A Direct Search Optimization Method That Models the Objective and Constraint Functions by Linear Interpolation

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An iterative algorithm is proposed for nonlinearly constrained optimization calculations when there are no derivatives. Each iteration forms linear approximations to the objective and constraint functions by interpolation at the vertices of a simplex and a trust region bound restricts each change to the variables. Thus a new vector of variables is calculated, which may replace one of the current vertices, either to improve the shape of the simplex or because it is the best vector that has been found so far, according to a merit function that gives attention to the greatest constraint violation. The trust region radius ρ is never increased, and it is reduced when the approximations of a well-conditioned simplex fail to yield an improvement to the variables, until ρ reaches a prescribed value that controls the final accuracy. Some convergence properties and several numerical results are given, but there are no more than 9 variables in these calculations because linear approximations can be highly inefficient. Nevertheless, the algorithm is easy to use for small numbers of variables.