Variable metric algorithms: Necessary and sufficient conditions for identical behavior of nonquadratic functions

Dixon, L. C. W.

doi:10.1007/bf00934961

Cited by 76 publications

(23 citation statements)

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“…(Dixon [7,8] The theorem formulated above gives us a natural extension of the wellknown Powell's result on the global convergence and local Q-superlinear convergence of the DFP-method 1-15, 16] to all Huang's methods with pk = 1 for all k>0, which generate the sequences of nonsingular matrices {Vk}.…”

Section: Basic Properties Of Considered Methodsmentioning

confidence: 97%

Superlinear convergence of symmetric Huang's class of methods

Stachurski

1981

Numer. Math.

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Summary.In this paper the problem of minimizing the functional f: D~R"~R is considered. Typical assumptions on fare assumed. A class of Quasi-Newton methods, namely Huang's class of methods is used for finding an optimal solution of this problem. A new theorem connected with this class is presented. By means of this theorem some convergence results known up till now only for the methods which satisfy Quasi-Newton condition are extended, that is the results of superlinear convergence of variable metric methods in the cases of exact and asymptotically exact minimization and the so-called direct-prediction case. This theorem allows to interpretate one of the parameters as the scaling parameter.

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Section: Basic Properties Of Considered Methodsmentioning

confidence: 97%

Superlinear convergence of symmetric Huang's class of methods

Stachurski

1981

Numer. Math.

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“…Dixon (1972) notes that with exact line search, all class members yield same iterates and their performance varies markedly. We assume step-size α k is chosen by an inexact line search satisfying two conditions as thus:…”

Section: Quasi-newton Methodsmentioning

confidence: 99%

Stochastic Solution on Convergence Property of Quadratic Functions

Adekunle¹,

Seun²,

Yoro³

2016

CAE

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Constraints satisfaction problems (CSPs) aim at solution via algorithms that search a domain space for goal states. The solution of which must satisfy all constraints and guarantees explicit reasoning structure that conveys data about the problem to the algorithm. Thus, it assigns to an output, a set of variables that satisfies a set of constraints in its bid to prune off huge portion of the search space. This study presents solutions to quadratic functions via David Fletcher Powell method and stochastic method of optimization. To aim this purpose, hybrid neural networks are trained using DFP as a pre-processor to yield approximate solutions to the quadratic function. A trial solution of the quadratic equation is written as sum of two parts: (a) first part satisfies the initial condition for unconstrained optimization using DFP and hybrids as separate methods to solve a quadratic function; while (b) second part uses DFP as a pre-processor with adjustable parameters of for the ANN-TLRN hybrid. Results show that presented method introduces a closer form to the analytic solution. These present method is easily extended to solve a wide range of problems.

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“…It has been shown that the BFGS method is very efficient for solving unconstrained optimization problems (1.1) [11,14,47]. For convex minimization problems, the BFGS method are global convergence if the exact line search or some special inexact line search is used [1,2,4,12,16,32,33,38] and its local convergence has been well established [9,10,17]. For general function f, Dai [5] have constructed an example to show that the standard BFGS method may fail for non-convex functions with inexact line search, Mascarenhas [29] showed that the nonconvergence of the standard BFGS method even with exact line search.…”

Section: 2)mentioning

confidence: 99%