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In this paper, we develop a new approach for solving a large class of global optimization problems for objective functions which are only continuous on a rectangle of R n. This method is based on the reducing transformation technique by running in the feasible domain a single parametrized Lissajous curve, which becomes increasingly denser and progressively fills the feasible domain. By means of the one-dimensional Evtushenko algorithm, we realize a mixed method which explores the feasible domain. To speed up the mixed exploration algorithm, we have incorporated a DIRECT local search type algorithm to explore promising regions. This method converges in a finite number of iterations to the global minimum within a prescribed accuracy ε > 0. Simulations on some typical test problems with diverse properties and different dimensions indicate that the algorithm is promising and competitive. [ABSTRACT FROM AUTHOR]