A regularized Newton method for degenerate unconstrained optimization problems

In this paper, a new regularized factorization method is presented for solving unconstrained minimization problems in which the Hessian matrix may not be a positive definite or may be close to singular. With employing the modified quadrant interlocking factorization (QIF), an efficient algorithm is developed to find a suitable method. Moreover, under suitable conditions, the global convergence of the proposed method is established. Some interesting examples are solved by the proposed method and the improved Cholesky factorization method. The results show that the proposed method, in the most cases, the algorithm has a faster convergence than the improved Cholesky factorization method. KEYWORDSimproved Cholesky factorization, modified newton method, QIF, unconstrained optimization Int J Numer Model. 2019;32:e2580.wileyonlinelibrary.com/journal/jnm

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“…Proposition (Cheng and Higham and Shen et al) If

λ_{\min} false(\overset{D}{true} false) < 0

, then

false‖ E_{k} ‖_{2} = λ_{\max} false(E_{k} false) \leq λ_{\max} false(W_{k} W_{k}^{T} false) ((), δ_{k} - \frac{λ_{\min} false(\nabla^{2} f_{k} false)}{λ_{\min} false(W_{k} W_{k}^{T} false)}) .

0 \leq λ_{\min} false(\overset{D}{true} false) < δ_{k}

, then

false‖ E_{k} ‖_{2} = λ_{\max} false(E_{k} false) \leq λ_{\max} false(W_{k} W_{k}^{T} false) δ_{k} .

λ_{\min} false(\overset{D}{true} false) \geq δ_{k}

, then ‖ E k ‖ 2 = 0, which is true if

λ_{\min} false(\nabla^{2} f_{k} false) \geq λ_{\max} false(W_{k} W_{k}^{T} false) δ_{k}

. Finally,

λ_{\min} false(B_{k} false) \geq λ_{\max} false(W_{k} W_{k}^{T} false) δ_{k}

in all cases.…”

Section: Convergencementioning

confidence: 98%

Using a new regularized factorization method for unconstrained optimization problems

Niri

Fazeli

Hosseini

2019

Int J Numerical Modelling

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“…Unconstrained optimization is the aim of many papers, and it has a variety of applications (see, for example, [1][2][3][4][5]). Nevertheless, the existing numerical methods for solving the general unconstrained optimization problem up to the second order have a very low convergence rate in the case of degenerate problems [6][7][8][9][10][11][12][13][14][15][16][17] since for increasing the convergence rate, it is necessary to use derivatives of orders greater than two [6,7]. At the same time, using derivatives of the 3 rd and 4 th orders makes a numerical method very time-consuming.…”

Section: Introductionmentioning

confidence: 99%

Higher-Order Optimality Conditions for Degenerate Unconstrained Optimization Problems

Zadachyn

2022

JODEA

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In this paper necessary and sufficient conditions of a minimum for the unconstrained degenerate optimization problem are presented. These conditions generalize the well-known optimality conditions. The new optimality conditions are presented in terms of polylinear forms and Hesse's pseudoinverse matrix. The results are illustrated by examples.The formulation and appearance of these conditions differ from high-order optimality conditions by other authors. The suggested representation of high-order optimality conditions makes them convenient for the evaluation of the convergence rate for unconstrained optimization methods in the case of a singular minimum point, for example, for the analysis of Newton's and quasi-Newton's methods.

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“…We consider the unconstrained optimization problem [1]- [3] ( ) min , ) H x respectively. Throughout this paper, we assume that the solution set of (1.1) is nonempty and denoted by X, and in all cases ⋅ refers to the 2-norm.…”

Section: Introductionmentioning

confidence: 99%

A Regularized Newton Method with Correction for Unconstrained Convex Optimization

Li¹,

Qin²,

Wang³

2016

OJOp

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In this paper, we present a regularized Newton method (M-RNM) with correction for minimizing a convex function whose Hessian matrices may be singular. At every iteration, not only a RNM step is computed but also two correction steps are computed. We show that if the objective function is LC 2 , then the method posses globally convergent. Numerical results show that the new algorithm performs very well.

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A regularized Newton method for degenerate unconstrained optimization problems

Cited by 19 publications

References 14 publications

Using a new regularized factorization method for unconstrained optimization problems

Using a new regularized factorization method for unconstrained optimization problems

Higher-Order Optimality Conditions for Degenerate Unconstrained Optimization Problems

A Regularized Newton Method with Correction for Unconstrained Convex Optimization

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