Properties and numerical testing of a parallel global optimization algorithm

Abstract

In the framework of multistart and local search algorithms that find the global minimum of a real function f(x), $ x \in S\subseteq R^n$ , Gaviano et alias proposed a rule for deciding, as soon as a local minimum has been found, whether to perform or not a new local minimization. That rule was designed to minimize the average local computational cost $eval_1(\cdotp)$ required to move from the current local minimum to a new one. In this paper the expression of the cost $eval_2(\cdotp)$ of the entire process of getting a global minimum is found and investigated; it is shown that $eval_2(\cdotp)$ has among its components $eval_1(\cdotp)$ and can be only monotonically increasing or decreasing; that is, it exhibits the same property of $eval_1(\cdotp)$ . Moreover, a counterexample is given that shows that the optimal values given by $eval_1(\cdotp)$ and $eval_2(\cdotp)$ might not agree. Further, computational experiments of a parallel algorithm that uses the above rule are carried out in a MatLab environment.