Effect of rounding errors on the variable metric method

Two variants of the extended Rosenbrock function are analyzed in order to find the stationary points. The first variant is shown to possess a single stationary point, the global minimum. The second variant has numerous stationary points for high dimensionality. A previously proposed method is shown to be numerically intractable, requiring arbitrary precision computation in many cases to enumerate candidate solutions. Instead, a standard Newtonian method with multi-start is applied to locate stationary points. The relative magnitude of the negative and positive eigenvalues of the Hessian is also computed, in order to characterize the saddle points. For dimensions up to 100, only two local minimizers are found, but many saddle points exist. Two saddle points with a single negative eigenvalue exist for high dimensionality, which may appear as “near” local minima. The remaining saddle points we found have a predictable form, and a method is proposed to estimate their number. Monte Carlo simulation indicates that it is unlikely to escape these saddle points using uniform random search. A standard particle swarm algorithm also struggles to improve upon a saddle point contained within the initial population.

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“…We subsequently analyze two commonly encountered variants. Dixon and Mills (1994), among others, propose…”

Section: Rosenbrock Variantsmentioning

confidence: 99%

Locating and Characterizing the Stationary Points of the Extended Rosenbrock Function

Kok

Sandrock

2009

Evolutionary Computation

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“…Over the last five years several papers have been published focusing on this topic, e. g. Robertazzi and Schwartz, [10], Dixon and Mills, [3], Goldberg, [5], and Higham, [6]. Comprehensive lists of references can be found in [5,6].…”

Section: Introductionmentioning

confidence: 99%

On Floating-Point Summation

Espelid¹

1995

SIAM Rev.

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“…These problems were eliminated by accumulating separately positive and negative terms (for each component of Vf) in the sum (4.1), adding them together only after all M terms had been processed. " (2) Dixon and Mills [7] This function and its derivatives possess certain symmetries; for example, F(a, b, c, d) F(c, d, a, b) when n 4. It is observed in [7] that expected symmetries in the search (1.2) fl(x op y)= (z op y)(X + 6), 161 < u, op +,-, ,,/.…”

mentioning

confidence: 99%

“…" (2) Dixon and Mills [7] This function and its derivatives possess certain symmetries; for example, F(a, b, c, d) F(c, d, a, b) when n 4. It is observed in [7] that expected symmetries in the search (1.2) fl(x op y)= (z op y)(X + 6), 161 < u, op +,-, ,,/. This model is violated by machines that lack a guard digit, so we explain in 5 how our analysis has to be modified to accommodate such machines.…”

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confidence: 99%

“…It appears that there is some feature of this application, not apparent from [24], that encourages the +/-method to perform better than recursive summation with the natural ordering. The loss of symmetry in a quasi-Newton method that was observed in [7] is easier to understand. For example, symmetries in F in (1.1) can be preserved by using any summation method whose computed answer does not depend on the given order of the data--such as recursive summation with the increasing ordering and with elements of equal magnitude ordered by sign.…”

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confidence: 99%

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The Accuracy of Floating Point Summation

Higham¹

1993

SIAM J. Sci. Comput.

230

197

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The usual recursive summation technique is just one of several ways of computing the sum of n floating point numbers. Five summation methods and their variations are analyzed here. The accuracy of the methods is compared using rounding error analysis and numerical experiments. Four of the methods are shown to be special cases of a general class of methods, and an error analysis is given for this class. No one method is uniformly more accurate than the others, but some guidelines are given on the choice of method in particular cases.

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Effect of rounding errors on the variable metric method

Cited by 17 publications

References 1 publication

Locating and Characterizing the Stationary Points of the Extended Rosenbrock Function

Locating and Characterizing the Stationary Points of the Extended Rosenbrock Function

On Floating-Point Summation

The Accuracy of Floating Point Summation

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