Usage of the local information in Lipschitz global optimization

Global optimization problems, derived from high complexity industrial applications (see, e.g., [1, 2, 17, 19, 21, 22]), are often “black box” and determined by multiextremal objective functions. Solving efficiently this type of problems is a great challenge, since they present a high number of local minima, often with extremely different function values, and do not present a simple mathematical description of the global optima. One of the natural and powerful (from both the theoretical and the applied points of view) assumptions on these problems is that the objective function has bounded slopes. In this case, the methods of the Lipschitz global optimization can be applied (see, e.g., [2, 5, 10, 17, 22, 23]). This kind of global optimization problems is very frequent in practice. Let us refer only to the following examples: general (Lipschitz) nonlinear approximation; solution of nonlinear equations and inequalities; calibration of complex nonlinear system models; black-box systems optimization; optimization of complex hierarchical systems (related, for example, to facility location, mass-service systems); etc. (see, e.g., [2, 4, 5, 8, 15, 17, 22] and the references given therein). As well known, the usage of the only global information about behavior of the objective function during its optimization can lead to a slow convergence of algorithms to global optimum points. Therefore, particular attention should be paid to the usage of a local information in global optimization methods, as well. One of the traditional ways in this context (see, e.g., [5, 9, 10]) recommends stopping the global procedure and switching to a local optimization method in order to improve the solution and to accelerate the search during its final phase. Applying this technique can lead to some problems related to the combination of global and local phases, the main problem being that of determining when to stop the global procedure and start the local one. A premature arrest can provoke the loss of the global solution whereas a late one can slow down the search. In this talk, another fruitful approaches will be discussed. The first one (the so-called local tuning approach, see [12,13,17,18,20,22]) allows global optimization algorithms to tune their behavior to the shape of the objective function at different subintervals of the admissible domain by estimating the local Lipschitz constants. The second one regards a continual local improvement of the current best solution incorporated in a global search procedure (see [16,17]). These techniques become even more efficient when information about the objective function derivatives is available (see [3, 11, 14]). Several Lipschitz global optimization methods illustrating the above-mentioned concepts will be considered and compared numerically with some known algorithms (see [6, 7, 11, 16, 17]).