**5. Global optimization**

Along the investigation of the optimization problem, there are two kinds of solution points (Luenberger, 1984): *local minimum points*, and *global minimum points*. A point **u***\** is said to be a *global minimum point* of *f* over if *f*(**u**) *≥ f*(**u***\**) for all **u**. If *f*(**u**)*> f*(**u***\**) for all **u**, **u** *≠* **u***\**, then **u***\** is said to be a *strict global minimum point* of *f* over .

There are several search methods devoted to find the global minimum of a nonlinear objective function, since it is not an easy task. Well known methods such as genetic algorithms (Vose, 1999), differential evolution algorithm (Price *et al.*, 2005) and simulated annealing (Kirkpatrick, 1983) could be used in this case. The main characteristic of these methods is that the global (or near global) optimum is obtained through a high number of functional evaluations.

As proposed by Santos *et al.* (2005), to use the best feature of local optimization method (low computational cost) and global optimization method (global minimum), it is considered using the so-called tunneling strategy (Levy and Gomez, 1985) (Levy and Montalvo, 1985), a methodology designed to find the global minimum of a function. It is composed of a sequence of cycles, each cycle consisting of two phases: a minimization phase having the purpose of lowering the current function value, and a tunneling phase that is devoted to find a new initial point (other than the last minimum found) for the next minimization phase. This algorithm was first introduced in Levy and Gomez (1985), and the name derives from its graphic interpretation.

The first phase of the tunneling algorithm (minimization phase) is focused on finding a local minimum **u***\** of Equation (21), while the second phase (tunneling phase) generates a new initial point **u***0***u***\** where *f*(**u***0*) *f*(**u***\**). In summary, the computation evolves through the following phases:


In other words, there exists **u***0 Z* = {**u**  – {**u***\**} | *f*(**u**) *f*(**u***\**) }. To move from **u***\** to **u***<sup>0</sup>* along the tunneling phase a new initial point **u** = **u***\** + , **u** is defined and used in the auxiliary function

$$F(\mathbf{u}) = \frac{f(\mathbf{u}) - f(\mathbf{u}^\*)}{[(\mathbf{u} - \mathbf{u}^\*)^T (\mathbf{u} - \mathbf{u}^\*)]^\gamma} \tag{29}$$

that has a pole in **u***\** for a sufficient large value of �. By computing both phases iteratively, the sequence of local minima leads to the global minimum. Different values for � are suggested (Levy and Gomez, 1985) (Levy and Montalvo, 1985) for being used in Equation (29) to avoid undesirable points and prevent the search algorithm to fail.
