Global Optimization Portal - Algorithm | ||
The surrogate model software program developed at GOP is based on the methodolgy, mathematical formulation and software presented in RDIC.
In addition, a standard gradient-descent optimization routine is developed to carry out the search of global optimum location. The overview of the entire surrogate model based optimization process is given in the chart below. The structure chart of surrogate optimization process in terms of its components is presented in the following.  Dirac delta function is employed as a weighting function in the surrogate model. The Lorentz distribution function is chosen to approximate Dirac delta function in the computation process. easy to use. For example, in two dimensions at sample location (x', y'), the Dirac delta function can be written as,  Plotting the contour chart of the above formula by setting delta width Δ to 1 and rescaling by (π)2, it gives,  The center origin is located at the sample location (x', y') in the above chart. The abscissa is x-x' and the ordinate is y-y'. It is interesting to see that the contour lines near the origin form circular patterns (like L2 norm, the Euclidean norm) and beyond the origin change into shapes similar to diamond (like L1 norm, the Manhattan/taxicab norm). In contrast, IDW (Inverse Distance weighting) employs the L2 norm without pattern changes. Dirac delta function is symmetric. That is, δΔ(x'-x) = δΔ(x-x'). The surrogate model for the entire design space is constructed by using the target function values collected at the sample locations together with the weighting Dirac delta function, similar to the construction usd in IDW (inverse distance weighting) interpolation algorithm. However, unlike IDW which is ad-hoc, the mathematical construction of GOP surrogate model is based upon Monte Carlo method and the Central Limit Theorem. For details, check RDIC and the references listed there. For illustration, the surrogate model in one dimension can be written as follows.  |
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