The tunnelling method for solving the constrained global optimization problem with several non-connected feasible regions

Abstract

The problem of finding the global minimum of f(x) subject to the constraints g(x)≥0 and h(x)=0 is considered, where the feasible region generated by the constraints could be non-convex, and could even be several nonconnected feasible regions.

An algorithm is developed which consists of two phases, a “minimization phase” which finds a local constrained minimum and a “tunnelling phase” whose starting point is the local minimum just found, and where a new feasible point is found which has a function value equal to or less than at the starting point and this point is taken as the starting point of the next minimization phase, thus the algorithm approaches the global minimum in an orderly fashion, tunnelling below irrelevant local constrained minima. When nonconnected feasible regions are present the tunnelling phase moves from the minimum of a feasible set to another nonconnected feasible set where the function has lower or equal value. It ignores completely all feasible sets where the function attains a higher value.

Numerical results are presented for 6 examples, for the tunnelling algorithm and a multiple random method. These results show that the method presented here is a robust method and its robustness increases with the density of local minima, the dimension, and the number of nonconnected feasible regions.