National University of Mexico

We will present the basic ideas of a deterministic gradient-based method called the Tunneling Method (Levy-Montalvo, Levy-Gomez), which has the capability to pass from one valley to another of the objective non-convex function, in order to find a monotonic sequence of local minima with lower value of the objective function, towards the global minimum. It has also the ability to find local or global minima at the same value of the objective function.

It has been applied to solve several industrial problems, where the coefficients of a system of partial differential equations that simulate the process, have to be identified solving data fitting inverse problems. In some of these problems, the traditional local optimization methods, are unable to find with sufficient precision, one local minimum due to the shape of the objective function, whereas the Tunneling method, is able to find a sequence of these highly precise minima. It has also been used to find novel molecular structures, with up to 400 variables and chemical processes.

We will also present an Evolutionary method (not requiring derivatives), and describe some of the applications that have been solved using this method. In some cases, we will compare the results between the Tunneling and the Evolutionary methods.

Sequential and parallel versions of the methods will be discussed.