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Project titleNovel efficient methods and software tools for the time consuming decision making problems with using supercomputers of superior performance

Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-206 - Computational mathematics

Keywordsmulti-objective optimization, multiextremal optimization, multidimensional problems, non-convex constraints, dimension reduction, parallel computations, supercomputing, computing accelerators

Annotation
The project is aimed at the development of an integrated system of models, methods, and software tools for the investigation of complex decision making problems formulated as the problems of multicriterial multiextremal optimization. These problems are the fundamental ones and have found a lot of applications (optimal design, parameter identification, etc.) and are featured by high computation costs. The decision making problems are the field of intense scientific investigations. For example, 2200 scientists and experts from about 100 all over the world are registered in International Society on Multiple Criteria Decision Making (http://www.mcdmsociety.org). The project team has a long (more than 40 years) research experience in this direction in widely recognized in Russia as well as all over the world Nizhni Novgorod school of global optimization. One of the project participants, Prof. V.V. Toropov (at present, professor at School of Engineering and Materials Science, Queen Mary University of London) is a recognized in Great Britain expert in solving the complex decision making problems. Prof. V.V. Toropov has graduated from N.I. Lobachevsky University of Nizhni Novgorod (UNN), and occupied the position of associate professor at UNN from 1983 to 1992. His participation in the project will allow obtaining a synergetic effect from using the experience of UNN and QMUL in the development of the algorithms for solving the complex decision making problems. The achievement of the stated project goal includes the development of a system of the mathematical models, methods, and software tools covering the already known as well as the novel optimization problem statements (with partially computable, incompletely defined, discontinuous functionals). The possibility to alter the decision making problem statements in the course of computations will be the main feature of the models to be developed. All developed algorithms will be substantiated theoretically and tested experimentally in the solving of the test and applied problems. The developed methods and the software implementation of these ones will allow increasing the complexity of the optimization problems being solved essentially. As the result of the project fulfillment, a prototype parallel software system for solving the decision making problems will be developed. This software system will be oriented on the use in the supercomputer systems with superior performance. The scientific results obtained during the fulfillment of the project will be implemented into the educational process at UNN (development of the special courses for the Master program) as well as into the high level experts training system (the development of the retraining programs). The problem of development of the models and methods of support of the processes of superior is a field of intense scientific and applied investigations. The superior problem is immanent to almost all fields of human activities. This determines, in particular, a large number of scientific publications devoted to this problem and a large number of the scientific conferences (such as International Conference on Computational Science, International Conference on Engineering Optimization, etc.) and scientific journals (such as Optimization, Optimization Methods and Software, Journal of Global Optimization, SIAM Journal on Optimization and many more). The novelty of investigations proposed in the project consists, first of all, in the orientation on the most time consuming decision making problems, in the statement of the problem of the development of a general mathematical model for the problems of search for the optimal variants, in the inclusion of a principally important possibility to alter the problem statements in the course of computations into the decision making model, in the development of the computational schemes for the parallel computing, efficiently scalable up to use of several hundred thousand computational cores, and in a substantial expanding of the class of the solvable problems by means of the maximum use of the computational potential of the supercomputer systems of new transpetaflop level of performance.

Expected results
The main planned results of the project and the importance of these ones are as follows: 1. A mathematical model of choice of the optimal variants, covering many already known optimization problem statements. The statement of the problem of the development of such a general model is undertaken in the first time. This result is an important one since it will allow formulating a unified approach to solving the decision making problems instead of using a number of particular problem statements. 2. A mathematical model of the process of search for the optimum variants, providing the possibility to alter the decision making problem statement (the set of criteria and constraints, the varied parameters, the search domain). The statement of the problem of possibility to alter the time consuming decision making problem statements in the course of computations is undertaken for the first time, since the solving of new arising decision making problems without the reuse search information results in the necessity of so long computations that are inadmissible in fact from the practical viewpoint. The formulation of such a model will allow carry out the modeling of the process of choice for the optimal variants in the complex decision making situations more adequately. 3. A novel approach to solving the multicriterial optimization problems with the multiextremal criteria and nonlinear constraints, allowing obtaining the particular as well as the complete estimates of the set of the unimprovable (Pareto optimal) variants. The novelty of this approach consists in using the search information obtained during all previous computations in solving new arising scalar optimization problems. This approach will allow reducing the required amount of computations essentially that will provide the possibility to analyze a wider spectrum of variants to choose the ones most suiting the concept of optimum. 4. A model of the information support of the process of search for the optimal variants, providing the accumulation, storing, efficient processing, and re-use of all search information obtained during the computation process. This result is one of the key issues within the framework of the project since it provides a practical base for application of all developed models and methods. The solving of even single decision making problem statement is a very difficult computational problem; the solving of all problems arising when altering the problem statements is possible only re-using all search information obtained in the course of computation. The statement of the problem of development of a model of the information support of the time consuming choice problems is undertaken for the first time. 5. An integrated family of the methods for solving the multicriterial multiextremal optimization problems with non-convex constraints, allowing the re-use of the available search information and which can be parallelized efficiently for the supercomputer systems with a new level of possible parallelism (up to millions computational units). As it has been already mentioned above, the statement of problem of re-use of the search information is a novel one. The problem of practical use of the computational potential of modern supercomputer systems is of a principal importance for practically a whole spectrum of computer sciences. 6. A scheme of parallel computations in the multicriterial multiextremal optimization problems oriented onto the massive parallelism for the supercomputer systems of new transpetaflop level of performance. This result can be also qualified as one of the especially valuable within the framework of the project since it will allow using the great computational potential of modern supercomputer systems in practice. The developed models and methods along with the organization scheme of the efficient parallel computations will allow increasing the complexity of the solvable problems of the choice of the optimal variants no less than by an order of magnitude. This result is a novel one and will overcome the results achieved by Russian and international researchers in the field of theory and practice of the decision making problems. 7. An integrated software system implementing the models and methods for solving the decision making problems planned to be developed within the framework of the project. The novelty of the software system to be developed consists, first of all, in its mathematical and algorithmic content. As a result of using the system, the supported processes of search of the optimal variants will agree to the theory and practice of solving the decision making problems more adequate. The possibility to use a considerable number of computational units (up to several hundred thousand computational cores) is a valuable result and will overcome all results achieved by Russian and intrnational researchers in the field of theory and practice of the decision making problems. The developed software system can be implemented in the industrial applications for solving the important practical decision making problems in various application fields (for example, in the computer aided design systems). The results obtained during the fulfillment of the project will be published in a series of papers in the scientific publications indexed in Web of Science and Scopus (no less than 15); in the publications indexed in Russian Index of Scientific Citations (not less than 8).

Annotation of the results obtained in 2018
The project “Novel efficient methods and software tools for the time consuming decision making problems with using supercomputers of superior performance” is aimed at the development of an integrated system of models, methods, and software tools for the investigation of complex decision making problems formulated as the problems of multiobjective multiextremal optimization. These problems are the fundamental ones and have found a lot of applications (optimal design, parameter identification, etc.) and are featured by high computation costs. The novelty of investigations proposed in the project consists in the following: - orientation on the most time consuming decision making problems and development of a general mathematical model for the problems of search for the optimal variants for these problems - inclusion of a principally important possibility to alter the problem statements in the course of computations into the decision making model - development of the computational schemes for the parallel computing, efficiently scalable up to use of several hundred thousand computational cores - expanding of the solvable problems class by means of the maximum use of the computational potential of the supercomputer systems In 2018, during the fulfillment of the third stage of the project, the following scientific results have been obtained. 1. A mathematical model of the decision-making problems has been developed, which covers many existing optimization problem statements including the global optimization problems, the nonlinear programming ones, the constrained multiple-criteria optimization problems. Within the framework of developed model, the specified decision making problem statement can be adjusted if the requirements to the optimality are changed. For example, in the case of redundancy of the set of the partial criteria, it may appear to be reasonable to transform some criteria into the constraints. Or, vice versa, if a small feasible domain is obtained, it may be required to loosen the tolerances or to transform some constraints into the criteria, etc. When fulfillment the workplan in 2018, the developed model has been expanded by the opportunity of simultaneous formulating and solving a set of multiple-criteria optimization problems, which can change in the course of computations by adding new problems or removing the existing ones. The solving of the formed set of problems can be performed sequentially or simultaneously in the time-sharing mode or in parallel when several computational devices are available. The approach developed within the framework of the project (the general model of the decision-making problems, the opportunity of altering the problem statement in the course of computations, the formation of a set of problems solved simultaneously) is a novel one, allowing modeling the processes of choice of the optimal variants in various fields of application of science and technology adequately. 2. The parallel algorithm for the construction of a uniform approximation of the set of weakly efficient solutions in the multicriterial optimization problems has been studied. The absence of a guarantee of a full coverage of the Pareto-optimal set or of the Slater-optimal set on a wide class of problems including the ones with the multiextremal criteria is the main problem in the use of the known scalarization methods (based on the idea of the criteria convolution). A method providing a uniform convergence to the Slater set for the Lipschitzian multiextremal optimization problems has been developed by prof. R. G. Strongin on the basis of the information-statistical approach. A scalarization scheme for the multicriterial problems has been substantiated, when using which the set of the weakly efficient solutions of the initial problem would coincide to the set of the globally-optimal solutions of the reduced scalar problem. The method of solving of the arising scalar problem is based on the information-statistical global optimization algorithm (when solving the unconstrained problems) or on the index algorithm (when solving the constrained problems). These methods belong to the class of the characteristical algorithms that allows applying the parallelization technique based on the choice of the most promising subsets of the search domain and on the parallel execution of the scheduled iterations in these subsets. The numerical experiments with the parallel algorithm carried out using the sets of test multicriterial optimization problems have demonstrated the speedup close to linear one (in the number of the values of the objective functions computed). 3. A detailed model for the information support of the process of solving the multicriterial optimization problems has been worked out. The developed model is oriented onto the reuse and optimal storage of the search information obtained in the course of computations. When solving the complex problems of choice the optimal variants, the volume of the search information is considerable (hundreds of gigabytes). In order to provide efficient storage and processing of such volume of information, a block method of representation of the search information has been developed. This approach allows reducing the global procedures of the data analysis to the local operations of processing of separate small blocks of the search information. The block representation allows using an external memory of practically unlimited volume for storing the search information, and loading only a small number of blocks, which are necessary for performing the processing of search information at every current moment, into the random access memory. The storing of the search information with the use the external memory increases the reliability of computations and allows organizing a suspense of computations with an opportunity of further continuation. The numerical experiments on the Lobachevskii supercomputer confirming a high efficiency of the proposed model of processing the search information have been performed. 4. The development of the family of methods for solving the multiple multicriterial global search problems has been continued. The employed algorithms have been developed within the framework of the information-statistical theory of global search and provide the efficient solving of the whole spectrum of problems considered in the framework of the project (the multicriterial optimization problems, the nonlinear programming with the non-convex constraints, the global optimization problems). The family of algorithms forms a single computational scheme for solving the multiple multicriterial global search problems that allows organizing the efficient reuse of the whole search information obtained in the course of computations. The reuse of the search information provides a basis for overcoming the essential increasing of the computational complexity at the variation of the statements of the problems being solved by means of continuing the global search taking into account all preceding computations. The developed computational schemes for the reuse of the search information were theoretically substantiated. The estimates of the reduction of the amount of computations at the multiple solving the multicriterial optimization problems have been made. The conditions of sufficiency of the search information have been revealed, when altering the problem statements of the problems being solved, the volume of the search information appears to be enough to obtain the estimates of the optimal variants without conducting any additional computations. The results have been published in Journal of Global Optimization (doi: 10.1007/s10898-018-0624-3). 5. Within the framework of the project, the decision making problems were considered, the mathematical models of which are the time-consuming global optimization problems. In the numerical solving of the problems of this class, the amount of necessary computations approaches the petaflop level even at relatively small number of variables. For solving such problems using the supercomputer systems of superior performance, the generalized computational schemes have been proposed, in the framework of which many efficient parallel global optimization algorithms can be applied. The developed schemes include various methods of multilevel decomposition of the parallel computations that provides the opportunity of efficient application of the systems with shared and distributed memory as well as of the use of the accelerators of computations with a large number (tens and hundreds thousand) cores/processors for solving the global optimization problems. The description of the computational schemes of the parallel algorithms for the time-consuming global optimization problems is presented in Lobachevskii Journal of Mathematics (doi: 10.1134/S1995080218040133). For testing the developed approach, the large-scale numerical experiments using the high-performance supercomputer systems (with the use of more than 250 thousand computational cores) have been carried out. 6. A parallel software system Globalizer intended for the solving of the global optimization problems using modern supercomputers with the heterogeneous architectures has been developed. The system Globalizer implements the general scheme of parallel solving the globally-optimal choice problems proposed in the framework of the project with the use of a wide spectrum of the computer systems: with the shred and distributed memory, NVIDIA graphical processors and Intel Xeon Phi co-processors. The system Globalizer can be used in two different variants: - GlobalizerPro is a research version of the system provides the full set of options and is oriented onto the solving of the most complex optimal choice problems. - GlobalizerLite is a limited version of the system; it is accessible for free use for solving the problems of the medium level of complexity. The description of the software system was presented in the journal Numerical Algebra, Control, and Optimization (doi: 10.3934/naco.2018003). The project results are presented in 13 publications, including - 12 papers in the journals indexed in Web of Science and/or Scopus. - 1 publication in the sources indexed in Russian Index of Scientific Citations. The information on the project is presented on the website http://hpc-education.unn.ru/исследования/гранты/принятие-решений.

Publications

1. Barkalov K.A., Gergel V.P. High performance computing for global optimization problems Сборник трудов ИТНТ-2018, с. 2900-2905 (year - 2018)

2. Barkalov K.A., Sorvasov V.V., Lebedev I.G. Comparison of Dimensionality Reduction Schemes for Parallel Global Optimization Algorithms Communications in Computer and Information Science, vol. 965 (year - 2019) https://doi.org/10.1007/978-3-030-05807-4_5

3. Candelieri A., Giordani I., Archetti F., Barkalov K., Meyerov I., Polovinkin A., Sysoyev A., Zolotykh N. Tuning hyperparameters of a SVM-based water demand forecasting system through parallel global optimization Computers and Operations Research, - (year - 2018) https://doi.org/10.1016/j.cor.2018.01.013

4. Gergel V., Barkalov K., Lebedev I. A Global Optimization Algorithm for Non-Convex Mixed-Integer Problems Lecture Notes in Computer Science, vol. 11353, pp. 1-4 (year - 2019) https://doi.org/10.1007/978-3-030-05348-2_7

5. Gergel V., Barkalov K., Lebedev I., Rachinskaya M., Sysoyev A. A Flexible Generator of Constrained Global Optimization Test Problems AIP Conference Proceedings, - (year - 2018)

6. Gergel V., Barkalov K., Sysoyev A. A novel supercomputer software system for solving time-consuming global optimization problems Numerical Algebra, Control and Optimization, vol. 8(1), pp. 47-62 (year - 2018) https://doi.org/10.3934/naco.2018003

7. Gergel V., Kozinov E. Efficient multicriterial optimization based on intensive reuse of search information Journal of Global Optimization, Vol. 71 (1), pp. 73-90 (year - 2018) https://doi.org/10.1007/s10898-018-0624-3

8. Gergel V.P., Barkalov K.A., Kozinov E.A.,Toropov V.V. Parallel Multipoint Approximation Method for Large-Scale Optimization Problems Communications in Computer and Information Science, vol. 910, pp. 174-185 (year - 2018) https://doi.org/10.1007/978-3-319-99673-8_13

9. Gergel V.P., Kozinov E.A. GPU-based Parallel Computations in Multicriterial Optimization Communications in Computer and Information Science, vol. 965 (year - 2019) https://doi.org/10.1007/978-3-030-05807-4_8

10. Raheel M., Toropov V. Topology optimization of an aircraft wing with an outboard X-stabilizer Multidisciplinary Analysis and Optimization Conference, AIAA 2018-3885 (year - 2018) https://doi.org/10.2514/6.2018-3885

11. Sovrasov V.V. Comparison of dimensionality reduction schemes for derivative-free global optimization algorithms Procedia Computer Science, vol. 136, pp. 136-143 (year - 2018) https://doi.org/10.1016/j.procs.2018.08.246

12. Strongin R.G., Gergel V.P., Barkalov K.A., Sysoyev A.V. Generalized Parallel Computational Schemes for Time-Consuming Global Optimization Lobachevskii Journal of Mathematics, Vol. 39, No. 4, pp. 576–586 (year - 2018) https://doi.org/10.1134/S1995080218040133

13. Toropov V., Korolev Y., Barkalov K., Kozinov E., Gergel V. HPC Implementation of the Multipoint Approximation Method for Large Scale Design Optimization Problems Under Uncertainty Proceedings of the 6th International Conference on Engineering Optimization, pp. 296–306 (year - 2018) https://doi.org/10.1007/978-3-319-97773-7_27

Annotation of the results obtained in 2016
The project «Novel efficient methods and software tools for the time consuming decision making problems with using supercomputers of superior performance» is aimed at the development of an integrated system of models, methods, and software tools for the investigation of complex decision making problems formulated as the problems of multiobjective multiextremal optimization. These problems are the fundamental ones and have found a lot of applications (optimal design, parameter identification, etc.) and are featured by high computation costs. The novelty of investigations proposed in the project consists in the following: • orientation on the most time consuming decision making problems and development of a general mathematical model for the problems of search for the optimal variants for these problems • inclusion of a principally important possibility to alter the problem statements in the course of computations into the decision making model • development of the computational schemes for the parallel computing, efficiently scalable up to use of several hundred thousand computational cores • expanding of the solvable problems class by means of the maximum use of the computational potential of the supercomputer systems In 2016, when completing the first stage of the project, the following results have been obtained. 1. A general model for the choice of the optimal variants in the decision-making problems has been developed, which: a. Combines various optimization problem statements (with linear, convex, and non-convex functions, multiobjective problems, the multiparameter optimization problems), b. provides more adequate modeling of the decision making processes for complex problems of the choice of the optimal variants, allowing the possibility of correcting the statements of the problems being solved if the understanding of the optimality is changed. Such an expansion of the generality of the optimal choice model leads to a considerable increase of the computational complexity. 2. A new class of multiple global search problems generated when solving the optimal choice problems has been formulated. Multiple global search problems arise because of the necessity to search for several different variants when altering the statements of problems to be solved. 3. The necessary requirements to the methods and software systems for solving the multiple criteria global search problems to provide a possibility of the practical application of the developed general optimal choice model have been determined. 4. A novel method for solving the multiple criteria global search problems has been proposed as further development of multidimensional generalized global search algorithm developed earlier by Prof. R. G. Strongin. The decision rules of the method have been modified in such a way that all its properties proved earlier are preserved, and the method could be applied for solving several problems simultaneously. 5. The uniform convergence of the proposed algorithm of solving the multiple criteria global search problems has been proven. At the same time, a property of uniform distribution of the computational load among separate processors/cores of a computer system in the parallel of a series of the optimization problems has been established. 6. A computational scheme for the reuse of the search information obtained in the course of computations when solving a series of global optimization problems has been developed. The results of the computational experiments have shown that the reuse of the search information can reduce the amount of computations required for solving an optimal choice problem essentially. 7. In the course of the fulfillment of the first stage of the project, Globalizer Lite parallel system for solving the global optimization problems has been developed. The system is intended for solving the multidimensional global single criterion optimization problems. The method of solving is based on the dimensionality reduction with the use of the evolvents of the multidimensional search domain onto a one-dimensional interval. This reduction is performed with the use of the mutually unambiguous mappings like Peano space-filling curve. The software is oriented onto the systems with distributed memory (the implementation was performed with the use of MPI library). From the viewpoint of a user of the system, the statement of an optimization problem consists in linking a dynamic link library, which the optimized function should be implemented in. In general, the system is a cross-platform (Windows, Linux) console application. Globalizer has been used for the practical evaluation of the efficiency of the parallel solving the optimization problems with the use of the method of operation characteristics. 8. A method for carrying out response surface-based multidisciplinary optimization, using metamodels built in spaces of reduced dimensionality was proposed and followed by a software implementation . The dimensionality of the space in which the metamodels are built is separate for each computational model, and based on a sub-set of the design variables. The chosen sub-set is limited to the design variables relevant to the responses associated with it. The benefit of the method is that since the dimensionality of the space in which an approximation is built is reduced, the computational budget required for building the approximations is also reduced. The method relies on the localised variable dependence of each response in the optimization. A publication (indexed in Scopus) will appear in early 2017. This is the foundation for the further work that will apply this approach (i) to complex engineering problems and (ii) generalised for the multiobjective problems. The project results are presented in 10 publications, including - 7 papers in the journals indexed in Web of Science and/or Scopus (5 of these ones have been published and 2 were accepted for publication). - 3 publications in the sources indexed in Russian Index of Scientific Citations (RISC). The information on the project is presented on the website http://hpc-education.unn.ru/исследования/гранты/принятие-решений

Publications

1. Gergel V.P., Kozinov E.A. Accelerating multicriterial optimization by the intensive exploitation of accumulated search data AIP Conference Proceedings, 1776, 090003-1 - 090003-4 (year - 2016) https://doi.org/10.1063/1.4965367

2. Gergel V.P., Kozinov E.A., Linev A.V., Shtanyk A.A. Educational and Research Systems for Evaluating the Efficiency of Parallel Computations Lecture Notes in Computer Science, vol. 10049, pp 278-290 (year - 2016) https://doi.org/10.1007/978-3-319-49956-7_22

3. Gergel V.P., Кустикова В.Д. Internet-Oriented Educational Course "Introduction to Parallel Computing" Communications in Computer and Information Science, - (year - 2016)

4. Sovrasov V.V. Local Tuning in Peano Curves-based Global Optimization Scheme Procedia Computer Science, vol. 101, 2016, pp. 27–34 (year - 2016) https://doi.org/10.1016/j.procs.2016.11.005

5. Strongin R.G., Barkalov K.A., Nikolaev K.A. New approach to parallel solving a set of global optimization problems AIP Conference Proceedings, 1776, 060002-1–060002-4 (year - 2016) https://doi.org/10.1063/1.4965336

6. Toropov V.V., Ollar J., Jones R.D. Sub-space Metamodel-based Multidisciplinary Optimization of an Aircraft Wing subjected to Bird Strike AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, - (year - 2016)

7. Zhbanova A.S. Development of Parallel Block Multistage Scheme of Dimension Reduction for Globalizer Lite Parallel Software System Procedia Computer Science, vol. 101, 2016, pp. 18–26 (year - 2016) https://doi.org/10.1016/j.procs.2016.11.004

8. Barkalov K.A., Nikolaev K.A. Распределение вычислительной нагрузки при параллельном решении серии задач оптимизации Суперкомпьютерные дни в России: Труды международной конференции (26-27 сентября 2016 г., г. Москва), c. 589-595 (year - 2016)

9. Gergel V.P., Kozinov E.A. Параллельные вычисления при поиске эффективных вариантов в задачах многокритериальной оптимизации Суперкомпьютерные дни в России: Труды международной конференции (26-27 сентября 2016 г., г. Москва), с. 447-453 (year - 2016)

10. Gergel V.P., Kustikova V.D. Интернет-ориентированный учебный курс «Основы параллельных вычислений»: понятно о сложном Суперкомпьютерные дни в России: Труды международной конференции (26-27 сентября 2016 г., г. Москва), с. 1033-1041 (year - 2016)

Annotation of the results obtained in 2017
The project “Novel efficient methods and software tools for the time consuming decision making problems with using supercomputers of superior performance” is aimed at the development of an integrated system of models, methods, and software tools for the investigation of complex decision making problems formulated as the problems of multiobjective multiextremal optimization. These problems are the fundamental ones and have found a lot of applications (optimal design, parameter identification, etc.) and are featured by high computation costs. The novelty of investigations proposed in the project consists in the following: • orientation on the most time consuming decision making problems and development of a general mathematical model for the problems of search for the optimal variants for these problems • inclusion of a principally important possibility to alter the problem statements in the course of computations into the decision making model • development of the computational schemes for the parallel computing, efficiently scalable up to use of several hundred thousand computational cores • expanding of the solvable problems class by means of the maximum use of the computational potential of the supercomputer systems. In 2017, when completing the second stage of the project, the following results have been obtained. 1. An approach to solving the multicriterial optimization problems has been proposed, providing the reduction of the computational costs by means of the reuse of the whole search information obtained in the course of solving the problem. Within the framework of this approach, a parallel algorithm has been developed based on the convolution of the criteria and on the reduction of the whole available search information (i.e. the points of iterations and the values of partial criteria computed at these points) to the values of the scalar criterion of the next scalar problem of nonlinear programming being solved. As a result of such recalculation of the values, at the transition to solving the next scalar subproblem, the search information accumulated earlier will be already available for the utilization by the algorithm solving the scalar problem without any time consuming computations. The constructed algorithm has been proven to be stable and the upper bound of the iterations number required to solve the next scalar subproblem has been done. The numerical experiments on Lobachevsky supercomputer confirming the theoretical statements have been carried out. 2. An efficient data structure for storing the search information based on the memory paging principle has been developed. Within this approach, the random access memory (RAM) used and the external memory (EM) are divided into some continuous parts (pages or blocks) of the same size. The processing of the search information is performed in the blocks, which are located in the RAM pages only. If a block requiring processing is located in the EM, this one is moved to RAM. At that, RAM may be full (all pages in RAM may be busy); in this case, some blocks should be moved from RAM back to EM. A strategy of replacement oriented onto the minimization of exchange of blocks between EM and RAM has been developed. The introduction of the block representation of the accumulated search information allows reducing the global operations of its processing to the local processing of separate blocks. 3. A scheme for efficient utilization of the heterogeneous computational resources of modern supercomputers for solving the global optimization problems has been developed. The proposed approach is based on the block recursive scheme of dimensionality reduction combining the use of Peano space filling curves and the nested optimization scheme. The block-recursive scheme is very flexible: one can vary the number of processors, apply various parallel search methods, and use various kinds of computational devices on different levels of recursion. 4. The theoretical investigation of the developed scheme of the reuse of the search information in solving the multicriterial optimization problems has been carried out. The upper bound of the number of iterations necessary for solving next scalar subproblem has been obtained, which demonstrates that upon achievement a high enough density of trials in the intervals of search domain, the number of additional iterations when solving the scheduled subproblems would decrease continuously because of the reuse of the search information. 5. A prototype parallel software system Globalizer intended for solving the global optimization problems on the supercomputer systems has been developed. The following main blocks of Globalizer have been implemented: a. The block of computing the values of the functionals (criteria and constraints) of the optimization problem being solved (provides the interaction between the external routines and Globalizer according to the predefined interface); b. The optimization subsystem, which provides solving the problems of global optimization, nonlinear programming, multicriterial optimization, and general problems of optimal choice (provides the sequential interaction scheme for the components implemented in this one – the optimal choice problems are solved with the use of the multicriterial optimization block, which, in turn, uses the block of nonlinear programming, etc.) c. The block of accumulation and processing of the search information; d. The block of dimensionality reduction (uses the combined scheme, combining the dimensionality reduction with the use of the space-filling curves and the nested optimization scheme); e. The block of control of the parallel processes when performing the global search (provides, in particular, the distribution of the processes among the computational elements, performs the balancing of the computational loads, etc.); f. A set of tools for the visual presentation of the global search results 6. The experimental evaluation of the efficiency of the developed global search algorithms with the use of Globalizer system (the methods listed above have been included into its algorithmic content) has been performed. Total 32 cluster nodes have been involved into the conducting of the experiments; each node included 3 NVIDIA Kepler K20Х accelerators (2688 stream processors). So far, total over 250 thousand computational cores with the overall speed of 115.8 TFlop/s have been employed. 7. A parallel algorithm for solving the large scale optimization problems based on constructing the metamodels of the objective function and constraints has been developed. This method is an iterative one and is based on the solving of a series of auxiliary problems approximating the initial problem. The approximating problem is constructed on the basis of the metamodels of the criteria and constraints constructed in the trust region, i.e. in a subdomain of the initial domain of the parameters variation, in which the constructed metamodels are considered to be the correct ones. New methods of constructing the metamodels based on the kriging method have been developed. Also, a strategy of the modification of the trust region has been proposed, which can account for the numerical error and occasional failures in the course of modeling necessary for computing the values of the criteria and constraints. The investigation of efficiency of the software implementations of the metamodel construction and solving the approximated problems has been performed; the most time-consuming operations of the algorithm have been identified (with the use of Intel VTune Amplifier XE analyzing tool) and parallelized. The efficiency of the parallel algorithm was evaluated using the classical test problem of the optimal design – the minimization of the volume of a scalable cantilevered beam. The number of parameters in the problem varied from 100 up to 1000, the number of constraints varied from 200 to 2000. The project results are presented in 20 publications, including - 17 papers in the journals indexed in Web of Science and/or Scopus. - 3 publications in the sources indexed in Russian Index of Scientific Citations. The information on the project is presented on the website http://hpc-education.unn.ru/исследования/гранты/принятие-решений

Publications

1. Barkalov K.A., Lebedev I.G. Parallel Algorithm for Solving Constrained Global Optimization Problems Lecture Notes in Computer Science, vol. 10421, pp. 396–404 (year - 2017) https://doi.org/10.1007/978-3-319-62932-2_38

2. Barkalov K.A., Lebedev I.G. Comparing Two Approaches for Solving Constrained Global Optimization Problems Lecture Notes in Computer Science, vol. 10556, pp. 301–306 (year - 2017) https://doi.org/10.1007/978-3-319-69404-7_22

3. Barkalov K.A., Strongin R.G. Test Problems for Parallel Algorithms of Constrained Global Optimization Lecture Notes in Computer Science, vol.10556, pp.18-33 (year - 2017) https://doi.org/10.1007/978-3-319-69404-7_2

4. Barkalov K.A., Strongin R.G. Solving a set of global optimization problems by the parallel technique with uniform convergence Journal of Global Optimization, - (year - 2017) https://doi.org/10.1007/s10898-017-0555-4

5. Gergel V.P. An Approach for Generating Test Problems of Constrained Global Optimization Lecture Notes in Computer Science, vol. 10556, pp. 314–319 (year - 2017) https://doi.org/10.1007/978-3-319-69404-7_24

6. Gergel V.P., Goryachih A.S. Global Optimization Using Numerical Approximations of Derivatives Lecture Notes in Computer Science, vol. 10556, pp. 320–325. (year - 2017) https://doi.org/10.1007/978-3-319-69404-7_25

7. Gergel, V., Kozinov, E. Parallel computing for time-consuming multicriterial optimization problems Lecture Notes in Computer Science, vol. 10421, pp. 446-458 (year - 2017) https://doi.org/10.1007/978-3-319-62932-2_43

8. Gergel, V., Kozinov, E. Efficient methods of multicriterial optimization based on the intensive use of search information Springer Proceedings in Mathematics and Statistics, vol. 197, pp. 27-45 (year - 2017) https://doi.org/10.1007/978-3-319-56829-4_3

9. Gergel, V., Kozinov, E. Accelerating Parallel Multicriterial Optimization Methods Based on Intensive Using of Search Information Procedia Computer Science, vol. 108, pp. 1463-1472 (year - 2017) https://doi.org/10.1016/j.procs.2017.05.051

10. Goryachih A.S. A Class of Smooth Modification of Space-Filling Curves for Global Optimization Problems Springer Proceedings in Mathematics and Statistics, vol. 197, pp. 57-65 (year - 2017) https://doi.org/10.1007/978-3-319-56829-4_5

11. Kalyulin S.L., Modorskii V.Ya., Barkalov K.A., Gergel V.P., Lapteva Yu.A., Kozinov E.A. Интеграция программных комплексов Globalizer и Ansys для оптимизации процессов охлаждения капли в потоке газа Научно-технический вестник Поволжья, т. 5, с.145-148 (year - 2017) https://doi.org/10.24153/2079-5920-2017-7-5-145-148

12. Kalyulin S.L.,Shavrina E.V., Modorskii, V.Y., Barkalov K.A., Gergel V.P. Optimization of Drop Characteristics in a Carrier Cooled Gas Stream Using ANSYS and Globalizer Software Systems on the PNRPU High-Performance Cluster Communications in Computer and Information Science, Volume 753, 2017, Pages 331-345 (year - 2017) https://doi.org/10.1007/978-3-319-67035-5_24

13. Koledina K.F., Sysoev A.V., Gubajdullin I.M., Ahmetov A.F., Zajnullin R.Z., Gergel V.P. Многомерный алгоритм глобального поиска при решении обратной кинетической задачи и его параллельная схема Параллельные вычислительные технологии – XI международная конференция, ПаВТ'2017, г. Казань, 3–7 апреля 2017 г. Челябинск: Издательский центр ЮУрГУ, 2017. 552 с., С. 366 - 376 (year - 2017)

14. Korolev Y.M., Toropov V.V., Shahpar S. Design Optimization Under Uncertainty Using the Multipoint Approximation Method IAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA SciTech Forum, paper no. 2017-1934 (year - 2017) https://doi.org/10.2514/6.2017-1934

15. Ollar J., Mortished Ch., Jones R., Sienz J., Toropov V.V. Gradient based hyper-parameter optimisation for well conditioned kriging metamodels Structural and Multidisciplinary Optimization, vol. 55, pp. 2029–2044 (year - 2017) https://doi.org/10.1007/s00158-016-1626-8

16. Sovrasov V.V. Parallel Multi-Objective Optimization Method for Finding Complete Set of Weakly Efficient Solutions CEUR Workshop Proceedings, vol. 1990, pp. 1-9 (year - 2017)

17. Sysoyev A.V., Barkalov K.A., Lebedev I.G., Sovrasov V. V., Gergel, V.P. Solving Time-Consuming Global Optimization Problems with Globalizer Software System Communications in Computer and Information Science, vol. 793, pp. 108-120 (year - 2017) https://doi.org/10.1007/978-3-319-71255-0_9

18. Sysoyev A.V., Barkalov K.A., Sovrasov V.V., Lebedev I.G., Gergel V.P. Globalizer – A Parallel Software System for Solving Global Optimization Problems Lecture Notes in Computer Science, vol. 10421, pp. 492-499 (year - 2017) https://doi.org/10.1007/978-3-319-62932-2_47

19. Sysoyev A.V., Zhbanova A.S., Barkalov K.A., Gergel V.P. Globalizer Lite: A Software System for Solving Global Optimization Problems Communications in Computer and Information Science, vol. 753, pp. 130-143 (year - 2017) https://doi.org/10.1007/978-3-319-67035-5_10

20. Barkalov K.A., Lebedev I.G. Исследование масштабируемости алгоритма глобальной оптимизации на классе задач варьируемой сложности Суперкомпьютерные дни в России: Труды международной конференции (25-26 сентября 2017 г., г. Москва). – М.: Изд-во МГУ, 2017. – 904 c., С. 753-754. (year - 2017)