Basin Analysis via Simulation
We illustrate simulated basins of attraction for four example objective functions applying the same four different numerical methods to each (coordinate-search, and steepest-descent with three different line-searches). We define basin size and basin entropy (quantifying basin complexity) and compute each for our examples. We apply the Nelder–Mead method to all four example functions in a separate section because its iterate-multisets are non-singleton, which requires more complicated illustrations. We use the same tools to investigate the practical significance of two well-known counterexamples to good convergence behavior in numerical minimization: the canoe function with coordinate-search and McKinnon’s function (McKinnon, SIAM J. Optim. 9, 148–158 (1998)) with Nelder–Mead. We use our notions of basin size and basin entropy to quantify the extent to which initial data are likely to lead to undesirable consequences.
- 10.Dennis Jr., J.E., Woods, D.J.: Optimization on microcomputers: the Nelder-Mead simplex algorithm. In: Wouk, A. (ed.) New Computing Environments: Microcomputers in Large-Scale Computing. SIAM, Philadelphia (1987)Google Scholar
- 23.Torczon, V.: Multi-directional search: a direct search algorithm for parallel machines. Ph.D. thesis, Rice University, Houston, TX (1989)Google Scholar