Optimization via gradient oriented polar random search

Search algorithms are often used for optimization problems where its mathematical formulation is difficult to be analyzed, e.g., simulation optimization. In literature, search algorithms are either driven by gradient or based on random sampling within specified neighborhood, but both methods have limitation as gradient search can be easily trapped at a local optimum and random sampling loses efficiency by not utilizing local information such as gradient direction that might be available. A combination of the two is believed to overcome both disadvantages. However, the main difficulty is how to incorporate and control randomness in a direction instead of a point. Thus, this paper makes use of a polar coordinate representation in any high dimension to randomly generate directions where the concentration can be explicitly controlled, based on which a brand new Gradient Oriented Polar Random Search (GO-POLARS) is designed and proved to satisfy the conditions for strong local convergence.