A modified Newton method for minimization

Fletcher, R.; Freeman, T. L.

doi:10.1007/bf00933446

Cited by 42 publications

(10 citation statements)

References 7 publications

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“…The idea of selecting either a Newton-type direction or a direction which contains negative curvature information of the objective function, at each iteration, is not new. In particular the algorithms proposed in [6] and [17] are similar in aim to our approach. Both these algorithms use as criterion for selecting a Newton-type direction or an alternative the simple fact that a negative curvature direction exists (or that the Newton-type direction can not be computed).…”

Section: Introductionmentioning

confidence: 87%

Exploiting negative curvature directions in linesearch methods for unconstrained optimization

Gould

Lucidi

Roma

et al. 2000

Optimization Methods and Software

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In this paper we propose efficient new linesearch algorithms for solving large scale unconstrained optimization problems which exploit any local nonconvexity of the objective function. Current algorithms in this class typically compute a pair of search directions at every iteration: a Newton-type direction, which ensures both global and fast asymptotic convergence, and a negative curvature direction, which enables the iterates to escape from the region of local non-convexity. A new point is generated by performing a search along a line or a curve obtained by combining these two directions. However, in almost all if these algorithms, the relative scaling of the directions is not taken into account.We propose a new algorithm which accounts for the relative scaling of the two directions. To do this, only the most promising of the two directions is selected at any given iteration, and a linesearch is performed along the chosen direction. The appropriate direction is selected by estimating the rate of decrease of the quadratic model of the objective function in both candidate directions. We prove global convergence to second-order critical points for the new algorithm, and report some preliminary numerical results.

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Section: Introductionmentioning

confidence: 87%

Exploiting negative curvature directions in linesearch methods for unconstrained optimization

Gould

Lucidi

Roma

et al. 2000

Optimization Methods and Software

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“…After finitely many reductions of the trust region radius 6 either step (5) yields a point satisfying the admissibility test in step (8) or one gets 6 with 0 < ~ < IIx(?)-~11. But then, by Lemma 2.2, step (5) yields a point x(i') with x(~') # ~, IIx(f)-~ll = and 0< ~'<?.…”

Section: A General Algorithmmentioning

confidence: 99%

Curvilinear path and trust region in unconstrained optimization: A convergence analysis

Bulteau

Vial

1987

Mathematical Programming Studies

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In this paper we propose a general algorithm for solving unconstrained optimization problems. The basic step of the algorithm consists in finding a "good" successor point to the current iterate by choosing it along a curvilinear path and within a trust region. This scheme is due to Powell and it has been applied by Sorensen to a particular type of path. We give a series of properties that an arbitrary path should satisfy in order to achieve global convergence and fast asymptotic convergence. We review various paths that have been proposed in the literature and study the extent to which they satisfy our properties.

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“…More recently, the use of this information has been studied by Fletcher and Freeman [9], Gill and Murray [11], or Mukai and Polak [22], among others. In a linesearch context the work of McCormick [18] is particularly relevant.…”

Section: Introductionmentioning

confidence: 99%

Nonconvex optimization using negative curvature within a modified linesearch

Olivares

Moguerza

Prieto

2008

European Journal of Operational Research

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This paper describes a new algorithm for the solution of nonconvex unconstrained optimization problems, with the property of converging to points satisfying second order necessary optimality conditions. The algorithm is based on a pro cedure which, from two descent directions, a Newton type direction and a direction of negative curvature, selects in each iteration the linesearch model best adapted to the properties of these directions. The paper also presents results of numeri cal experiments that illustrate its practical efficiency.

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A modified Newton method for minimization

Cited by 42 publications

References 7 publications

Exploiting negative curvature directions in linesearch methods for unconstrained optimization

Exploiting negative curvature directions in linesearch methods for unconstrained optimization

Curvilinear path and trust region in unconstrained optimization: A convergence analysis

Nonconvex optimization using negative curvature within a modified linesearch

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