Simpat: Self-Bounding Direct Search Method for Optimization Donald 1. Keeferl Gulf Research & Development Co., Pittsburgh, Pa. 16230
The Simplex search method has been modified to include a new procedure for dealing with bounds on the independent variables. Through recursive partitioning of the independent variable set, Pattern search is merged with Simplex search to form a composite hillclimber named Simpat. In Simpat, Pattern search is used for those variables which are at or are very near their bounds, while the Simplex method is applied to the remaining variables. Consequently, Simpat is self-bounding and is a direct search method in that it requires only values for the objective function in order to proceed. Results on test problems have been most encouraging. In conjunction with a penalty function straiegy for handling constraints, Simpat has proved extremely successful in optimizing realistic engineering-economic models-including several which were exceedingly large and complex.
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the Complex method. We compare t’he new method with the popular Variable Metric and Pattern search algorithms and show that, when coupled with a n exterior penalty function scheme, Simpat is a n extremely effective means of solving practical engineering-economic problems. We begin by examining the Simplex technique, which forms the basis for t’he new method.
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Simplex Method
T h e general nonlinear programming problem can be stated as follows: 3
maximize/minimize y (9)
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subject to the constraints 3
g,(X)
=
0,j
=
3
g , ( X ) F: 0 , j
=
k
1, 2 ,
...)
k, k
