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CNAA / Theses / 2010 / May /

Deterministic and stochastic methods for solving extremal problems


Author: Balan Pavel
Degree:doctor of physics and mathematics
Speciality: 01.01.09 - Mathematical cybernetics and operation research
Year:2010
Scientific adviser: Anatol Godonoagă
doctor, associate professor (docent), Academy of Economic Studies of Moldova
Institution: Moldova State University
Scientific council: DH 30-01.01.09
Moldova State University

Status

The thesis was presented on the 12 May, 2010
Approved by NCAA on the 5 July, 2010

Abstract

Adobe PDF document0.31 Mb / in romanian

Keywords

convex optimization, differentiable and nondifferentiable functions, gradient, subgradient, deterministic and stochastic methods, minimax problem, almost certain convergence (with probability 1)

Summary

The thesis contains the introduction, three chapters, conclusions and recommendations, the bibliography (108 titles) , 2 annexes and consists of 153 pages, from which 97 pages of the main part, including 6 figures and 14 tables. Obtained results are published in 11 scientific papers.

The area of study refers to the research and solution of a set of problems from mathematical optimization. These problems are described by convex nonlinear models. Minimax problem has an important place among them.

The aim of this work consists in following: development of deterministic and stochastic methods for solving nonlinear optimization models of general form; description of fundamental differences between elaborated methods and existing ones; to expose the range of the problems that fit best for solving using described methods in comparison with existing methods; description of theoretical aspects of proposed methods; development of software.

The scientific novelty and originality of obtained results consist in: presentation of new methods for solving a problem characterized by a differentiable convex model (all these methods are generalizing gradient method); description of two methods based on the subgradient for solving a minimax nondifferentiable problem with restrictions; proposed methods are of deterministic or stochastic nature.

The theoretical signification consists in: proposed methods solve problems of convex programming of general form, inclusively a minimax problem. Such problems are often encountered in applied domains. Described methods contain rigorous theoretical argumentation. Convergence conditions have been formulated with detailed demonstrations for all of them.

The practical value of the work: for every proposed method has been developed a program. Elaborated software is simple, efficient and does not require considerable computing resources.