Rudyuk L. Mathematical model and computational methods for solving a problem of rectangles placement optimization

Object of research is considered a process of mathematical model formalization of problem of placement of geometric rectangular objects. Research goal is considered the improvement of mathematical model and development of numerical methods for solving problem of optimal rectangles placement with special criterions functional. Methods of research are considered instrument of linear inequality structures for task mathematical model building; the methods of conditional optimization and univariate optimization methods for developing numerical methods of optimization problem solving; a mathematical analysis, complexity theory, methods of mathematical statistics, approximation and interpolation of functions for researching convergence and timing complexity of developed methods. Theoretical results of research are considered in development new, more effective on timing complexity numerical methods for solving the task of optimal rectangles placement with continuously differential criterions functional. Practical results of research are considered in creation of software for solving the task of optimal rectangles placement with continuously differential criterions functional. Scientific newsness: evaluated a mathematical model of problem of optimization of geometrical objects of rectangular form accounting continuously differential criterions functional; first researched a gradient projection method modification, so-called G-projection method, that accounts a specificity of feasible set of the problem set by; proved a theorem about a theoretical evaluation and given a statistical evaluation of timing complexity of G-projection method; first proved a theorem about convergence of constructed G-projection method regarding a stationary point; first researched a method of a directional selection of constructed convex subsets for solving an initial problem of optimization with continuously differential criterions functional and given a statistic evaluation of its timing complexity. The results of scientific researchesare inculcated in lecturing the course "Mathematical methods of operations researchment" in Zhytomyr State Technological University. The results achieved in dissertation are used for solving the tasks of optimization in diploma projecting of specialists and examination works of holders of master's degree in the chair of software of Zhytomyr State Technological University. The inculcation for adoption of applied researchments in G.E.Pukhov's Institute of Modelling Problems in Power Engineering of the National Academy of Sciences of Ukraine (Kyiv), The Institute of Mathematical Machines and Systems Problems of the Ukraine National Academy of Science (Kyiv), Glushkov Institute of Cybernetics of the Ukraine National Academy of Science (Kyiv) is planned. The researchments results can be used on practice for solving practical tasks that can be reduced to the tasks of optimal rectangles replacement with persistent differential quality criteria, in particular in radio electronics, ecology, building, etc.