Numerical verification of the efficiency of global optimization methods

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.

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Estimation of the numerical efficiency of global optimization methods

System technologies ◽

10.34185/1562-9945-3-134-2021-04 ◽

2021 ◽

Vol 3 (134) ◽

pp. 31-39

Author(s):

Anatolii Kosolap

Keyword(s):

Global Optimization ◽

Unconstrained Optimization ◽

Regularization Method ◽

Optimization Problems ◽

Optimization Methods ◽

Test Problems ◽

Numerical Efficiency ◽

Global Optimization Methods ◽

Search Program ◽

Conditional Optimization

Currently, test problems are used to test the effectiveness of new global optimization methods. In this article, we analyze test global optimization problems to test the numerical efficiency of methods for their solution. At present, about 200 test problems of unconditional optimization and more than 1000 problems of conditional optimization have been developed. We can find these test problems on the Internet. However, most of these test problems are not informative for testing the effectiveness of global optimization methods. The solution of test problems of conditional optimization, as a rule, has trivial solutions. This allows the parameters of the algorithms to be tuned before these solutions are obtained. In test problems of conditional optimization, the accuracy of the fulfillment of constraints is important. Often, small errors in the constraints lead to a significant change in the value of an objective function. Construction of a new package of test problems to test the numerical efficiency of global optimization methods and compare the exact quadratic regularization method with existing methods.The author suggests limiting oneself to test problems of unconstrained optimization with unknown solutions. A package of test problems of unconstrained optimization is pro-posed, which includes known test problems with unknown solutions and modifications of some test problems proposed by the author. We also propose to include in this package J. Nie polynomial functions with unknown solutions. This package of test problems will simplify the verification of the numerical effectiveness of methods. The more effective methods will be those that provide the best solutions. The paper compares existing global optimization methods with the exact quadratic regularization method proposed by the author. This method has shown the best results in solving most of the test problems. This paper presents some of the results of the author's numerical experiments. In particular, the best solutions were obtained for test problems with unknown solutions. This method allows solving multimodal problems of large dimensions and only a local search program is required for its implementation.

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INFLUENCE OF LIPSCHITZ BOUNDS ON THE SPEED OF GLOBAL OPTIMIZATION

Technological and Economic Development of Economy ◽

10.3846/20294913.2012.661170 ◽

2012 ◽

Vol 18 (1) ◽

pp. 54-66 ◽

Cited By ~ 5

Author(s):

Remigijus Paulavičius ◽

Julius Žilinskas

Keyword(s):

Global Optimization ◽

Branch And Bound ◽

Lipschitz Function ◽

Optimization Problems ◽

Optimization Methods ◽

Branch And Bound Algorithm ◽

Test Problems ◽

Global Optimization Methods

Global optimization methods based on Lipschitz bounds have been analyzed and applied widely to solve various optimization problems. In this paper a bound for Lipschitz function is proposed, which is computed using function values at the vertices of a simplex and the radius of the circumscribed sphere. The efficiency of a branch and bound algorithm with proposed bound and combinations of bounds is evaluated experimentally while solving a number of multidimensional test problems for global optimization. The influence of different bounds on the performance of a branch and bound algorithm has been investigated.

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