Various Degrees of Steadiness in NSGA-II and Their Influence on the Quality of Results

ABSTRACT Steady-state evolutionary algorithms are often favoured over generational ones due to better scalability in parallel and distributed environments. However, in certain conditions they are able to produce results of better quality as well. We consider several ways to introduce various "degrees of steadiness" in the NSGA-II algorithm, some of which have not been known in literature, and show experimentally (on a corpus of 21 test problems) the presence of a general trend: algorithms with more steadiness yield better results

Hybridizing Non-dominated Sorting Algorithms: Divide-and-Conquer Meets Best Order Sort

Many production-grade algorithms benefit from combining an asymptotically efficient algorithm for solving big problem instances, by splitting them into smaller ones, and an asymptotically inefficient algorithm with a very small implementation constant for solving small subproblems. A well-known example is stable sorting, where mergesort is often combined with insertion sort to achieve a constant but noticeable speed-up. We apply this idea to non-dominated sorting. Namely, we combine the divide-and-conquer algorithm, which has the currently best known asymptotic runtime of $O(N (\log N)^{M - 1})$, with the Best Order Sort algorithm, which has the runtime of $O(N^2 M)$ but demonstrates the best practical performance out of quadratic algorithms. Empirical evaluation shows that the hybrid's running time is typically not worse than of both original algorithms, while for large numbers of points it outperforms them by at least 20%. For smaller numbers of objectives, the speedup can be as large as four times.Comment: A two-page abstract of this paper will appear in the proceedings companion of the 2017 Genetic and Evolutionary Computation Conference (GECCO 2017

МОДИФИКАЦИЯ МЕТАЭВРИСТИЧЕСКОГО МЕТОДА ФЕЙЕРВЕРКОВ ДЛЯ ЗАДАЧ МНОГОКРИТЕРИАЛЬНОЙ ОПТИМИЗАЦИИ НА ОСНОВЕ НЕДОМИНИРУЕМОЙ СОРТИРОВКИ

The article suggests a modification for numerical fireworks method of the single-objective optimization for solving the problem of multiobjective optimization. The method is metaheuristic. It does not guarantee finding the exact solution, but can give a good approximate result. Multiobjective optimization problem is considered with numerical criteria of equal importance. A possible solution to the problem is a vector of real numbers. Each component of the vector of a possible solution belongs to a certain segment. The optimal solution of the problem is considered a Pareto optimal solution. Because the set of Pareto optimal solutions can be infinite; we consider a method for finding an approximation consisting of a finite number of Pareto optimal solutions. The modification is based on the procedure of non-dominated sorting. It is the main procedure for solutions search. Non-dominated sorting is the ranking of decisions based on the values of the numerical vector obtained using the criteria. Solutions are divided into disjoint subsets. The first subset is the Pareto optimal solutions, the second subset is the Pareto optimal solutions if the first subset is not taken into account, and the last subset is the Pareto optimal solutions if the rest subsets are not taken into account. After such a partition, the decision is made to create new solutions. The method was tested on well-known bi-objective optimization problems: ZDT2, LZ01. Structure of the location of Pareto optimal solutions differs for the problems. LZ01 have complex structure of Pareto optimal solutions. In conclusion, the question of future research and the issue of modifying the method for problems with general constraints are discussed.В работе предлагается модификация численного метода фейерверков однокритериальной оптимизации для решения задач многокритериальной оптимизации. Метод относится к метаэвристическим алгоритмам, он не гарантирует нахождения точного решения, но может найти достаточно хорошее приближенное решение. Рассматриваются многокритериальные задачи оптимизации с числовыми критериями, имеющими одинаковую важность. Допустимое решение задачи представляется вектором из действительных чисел, значение каждой компоненты которого принадлежит определенному отрезку. Под оптимальным решением понимается решение, оптимальное по Парето. Так как точных решений, оптимальных по Парето, может быть бесконечно много, рассматривается способ нахождения приближения, состоящего из конечного числа решений, оптимальных по Парето. Модификация основана на процедуре недоминируемой сортировки, которая является основной процедурой для управления процессом поиска приближенного решения. Недоминируемая сортировка – это ранжирование решений на основе значений компонент числового вектора, полученных с помощью вычисления критериев. Каждая компонента соответствует определенному критерию, а множество решений разбивается на непересекающиеся подмножества. Первое подмножество – это решения, оптимальные по Парето, второе подмножество – это решения, оптимальные по Парето, если не учитывать первое подмножество, последнее подмножество – это решения, оптимальные по Парето, если не учитывать все предыдущие подмножества. После такого разбиения принимается решение о генерировании новых допустимых решений. Работа метода протестирована на общеизвестных задачах многокритериальной оптимизации с двумя критериями: ZDT2, LZ01. Задачи отличаются структурой расположения решений, оптимальных по Парето. Так LZ01 имеет достаточно сложную структуру решений, оптимальных по Парето. В заключении обсуждаются вопросы о дальнейшем направлении исследований и о возможности модификации метода для задач многокритериальной оптимизации с произвольными, а не параллелепипедными ограничениями

Incremental non-dominated sorting with O(N) insertion for the two-dimensional case.

Abstract-We propose a new algorithm for incremental nondominated sorting of two-dimensional points. The data structure which stores non-dominating layers is based on a tree of Cartesian trees. If there are N points in M layers, the running time for of an insertion is O(M (1 + log(N/M )) + log M log(N/ log M )), which is O(N ) in the worst case. This algorithm can be a basic building block for efficient implementations of steady-state multiobjective algorithms such as NSGA-II

Quality-driven Multi-objective Optimization of Software Architecture Design: Method, Tool, and Application

Software architecting is a non-trivial and demanding task for software engineers to perform. The architecture is a key enabler for software systems. Besides being crucial for user functionality, the software architecture has deep impact on software qualities such as performance, safety, and cost. In this dissertation, an automated approach for software architecture design is proposed that supports analysis and optimization of multiple quality attributes:First of all, we demonstrate an optimization approach for automated software architecture design. It reports the results of applying our architecture optimization framework to an automotive sub-system that was conducted based on a large-scale real world case study. Moreover, we introduce two novel degrees of freedom which demonstrate how the number of processing nodes and their interconnecting network can be codified to fit into a genetic algorithm. Our studies show that these extra degrees of freedom lead to better overall software architecture optimization. Finally, we propose a new search-based approach for generating a set of optimal software architectural solutions for use in software product lines. Our new approach analyses the commonality of the found optimal solutions and proposes a set of solutions which are suitable for the range of products defined by various feature combinations.Algorithms and the Foundations of Software technolog