On the optimal parameter values of the Nelder–Mead simplex algorithm

The paper outlines the derivation of optimal parameter values for the Nelder–Mead simplex algorithm as a function of the optimization problem’s dimension. The derivation applies to a general, strictly convex quadratic objective function, under the assumption that the simplex’s centroid probability density function within the ellipsoid defined by the simplex’s worst vertex is independent of the centroid’s distance to the worst vertex. The derived dependences of the Nelder–Mead simplex algorithm parameters show similarities with the heuristic solutions proposed so far. The algorithm’s performance, relative to its default parameter settings, was tested on a quadratic function in 10, 20, 50, and 100 dimensions.