Global Optimization: Software, Test Problems, and Applications

Global optimization of mechanical design problems using heuristic methods such as Simulated annealing (SA) and genetic algorithms (GAs) have been able to find global or near-global minima where prior methods have failed. The use of these nongradient based methods allow the broad efficient exploration of multimodal design spaces that could be continuous, discrete or mixed. From a survey of articles in the ASME Journal of Mechanical Design over the last 10 years, we have observed that researchers will typically run these algorithms in continuous mode for problems that contain continuous design variables. What we suggest in this paper is that computational efficiencies can be significantly increased by discretizing all continuous variables, perform a global optimization on the discretized design space, and then conduct a local search in the continuous space from the global minimum discrete state. The level of discretization will depend on the complexity of the problem, and becomes an additional parameter that needs to be tuned. The rational behind this assertion is presented, along with results from four test problems.

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GOPS: efficient RBF surrogate global optimization algorithm with high dimensions and many parallel processors including application to multimodal water quality PDE model calibration

Optimization and Engineering ◽

10.1007/s11081-020-09556-1 ◽

2020 ◽

Author(s):

Wei Xia ◽

Christine Shoemaker

Keyword(s):

Water Quality ◽

Global Optimization ◽

Objective Function ◽

Black Box ◽

High Dimensional ◽

Function Evaluation ◽

Test Problems ◽

Quality Model ◽

Lake Water Quality ◽

Dimensional Parameter

Abstract This paper describes a new parallel global surrogate-based algorithm Global Optimization in Parallel with Surrogate (GOPS) for the minimization of continuous black-box objective functions that might have multiple local minima, are expensive to compute, and have no derivative information available. The task of picking P new evaluation points for P processors in each iteration is addressed by sampling around multiple center points at which the objective function has been previously evaluated. The GOPS algorithm improves on earlier algorithms by (a) new center points are selected based on bivariate non-dominated sorting of previously evaluated points with additional constraints to ensure the objective value is below a target percentile and (b) as iterations increase, the number of centers decreases, and the number of evaluation points per center increases. These strategies and the hyperparameters controlling them significantly improve GOPS’s parallel performance on high dimensional problems in comparison to other global optimization algorithms, especially with a larger number of processors. GOPS is tested with up to 128 processors in parallel on 14 synthetic black-box optimization benchmarking test problems (in 10, 21, and 40 dimensions) and one 21-dimensional parameter estimation problem for an expensive real-world nonlinear lake water quality model with partial differential equations that takes 22 min for each objective function evaluation. GOPS numerically significantly outperforms (especially on high dimensional problems and with larger numbers of processors) the earlier algorithms SOP and PSD-MADS-VNS (and these two algorithms have outperformed other algorithms in prior publications).

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A Convex Cutting Plane Algorithm for Global Solution of Generalized Polynomial Optimal Design Models

Journal of Mechanical Design ◽

10.1115/1.2826861 ◽

1996 ◽

Vol 118 (1) ◽

pp. 82-88 ◽

Cited By ~ 6

Author(s):

Chihsiung Lo ◽

P. Y. Papalambros

Keyword(s):

Global Optimization ◽

Cutting Plane ◽

Test Problems ◽

Cutting Plane Algorithm ◽

Computational Results ◽

Design Models ◽

Generalized Polynomial ◽

Speed Reducer ◽

Improved Performance ◽

Search Approach

Global optimization algorithms for generalized polynomial design models using a global feasible search approach was discussed in a previous article. A new convex cutting plane algorithm (CONCUT) based on global feasible search and with improved performance is presented in this sequel article. Computational results of the CONCUT algorithm compared to one using linear cuts (LINCUT) are given for various test problems. A speed reducer design example illustrates the application of the algorithms.

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