# A new method for solving a series of global optimization problems developed

*Lobachevsky University scientists have experimentally proved the theorem on the uniform convergence of the new algorithm*

To create highly effective technical systems and technological processes, in addition to the use of new principles, new materials, new physical effects and other new solutions that determine the overall structure of the object being created, one has to choose the best combination of this object’s parameters (geometric dimensions, electrical characteristics, etc.), since any changes in the parameters with a fixed overall object structure can significantly affect the effectiveness indicators.

In computer-aided design, the testing of possible options can be carried out without involving the actual object, by analyzing its mathematical model for different parameter values. As the models become more and more complex, the need arises for a targeted choice of options in the search for the optimal (most effective) solution.

A team of scientists from the Lobachevsky State University of Nizhny Novgorod (UNN) lead by Professor Roman Strongin has been active in studying targeted choice of options when searching for the optimal solution. It involves an analysis of a small part of the possible options with the aim to exclude from consideration many obviously unpromising cases and to concentrate the search on the subset containing the best option.

"When mathematical models of the processes that occur in an object become more complicated, the efficiency characteristics will not possess the property of monotonicity, that is why local search methods are insufficient to evaluate the best option," says Professor Roman Strongin.

The global search procedures that are used in such problems (also called multi-extremal optimization problems) ensure that the search is targeted by taking into account the limited nature of the change in the object’s characteristics when the changes in its parameters are limited. The mathematical formulation of this fact can take the form of the Lipschitz condition, the uniform Hölder condition, etc.

Multi-extremal optimization problems offer a narrow scope of analytical research opportunities, and they become computationally expensive when trying to get numerical solutions, since in the latter case computational costs grow exponentially with increasing dimension of the problem.

According to Konstantin Barkalov, Associate Professor of the UNN Department of Software and Supercomputer Technologies, the use of modern parallel computing systems expands the scope of global optimization methods. However, at the same time, it poses a challenge of effectively parallelizing the search process.

"That is why the main efforts in this area should be concentrated on developing efficient parallel methods for numerically solving multi-extremal optimization problems and creating appropriate software for modern computing systems on the basis of such methods," emphasizes Konstantin Barkalov.

Usually, the methods of global optimization (both sequential and parallel) are intended to solve a single optimization problem. If one needs to solve a series of q problems, the problems in the series are solved sequentially, one after another. Therefore, the optimum estimation in the i-th problem in the series remains undefined until all preceding problems of the series (with the indices 1, 2, . . . , i − 1) have been completely solved. In the case of limited computational resources, the optima estimates in the problems i + 1, . . . , q will not be obtained if the computation resources are exhausted while solving the i-th problem.

Situations when a series of q problems have to be solved are not extraordinary. For example, a series of scalar problems arises when one needs to find a Pareto set in solving multi-objective optimization problems. In this case, the solution of a single scalar problem corresponds to one of the Pareto optimal points of a multi-objective problem. A series of optimization problems also arises when using dimension reduction methods to solve multidimensional optimization problems. Finally, a series of test problems can also be obtained with the help of a particular test problem generator.

Scientists believe that when solving a set of problems, it is necessary to have initially the estimates of solutions for all problems at once, so that at any time it is possible to evaluate the expediency of continuing the search. In this case, it is desirable to have the optimum estimates for all problems with the same accuracy.

Running many independent processes in a parallel computing system, each of the processes solving one problem from a series, has a number of drawbacks. First, a workload imbalance between the processors will occur. If solving the i -th problem requires considerably fewer iterations of the method than solving the j -th problem, the processor tasked with handling the i -th problem would remain idle after completing the task. Second, the estimates of the optima will be obtained with different precision in different problems. Simpler problems will be solved with higher precision, whereas precision will be lower for more complex problems.

The aim of the research performed at the Lobachevsky University was to develop a new method for solving a series of global optimization problems that would ensure: (i) a uniform loading of all the cores/processors employed; (ii) a uniform convergence to the solutions of all problems in the series.

In the theoretical part of their paper, UNN scientists also proved the theorem on uniform convergence of the new algorithm. The experimental part of the work consisted in solving a series of several hundred test problems of different dimensions, and the results have convincingly demonstrated the presence of uniform convergence.

The new method is an extension of the global optimization algorithm and its parallel generalization, which are described in detail in the monographs: Strongin R.G., Sergeev Ya.D. Global optimization with non-convex constraints. Sequential and parallel algorithms. - Dordrecht: Kluwer Academic Publishers. - 2000. - 704 p.; Strongin R.G., Gergel V.P., Grishagin V.A., Barkalov K.A. Parallel computations in global optimization problems - Moscow: Moscow University Publishing House. - 2013. - 280 p.