Организация вычислительных процедур для сложных управляемых систем
In this research we analyzed the problem of solving transcendental equations' systems. More detailed is analyzed the approach of solution based on genetic algorithms because it is less examined than the one based on numerical methods. The research will be useful for different kinds of physical and mathematical calculations containing transcendental equations of high complexity. The research is available for students and graduates who are familiar with the basics of numerical methods, mathematical analysis, discrete mathematics and combinatorial algorithms.
Most of the existing books on optimization focus on the problem of computing locally optimal solutions. Global optimization is concerned with the computation and characterization of global optima of nonlinear functions. Global optimization problems are widespread in the mathematical modeling of real world systems for a very broad range of applications. During the past three decades many new theoretical, algorithmic, and computational contributions have helped to solve globally multi-extreme problems arising from important practical applications. Introduction to Global Optimization is the first comprehensive textbook that covers the fundamentals in global optimization. The second edition includes algorithms, applications, and complexity results for quadratic programming, concave minimization, DC and Lipshitz problems, decomposition algorithms for nonconvex optimization, and nonlinear network flow problems. Each chapter contains illustrative examples and ends with carefully selected exercises, which are designed to help the student to get a grasp of the material and enhance their knowledge of global optimization methods. Audience: This textbook is addressed not only to students of mathematical programming, but to all scientists in various disciplines who need global optimization methods to model and solve problems.
It is shown that genetic algorithms can be used successfully in problems of definite integral calculation especially when an integrand has a primitive which can't be expressed analytically through elementary functions. A testing of the program, which uses the genetic algorithm developed by authors, showed that the best results are reached if the size of population makes 30-50 chromosomes, approximately 40-60% of its take a part in crossover, and the program stops if the population's leader didn't change during 5‑10 generations. An answer of genetic algorithm is more exact than answer received by the classical numerical methods, even if a quantity of partition’s points into segment is small or if an integrand is quickly oscillating. So genetic algorithms can compete both on the accuracy of calculations and on operating time with well-known classical numerical methods such as midpoint approximation, top-left corner approximation, top-right corner approximation, trapezoidal rule, Simpson's rule.
The companies that are IT-industry leaders perform from several tens to several hundreds of projects simultaneously. The main problem is to decide whether the project is acceptable to the current strategic goals and resource limits of a company or not. This leads firms to an issue of a project portfolio formation; therefore, the challenge is to choose the subset of all projects which satisfy the strategic objectives of a company in the best way. In this present article we propose the multi-objective mathematical model of the project portfolio formation problem, defined on the fuzzy trapezoidal numbers. We provide an overview of methods for solving this problem, which are a branch and bound approach, an adaptive parameter variation scheme based on the epsilon-constraint method, ant colony optimization method and genetic algorithm. After analysis, we choose ant colony optimization method and SPEA II method, which is a modification of a genetic algorithm. We describe the implementation of these methods applied to the project portfolio formation problem. The ant colony optimization is based on the max min ant system with one pheromone structure and one ant colony. Three modification of our SPEA II implementation were considered. The first adaptation uses the binary tournament selection, while the second requires the rank selection method. The last one is based on another variant of generating initial population. The part of the population is generated by a non-random manner on the basis of solving a one-criterion optimization problem. This fact makes the population more strongly than an initial population, which is generated completely by random. Comparing of ant colony optimization algorithm and three modifications of a genetic algorithm was performed. We use the following parameters: speed of execution and the C-metric between each pair of algorithms. Genetic algorithm with non-random initial population show better results than other methods. Thus, we propose using this algorithm for solving project portfolio formation problem.