Approaches to discrete optimization problems with interval objective function

В работе дается обзор подходов к решению задач дискретной оптимизации с интервальной целевой функцией. Эти подходы рассматриваются в общем контексте исследований оптимизационных задач с неопределенностями в постановках. Приводятся варианты концепций оптимальности решений для задач дискретной оптимизации с интервальной целевой функцией - робастные решения, множества решений, оптимальных по Парето, слабые и сильные оптимальные решения, объединенные множества решений и др. Оценивается предпочтительность выбора той или иной концепции оптимальности при решении задач и отмечаются ограничения для применения использующих их подходов Optimization problems with uncertainties in their input data have been investigated by many researchers in different directions. There are a lot of sources of the uncertainties in the input data for applied problems. Inaccurate measurements and variability of the parameters with time are some of such sources. The interval of possible values of uncertain parameter is the natural and the only possible way to represent the uncertainty for a wide share of applied problems. We consider discrete optimization problems with interval uncertainties in their objective functions. The purpose of the paper is to provide an overview of the investigations in this field. The overview is given in the overall context of the researches of optimization problems with uncertainties. We review the interval approaches for the discrete optimization problem with interval objective function. The approaches we consider operate with the interval values and are focused on obtaining possible solutions or certain sets of the solutions that are optimal according to some concepts of optimality that are used by the approaches. We consider the different concepts of optimality: robust solutions, the Pareto sets, weak and strong solutions, the united solution sets, the sets of possible approximate solutions that correspond to possible values of uncertain parameters. All the approaches we consider allow absence of information on probabilistic distribution on intervals of possible values of parameters, though some of them may use the information to evaluate the probabilities of possible solutions, the distribution on the interval of possible objective function values for the solutions, etc. We assess the possibilities and limitations of the considered approaches

Multi-Population Parallel Wolf Pack Algorithm for Task Assignment of UAV Swarm

The effectiveness of the Wolf Pack Algorithm (WPA) in high-dimensional discrete optimization problems has been verified in previous studies; however, it usually takes too long to obtain the best solution. This paper proposes the Multi-Population Parallel Wolf Pack Algorithm (MPPWPA), in which the size of the wolf population is reduced by dividing the population into multiple sub-populations that optimize independently at the same time. Using the approximate average division method, the population is divided into multiple equal mass sub-populations whose better individuals constitute an elite sub-population. Through the elite-mass population distribution, those better individuals are optimized twice by the elite sub-population and mass sub-populations, which can accelerate the convergence. In order to maintain the population diversity, population pretreatment is proposed. The sub-populations migrate according to a constant migration probability and the migration of sub-populations are equivalent to the re-division of the confluent population. Finally, the proposed algorithm is carried out in a synchronous parallel system. Through the simulation experiments on the task assignment of the UAV swarm in three scenarios whose dimensions of solution space are 8, 30 and 150, the MPPWPA is verified as being effective in improving the optimization performance.

Network Models for Multiobjective Discrete Optimization

This paper provides a novel framework for solving multiobjective discrete optimization problems with an arbitrary number of objectives. Our framework represents these problems as network models, in that enumerating the Pareto frontier amounts to solving a multicriteria shortest-path problem in an auxiliary network. We design techniques for exploiting network models in order to accelerate the identification of the Pareto frontier, most notably a number of operations to simplify the network by removing nodes and arcs while preserving the set of nondominated solutions. We show that the proposed framework yields orders-of-magnitude performance improvements over existing state-of-the-art algorithms on five problem classes containing both linear and nonlinear objective functions. Summary of Contribution: Multiobjective optimization has a long history of research with applications in several domains. Our paper provides an alternative modeling and solution approach for multiobjective discrete optimization problems by leveraging graphical structures. Specifically, we encode the decision space of a problem as a layered network and propose graph reduction operators to preserve only solutions whose image are part of the Pareto frontier. The nondominated solutions can then be extracted through shortest-path algorithms on such a network. Numerical results comparing our method with state-of-the-art approaches on several problem classes, including the knapsack, set covering, and the traveling salesperson problem (TSP), suggest orders-of-magnitude runtime speed-ups for exactly enumerating the Pareto frontier, especially when the number of objective functions grows.

Application Analysis on PSO Algorithm in the Discrete Optimization Problems

Particle Swarm Optimization (PSO) is kind of algorithm that can be used to solve optimization problems. In practice, many optimization problems are discrete but PSO algorithm was initially designed to meet the requirements of continuous problems. A lot of researches had made efforts to handle this case and varieties of discrete PSO algorithms were proposed. However, these algorithms just focus on the specific problem, and the performance of it significantly degrades when extending the algorithm to other problems. For now, there is no reasonable unified principle or method for analyzing the application of PSO algorithm in discrete optimization problem, which limits the development of discrete PSO algorithm. To address the challenge, we first give an investigation of PSO algorithm from the perspective of spatial search, then, try to give a novel analysis of the key feature changes when PSO algorithm is applied to discrete optimization, and propose a classification method to summary existing discrete PSO algorithms.

The Efficiency of Discrete Optimization Algorithm Portfolios

Introduction. Solving large-scale discrete optimization problems requires the processing of large-scale data in a reasonable time. Efficient solving is only possible by using multiprocessor computer systems. However, it is a daunting challenge to adapt existing optimization algorithms to get all the benefits of these parallel computing systems. The available computational resources are ineffective without efficient and scalable parallel methods. In this connection, the algorithm unions (portfolios and teams) play a crucial role in the parallel processing of discrete optimization problems. The purpose. The purpose of this paper is to research the efficiency of the algorithm portfolios by solving the weighted max-cut problem. The research is carried out in two stages using stochastic local search algorithms. Results. In this paper, we investigate homogeneous and non-homogeneous algorithm portfolios. We developed the homogeneous portfolios of two stochastic local optimization algorithms for the weighted max-cut problem, which has numerous applications. The results confirm the advantages of the proposed methods. Conclusions. Algorithm portfolios could be used to solve well-known discrete optimization problems of unprecedented scale and significantly improve their solving time. Further, we propose using communication between algorithms, namely teams and portfolios of algorithm teams. The algorithms in a team communicate with each other to boost overall performance. It is supposed that algorithm communication allows enhancing the best features of the developed algorithms and would improve the computational times and solution quality. The underlying algorithms should be able to utilize relevant data that is being communicated effectively to achieve any computational benefit from communication. Keywords: Discrete optimization, algorithm portfolios, computational experiment.

Identification Conditions for the Solvability of NP-complete Problems for the Class of Pre-fractal Graphs

Modern network systems (unmanned aerial vehicles groups, social networks, network production chains, transport and logistics networks, communication networks, cryptocurrency networks) are distinguished by their multi-element nature and the dynamics of connections between its elements. A number of discrete problems on the construction of optimal substructures of network systems described in the form of various classes of graphs are NP-complete problems. In this case, the variability and dynamism of the structures of network systems leads to an "additional" complication of the search for solutions to discrete optimization problems. At the same time, for some subclasses of dynamical graphs, which are used to model the structures of network systems, conditions for the solvability of a number of NP-complete problems can be distinguished. This subclass of dynamic graphs includes pre-fractal graphs. The article investigates NP-complete problems on pre-fractal graphs: a Hamiltonian cycle, a skeleton with the maximum number of pendant vertices, a monochromatic triangle, a clique, an independent set. The conditions under which for some problems it is possible to obtain an answer about the existence and to construct polynomial (when fixing the number of seed vertices) algorithms for finding solutions are identified.

On the Classical Version of the Branch and Bound Method

In the computer literature, a lot of problems are described that can be called discrete optimization problems: from encrypting information on the Internet (including creating programs for digital cryptocurrencies) before searching for “interests” groups in social networks. Often, these problems are very difficult to solve on a computer, hence they are called “intractable”. More precisely, the possible approaches to quickly solving these problems are difficult to solve (to describe algorithms, to program); the brute force solution, as a rule, is programmed simply, but the corresponding program works much slower. Almost every one of these intractable problems can be called a mathematical model. At the same time, both the model itself and the algorithms designed to solve it are often created for one subject area, but they can also be used in many other areas. An example of such a model is the traveling salesman problem. The peculiarity of the problem is that, given the relative simplicity of its formulation, finding the optimal solution (the optimal route). This problem is very difficult and belongs to the so-called class of NP-complete problems. Moreover, according to the existing classification, the traveling salesman problem is an example of an optimization problem that is an example of the most complex subclass of this class. However, the main subject of the paper is not the problem, but the method of its soluti- on, i.e. the branch and bound method. It consists of several related heuristics, and in the monographs, such a multi-heuristic branch and bound method was apparently not previously noted: the developers of algorithms and programs should have understood this themselves. At the same time, the method itself can be applied (with minor changes) to many other discrete optimization problems. So, the classical version of branch and bound method is the main subject of this paper, but also important is the second subject, i.e. the traveling salesman problem, also in the classical formulation. The paper deals with the application of the branch and bound method in solving the traveling salesman problem, and about this application, we can also use the word “classical”. However, in addition to the classic version of this implementation, we consider some new heuristics, related to the need to develop real-time algorithms.