Dynamic Hassan Nelder Mead with Simplex Free Selectivity for Unconstrained Optimization

We propose a free selective simplex for the downhill Nelder Mead simplex algorithm (1965), rather than the determinant simplex that forces its elements to perform a single operation, such as reflection. Unlike the Nelder-Mead algorithm, the elements of the proposed simplex select various operations of the algorithm to form the next simplex. In this way, we allow non-isometric reflections similar to that of the Nelder Mead, triangle simplex, but with rotation through an angle, permitting the proposed algorithm to have more control over the simplex, to change its size and direction for better performance. As a consequence, the solution that comes from the proposed simplex is always dynamic adaptive in size and orientation to different landscapes of mathematical functions. The proposed algorithm is examined in a large collection of different structures and classes of optimization problems. Additionally, comparisons are made with two enhanced, up-to-date versions of the Nelder-Mead algorithm. The numerical results show that Hassan Nelder Mead is stable due to non-dependence on the number of parameters processed. It also performs a higher accuracy for high dimensions compared with the other algorithms and a faster convergence rate toward global minima with respect to the number of simplex gradient estimates.