Global Optimization with Non-Convex Constraints

Global Search Based on Efficient Diagonal Partitions and a Set of Lipschitz Constants

“…Many decision-making problems arising in various fields of human activity (technological processes, economic models, etc.) can be stated as global optimization problems (see, e.g., [7,26,33]). Objective functions describing real-life applications are very often multiextremal, non-differentiable, and hard to be evaluated.…”

“…Objective functions describing real-life applications are very often multiextremal, non-differentiable, and hard to be evaluated. Numerical techniques for finding solutions to such problems have been widely discussed in literature (see, e.g., [5,14,15,26,33]). …”

“…In fact, it is based on a rather natural assumption that any limited change in the parameters of the objective function yields some limited changes in the characteristics of the object's performance. The knowledge of a bound on the rate of change of the objective function, expressed by the Lipschitz constant, allows one to construct global optimization algorithms and to prove their convergence (see, e.g., [14,15,26,33]). …”

“…Numerous algorithms have been proposed (see, e.g., [5,14,15,18,21,23,26,27,29,32,33]) for solving problem (1.1)-(1.3). One of the main questions to be considered in this occasion is: How can the Lipschitz constant L be specified?…”

“…The more promising and practical approaches are based on an adaptive estimation of L in the course of the search. In such a way, algorithms can use either a global estimate of the Lipschitz constant (see, e.g., [21,26,33]) valid for the whole region D from (1.3), or local estimates L i valid only for some subregions D i ⊆ D (see, e.g., [20,24,29,30,33]). …”