A Geometric Terrain Methodology for Global Optimization

Global optimization remains an important area of active research. Many macroscopic and microscopic applications in science and engineering still present formidable challenges to current global optimization techniques. In this work, a completely different, novel and general geometric framework for continuous global optimization is described. The proposed methodology is based on intelligent movement along the valleys and ridges of an appropriate objective function using downhill, local minimization calculations defined in terms of a trust region method and uphill integration of the Newton-like vector field combined with intermittent SQP corrector steps. The novel features of the proposed methodology include new rigorous mathematical definitions of valleys and ridges, the combined use of objective function and gradient surfaces to guide movement, and techniques to assist both exploration and termination. Collisions with boundaries of the feasible region, integral curve bifurcations, and the presence of non-differentiabilities are also discussed. A variety of examples are used to make key concepts clear and to demonstrate the reliability, efficiency and robustness of terrain methods for global optimization.