Проверка численной эффективности методов глобальной оптимизации

Abstract

In this paper, new difficult test problems are proposed to test the numerical efficiency of global optimization methods. These are problems of unconstrained optimization with unknown solutions. The proposed test problems are inseparable and have arbitrary dimensions. The author also proposes to include the test functions by J. Nie in the list of test functions for numerical verification of the effectiveness of methods. These functions are also inseparable functions of arbitrary dimensions with unknown solutions. The proposed test problems have many local extrema. Testing the effectiveness of global optimization methods for such functions is simplified. If the method allows improving the found solutions to test problems, then it will be more effective. The existing global optimization methods are compared with the exact quadratic regularization method developed by the author. This method is compared with known software packages that implement modern methods of global optimization. These packages include several methods. The best of them use convex relaxation of the problem to obtain estimates of solutions with subsequent use of local optimization programs. But even such powerful packages have difficulties in solving the considered test problems. Some test problems, for example, with the Rana or Egg Holder function, have been solved by different methods for over 20 years. During this time, no method has allowed obtaining results that are obtained by the method of exact quadratic regularization. For almost all complex test problems with unknown solutions, this method yielded better solutions. Sometimes the advantage of this method was significant, as is the case with the Rana test function. The essence of the exact quadratic regularization method is to transform any global optimization problem to maximize the square of the Euclidean norm of a vector on a convex set. This problem is computationally much simpler. Often, with such a transformation, the multimodal problem becomes unimodal, which is easy to solve. Keywords: test problems, global optimization, unimodal problems, multimodal problems, numerical methods.