDED-CONSTRAINED OPTIMIZATION

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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.

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 Nonmonotone Hybrid Method of Conjugate Gradient and Lanczos-type for Solving Nonlinear Systems

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Journal of the Operations Research Society of China 2014年第3期2卷 291-305页

作者： Chun-Xia Jia Jue-Yu Wang De-Tong Zhu Department of Mathematics Shanghai Normal UniversityShanghai 200234China Department of Mathematics Shanghai Normal UniversityGuilin Road 100Shanghai 200234China

In this paper,we construct a new algorithm which combines the conjugate gradient and Lanczos methods for solving nonlinear *** iterative direction can be obtained by solving a quadratic model via conjugate gradient and Lanczos *** the backtracking line search,we will find an acceptable trial step size along this direction which makes the objective function nonmonotonically decreasing and makes the norm of the step size monotonically *** convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable ***,we present some numerical results to illustrate the effectiveness of the proposed algorithm.

关键词： nonmonotonic technique Nonlinear systems Lanczos method Conjugate gradient

nonmonotonic TRUST REGION PROJECTED REDUCED HESSIAN ALGORITHM WITH TWO-PIECE UPDATE FOR CONSTRAINED OPTIMIZATION

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Journal of Systems Science & Complexity 2004年第3期17卷 332-348页

作者： ZHUDetong DepartmentofMathematics ShanghaiNormalUniversityShanghai200234China

This paper proposes a two-piece update of projected reduced Hessian algorithmwith nonmonotonic trust region strategy for solving nonlinear equality constrained optimizationproblems. In order to deal with large problems, a two-piece update of two-side projected reducedHessian is used to replace full Hessian matrix. By adopting the Fletcher's penalty function as themerit function, a nonmonotonic trust region strategy is suggested which does not require the meritfunction to reduce its value in every iteration. The two-piece update of projected reduced Hessianalgorithm which switches to nonmonotonic trust region technique possesses global convergence whilemaintaining a two-step Q-superlinear local convergence rate under some reasonable ***, one step Q-superlinear local convergence rate can be obtained if at least one of theupdate formulas is updated at each iteration by an alternative update rule. The numerical experimentresults are reported to show the effectiveness of the proposed algorithm.

关键词： trust region strategy nonmonotonic technique fletcher's penalty function two-piece update superlinear convergence