A Short Study on Minima Distribution

Global optimization is an imperative development of local optimization because there are many problems in artificial intelligence and machine learning requires highly acute solutions over entire domain. There are many methods to resolve the global optimization, which can be classified into three groups such as analytic methods (purely mathematical methods), probabilistic methods, and heuristic methods. Especially, heuristic methods like particle swarm optimization and ant bee colony attract researchers because their effective and practical techniques which are easy to be implemented by computer programming languages. However, these heuristic methods are lacking in theoretical mathematical fundamental. Fortunately, minima distribution establishes a strict mathematical relationship between optimized target function and its global minima. In this research, I try to study minima distribution and apply it into explaining convergence and convergence speed of optimization algorithms. Especially, weak conditions of convergence and monotonicity within minima distribution are drawn so as to be appropriate to practical optimization methods.