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A CLASS OF REVISED BROYDEN algorithms WITHOUT EXACT LINE SEARCH

Journal of Computational Mathematics 2004年第1期22卷 11-20页

作者： Ding-guoPu Sheng-huaGui Wei-wenTian DepartmentofMathematics ShanghaiTongjiUniversityShanghai200331China

In this paper, we discuss the convergence of the Broyden algorithms with revised search direction. Under some inexact line searches, we prove that the algorithms are globally convergent for continuously differentiable functions and the rate of local convergence of the algorithms is one-step superlinear and n-step second-order for uniformly convex objective functions.

关键词： Convergence Convergence rate Line search variable metric algorithms

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A quasi-second-order proximal bundle algorithm

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MATHEMATICAL PROGRAMMING 1996年第1期73卷 51-72页

作者： Mifflin, R Department of Pure and Applied Mathematics Washington State University Pullman USA

This paper introduces an algorithm for convex minimization which includes quasi-Newton updates within a proximal point algorithm that depends on a preconditioned bundle subalgorithm. The method uses the Hessian of a certain outer function which depends on the Jacobian of a proximal point mapping which, in turn, depends on the preconditioner matrix and on a Lagrangian Hessian relative to a certain tangent space. Convergence is proved under boundedness assumptions on the preconditioner sequence.

关键词： bundle methods convex minimization global convergence proximal point quasi-Newton variable metric algorithms

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A class of Broyden algorithms with revised search directions

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ASIA-PACIFIC JOURNAL OF OPERATIONAL RESEARCH 1997年第2期14卷 93-109页

作者： Pu, DG Tian, WW City Univ Hong Kong Dept Math Hong Kong Peoples R China Shanghai Univ Dept Math Jiading Peoples R China

In this paper we discuss the convergence of the Broyden algorithms with revised search direction. We prove that the algorithms are globally convergent for continuously differentiable functions and the rate of converge... 详细信息

关键词： variable metric algorithms line search convergence convergence rate

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SOME HISTORY OF THE CONJUGATE-GRADIENT AND LANCZOS algorithms - 1948-1976

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SIAM REVIEW 1989年第1期31卷 50-102页

作者： GOLUB, GH OLEARY, DP UNIV MARYLAND DEPT COMP SCI COLLEGE PK MD 20742 USA UNIV MARYLAND INST ADV COMP STUD COLLEGE PK MD 20742 USA

This paper gives some of the history of the conjugate gradient and Lanczos algorithms and an annotated bibliography for the period 1948-1976

关键词： 65F10 65F15 65H10 65F03 conjugate gradient algorithm Lanczos algorithm variable metric algorithms

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UPDATING CONJUGATE DIRECTIONS BY THE BFGS FORMULA

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MATHEMATICAL PROGRAMMING 1987年第1期38卷 29-46页

作者： POWELL, MJD 1.Department of Applied Mathematics and Theoretical Physics University of Cambridge England

Many iterative algorithms for optimization calculations form positive definite second derivative approximations,B say, automatically, butB is not stored explicitly because of the need to solve equations of the formBd--g. We consider working with matricesZ, whose columns satisfy the conjugacy conditionsZ ¹ BZ=1. Particular attention is given to updatingZ in a way that corresponds to revisingB by the BFGS formula. A procedure is proposed that seems to be much more stable than the direct use of a product formula [1]. An extension to this procedure provides some automatic rescaling of the columns ofZ, which avoids some inefficiencies due to a poor choice of the initial second derivative approximation. Our work is also relevant to active set methods for linear inequality constraints, to updating the Cholesky factorization ofB, and to explaining some properties of the BFGS algorithm.

关键词： Nonlinear programming conjugate directions updating variable metric algorithms

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variable metric METHOD FOR MINIMIZATION

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SIAM JOURNAL ON OPTIMIZATION 1991年第1期1卷 1-17页

作者： Davidon, William C. Haverford Coll Dept Math Haverford PA 19041 USA

This is a method for determining numerically local minima of differentiable functions of several variables. In the process of locating each minimum, a matrix which characterizes the behavior of the function about the minimum is determined. For a region in which the function depends quadratically on the variables, no more than N iterations are required, where N is the number of variables. By suitable choice of starting values, and without modification of the procedure, linear constraints can be imposed upon the variables.

关键词： variable metric algorithms quasi-Newton optimization

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UPDATING OF CONJUGATE DIRECTION MATRICES USING MEMBERS OF BROYDEN FAMILY

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MATHEMATICAL PROGRAMMING 1993年第2期60卷 167-185页

作者： SIEGEL, D 1. Department of Applied Mathematics and Theoretical Physics University of Cambridge Silver Street CB3 9EW Cambridge UK

Many iterative algorithms for optimization calculations use a second derivative approximation, B say, in order to calculate the search direction d = -B-1delf(x). In order to avoid inverting B we work with matrices Z, whose columns satisfy the conjugacy relations Z(T) BZ = I. We present an update of Z that is compatible with members of the Broyden family that generate positive definite second derivative approximations. The algorithm requires only 3n2 + O(n) flops for the update of Z and the calculation of d. The columns of the resultant Z matrices have interesting conjugacy and orthogonality properties with respect to previous second derivative approximations and function gradients, respectively. The update also provides a simple proof of Dixon's theorem. For the BFGS method we adapt the algorithm in order to obtain a null space method for linearly constrained calculations.

关键词： CONJUGATE DIRECTIONS LINEAR CONSTRAINTS MATRIX FACTORIZATIONS NONLINEAR OPTIMIZATION UPDATING variable metric algorithms

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CONVERGENCE PROPERTIES OF algorithms FOR NONLINEAR OPTIMIZATION

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SIAM REVIEW 1986年第4期28卷 487-500页

作者： POWELL, MJD Univ of Cambridge Cambridge Engl Univ of Cambridge Cambridge Engl

This paper reviews some of the most successful methods for unconstrained, constrained and nondifferentiable optimization calculations. Particular attention is given to the contribution that theoretical analysis has made to the development of algorithms. It seems that practical considerations provide the main new ideas, and that subsequent theoretical studies give improvements to algorithms, coherence to the subject, and better understanding.

关键词： 49D37 65K10 conjugate gradients constrained optimization nonlinear programming optimization rate of convergence trust regions variable metric algorithms

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MODIFYING THE BFGS UPDATE BY A NEW COLUMN SCALING TECHNIQUE

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MATHEMATICAL PROGRAMMING 1994年第1期66卷 45-78页

作者： SIEGEL, D 1. Department of Applied Mathematics and Theoretical Physics Silver Street Cambridge UK

Let B be a positive definite symmetric approximation to the second derivative matrix of the objective function of a minimization calculation. We study modifications of the BFGS method that apply a scaling technique to the columns of a conjugate direction matrix Z satisfying Z(T)BZ = I. For a simple scaling technique similar to the ones considered by Powell (1987) and (1989) we show that, due to a two iteration cycle, linear convergence can occur when the method is applied to a quadratic function and Wolfe's line search is employed, although for exact line searches quadratic termination can be proved. We then suggest a different scaling technique that prevents certain columns thought to contain important curvature information from being scaled. For this algorithm we prove global and superlinear convergence and demonstrate the superiority of our method over the BFGS formula on a range of ill-conditioned optimization problems. Moreover, we present an implementation of our algorithm tht requires only 3n2 + O(n) multiplications per iteration.

关键词： CONJUGATE DIRECTIONS MATRIX FACTORIZATIONS NONLINEAR OPTIMIZATION UPDATING variable metric algorithms

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Constrained Optimization: Projected Gradient Flows

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2009年第1期140卷 117-130页

作者： Shikhman, V. Stein, O. Rhein Westfal TH Aachen Dept Math C D-52056 Aachen Germany Univ Karlsruhe TH Sch Econ & Business Engn D-76128 Karlsruhe Germany

We consider a dynamical system approach to solve finite-dimensional smooth optimization problems with a compact and connected feasible set. In fact, by the well-known technique of equalizing inequality constraints using quadratic slack variables, we transform a general optimization problem into an associated problem without inequality constraints in a higher-dimensional space. We compute the projected gradient for the latter problem and consider its projection on the feasible set in the original, lower-dimensional space. In this way, we obtain an ordinary differential equation in the original variables, which is specially adapted to treat inequality constraints (for the idea, see Jongen and Stein, Frontiers in Global Optimization, pp. 223-236, Kluwer Academic, Dordrecht, 2003). The article shows that the derived ordinary differential equation possesses the basic properties which make it appropriate to solve the underlying optimization problem: the longtime behavior of its trajectories becomes stationary, all singularities are critical points, and the stable singularities are exactly the local minima. Finally, we sketch two numerical methods based on our approach.

关键词： Constrained optimization variable metric algorithms Quadratic slack variables Projected gradient algorithms Gradient systems

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