Finding and Choosing among Multiple Optima

Guenther, John; Lee, Herbert K. H.; Gray, Genetha Anne.

doi:10.4236/am.2014.52031

Cited by 4 publications

(16 citation statements)

References 21 publications

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“…The first step is to locate the optima in the input space. For this purpose there are several search methods for computer experiments, [2,1,14,15,16]. After finding all the optima of interest, pattern search is run using the user inputs for the weights of the four measures and the grid points for each optimum to maximize each optimum's utility and determine its tolerance center.…”

Section: A Robust Algorithm For Optimum Utilitymentioning

confidence: 99%

“…This summarizes the method for choosing optima as presented in [1]. Here there is no requirement for center of the tolerance region to be moved from the optimum point.…”

Section: Introductionmentioning

confidence: 99%

“…In viewing the graph one can see that this minima, although not the best in all measures, compares favorably in all measures to the other minima. This shows how optimum utilities are computed in [1] in the case where the measures are known. However, in the case where the minima are modeled by an emulator, the true values of the measures in the tolerance region are estimated by the emulator's predictions of the surface in the tolerance region.…”

Section: Introductionmentioning

confidence: 99%

“…Thus it is important to be able to accurately assess the utility of different local optima. Our previous work has focused more on location of optima [1,2], and assessment of utility under the assumption that the values of the objective function are spherically symmetric in the local region around an optimum. Here we provide a more robust algorithm for evaluating utility, in that it works well for both symmetric and non-symmetric optima.…”

Section: Introductionmentioning

confidence: 99%

“…This region is defined by the accuracy with which the influential variable's can be specified. [1] defines four measures of interest for a local minimum: 1) The minimum value (lower bound), 2) the mean value in the tolerance region, 3) the maximum value in the tolerance region (upper bound), and, 4) the range of values (upper bound minus lower bound) in the tolerance region. Figure 1 illustrates the different measures discussed above.…”

Section: Introductionmentioning

confidence: 99%

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A Robust Algorithm for Optimum Utility

Guenther¹,

Lee²

2017

JAAM

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The black box functions found in computer experiments often have multiple optima. When there are multiple optima, the goal of a computer experiment should be to determine the best optimum for the purposes of the experiment. This brings up the concept of the utility of an optimum: The degree to which the optimum fulfills the purposes of the experiment. The utility of an optima is based on measures of the variability of the surface in the tolerance region (region of interest around the optima) defined by the accuracy with which influential variables can be specified. This tolerance region can be symmetric or asymmetric. For symmetric optima the tolerance region that provides maximum utility is centered at the optimum point. For asymmetric optima, it may be advantageous to displace the center of the tolerance region from the optimum point. For example, a very skewed optimum (minimum in this case) can increase significantly in one direction but only slightly in the opposite direction from the minimum point. For such an optimum the center point of the tolerance region that provides maximum utility for the optimum is displaced from the optimum point along the direction that only slightly increases. The algorithm discussed in this paper locates the center point for the tolerance region to achieve the best utility for any optimum, symmetric or asymmetric. Making use of the surface predictions of the emulator around the minimum point, it employs pattern search to find the best center point for any optimum and for any dimensionality.

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Section: A Robust Algorithm For Optimum Utilitymentioning

confidence: 99%

“…This summarizes the method for choosing optima as presented in [1]. Here there is no requirement for center of the tolerance region to be moved from the optimum point.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Robust Algorithm for Optimum Utility

Guenther¹,

Lee²

2017

JAAM

Self Cite

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Active Learning for Enumerating Local Minima Based on Gaussian Process Derivatives

Sugita

Toyoura

et al. 2020

Neural Computation

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We study active learning (AL) based on gaussian processes (GPs) for efficiently enumerating all of the local minimum solutions of a black-box function. This problem is challenging because local solutions are characterized by their zero gradient and positive-definite Hessian properties, but those derivatives cannot be directly observed. We propose a new AL method in which the input points are sequentially selected such that the confidence intervals of the GP derivatives are effectively updated for enumerating local minimum solutions. We theoretically analyze the proposed method and demonstrate its usefulness through numerical experiments.

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Cluster Search Algorithm for Finding Multiple Optima

Guenther¹,

Lee²

2016

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The black box functions found in computer experiments often result in multimodal optimization programs. Optimization that focuses on a single best optimum may not achieve the goal of getting the best answer for the purposes of the experiment. This paper builds upon an algorithm introduced in [1] that is successful for finding multiple optima within the input space of the objective function. Here we introduce an alternative cluster search algorithm for finding these optima, making use of clustering. The cluster search algorithm has several advantages over the earlier algorithm. It gives a forward view of the optima that are present in the input space so the user has a preview of what to expect as the optimization process continues. It employs pattern search, in many instances, closer to the minimum's location in input space, saving on simulator point computations. At termination, this algorithm does not need additional verification that a minimum is a duplicate of a previously found minimum, which also saves on simulator point computations. Finally, it finds minima that can be "hidden" by close larger minima.

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Finding and Choosing among Multiple Optima

Cited by 4 publications

References 21 publications

A Robust Algorithm for Optimum Utility

A Robust Algorithm for Optimum Utility

Active Learning for Enumerating Local Minima Based on Gaussian Process Derivatives

Cluster Search Algorithm for Finding Multiple Optima

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