A second-order method for the general nonlinear programming problem

The aim of this paper is the study of different approaches to combine and scale, in an efficient manner, descent information for the solution of unconstrained optimization problems. We consider the situation in which different directions are available in a given iteration, and we wish to analyze how to combine these directions in order to provide a method more efficient and robust than the standard Newton approach. In particular, we will focus on the scaling process that should be carried out before combining the directions. We derive some theoretical results regarding the conditions necessary to ensure the convergence of combination procedures following schemes similar to our proposals. Finally, we conduct some computational

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“…and from (16) in Lemma 1 we also have min( k (0), k (0)) < 0 unless we are at a second-order critical point.…”

Section: Theorem 2 Under Assumptions A1 and A2 And Conditions (11)-(mentioning

confidence: 88%

Combining and scaling descent and negative curvature directions

et al. 2009

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“…This required us to assume that all the Kuhn-Tucker points in a certain set were strong local minimizers. In [19], by Mukai and Polak, we find an extension of Newton's method which converges only to stationary points that satisfy second order necessary conditions of optimality and hence are much more likely to be strong local minimizers. The Mukai-Polak algorithm in [19] requires that the function to be minimized have continuous second order derivatives.…”

Section: A Scheme For Automatic Penalty Limitationmentioning

confidence: 97%

“…As we saw in (9b), the Hessian of F is not defined everywhere and hence is discontinuous. Nevertheless, the M u k a i -P o l a k algorithm [19] can be extended to this case by suitably smearing the Hessian, as in methods of feasible directions [23]. We present this new algorithm in Appendix 2, where we see that it replaces the Hessian of F by the matrix, with/x > 0, n(x,X,~,c,~) = Oy(x) + E o:gJ(x) (XJ+c~(~))+…”

Section: A Scheme For Automatic Penalty Limitationmentioning

confidence: 99%

“…The following algorithm is an extension of an algorithm proposed by Mukai and Polak [19]. Since the augmented Lagrangian is in general not twice continuously differentiable, we have introduced a "smearing" scheme which is analogous to the use of e-active constraints in Polak and Zoutendijk methods of feasible directions [23,28] and is related to the e-bundles used in nondifferentiable optimization (see Demyanov [8,9], Polak-Sangiovanni-Vincentelli [24]).…”

Section: Appendix 1 Rate Of Convergencementioning

confidence: 99%

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A globally convergent, implementable multiplier method with automatic penalty limitation

Polak

Tits

1980

Appl Math Optim

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Abstract. This paper deals with penalty function and multiplier methods for the solution of constrained nonconvex nonlinear programming problems. Starting from an idea introduced several years ago by Polak, we develop a class of implementable methods which, under suitable assumptions, produce a sequence of points converging to a strong local minimum for the problem, regardless of the location of the initial guess. In addition, for sequential minimization type multiplier methods, we make use of a rate of convergence result due to Bertsekas and Polyak, to develop a test for limiting the growth of the penalty parameter and thereby prevent ill-conditioning in the resulting sequence of unconstrained optimization problems.

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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 second-order method for the general nonlinear programming problem

Cited by 15 publications

References 27 publications

Combining and scaling descent and negative curvature directions

Combining and scaling descent and negative curvature directions

A globally convergent, implementable multiplier method with automatic penalty limitation

Nonconvex optimization using negative curvature within a modified linesearch

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