Journal of Global Optimization 24: 219–236, 2002. 219  2002 Kluwer Academic Publishers. Printed in the Netherlands. New Classes of Globally Convexized Filled Functions for Global Optimization S. LUCIDI and V. PICCIALLI ` DIS, Universita ‘ La Sapienza’, Via Buonarroti 12, 00185, Rome, Italy E-mail : lucidi @dis.uniroma1.it piccialli @dis.uniroma1.it Abstract. We propose new classes of globally convexized filled functions. Unlike the globally convexized filled functions previously proposed in literature, the ones proposed in this paper are continuously differentiable and, under suitable assumptions, their unconstrained minimization allows to escape from any local minima of the original objective function. Moreover we show that the properties of the proposed functions can be extended to the case of box constrained minimization problems. We also report the results of a preliminary numerical experience. Key words: Filled functions; Global optimization; Nonlinear optimization 1. Introduction Several real world applications need the solution of global optimization problems. However the definition of an efficient method for such problems is still an open question. Many different approaches have been proposed in literature to solve this class of difficult problems. One of these is based on the use of the filled functions. These methods have been initially introduced in Ge (1990), Ge and Qin (1987, 1990), and recently reconsidered in Liu (2001). The idea behind the filled functions is to construct an auxiliary function that allows us to escape from a given local minimum x* of the original objective 1 function f ( x). In this work we try to extend the approach proposed in Ge and Qin (1990), since, in our opinion, the particular filled functions there introduced show interesting theoretical properties. This class of filled functions U( x, x*, t, r ) depends on the 1 local minimum x* of f ( x) and on two parameters, t, r . 0. If parameter r is chosen 1 properly and x* is not a global minimum of the objective function f ( x), then 1 ¯ ¯ U( x, x*, t, r ) has global minimum points x where f ( x ) , f ( x*). Moreover if 1 1 parameter t is greater than a threshold value, which depends on the behaviour of f ( x) ˆ on a compact set V, then U( x, x*, t, r ) has no unconstrained stationary points x [ V 1 ˆ where f ( x ) > f ( x*) except a prefixed point x . However the approach proposed in Ge 1 0 and Qin (1990) has some drawbacks: • the introduced filled functions are not smooth and therefore they are not easy to minimize by using standard code;