Methods Of Moving Asymptotes For Nonlinear Optimization And Their Applications

Nonlinear optimization is an important branch of operations research. In the engineering design, production management, transportation, government decision-making, economics, finance and other fields, many problems can be modeled as a constrained optimization problem. Especially in the oil exploration, atmospheric modeling, aerospace, data mining, economic planning, financial decisions,environmental engineering and lots of highly sophisticated scientific and technological fields, there are many large-scale constrained optimization questions, where the structure of objective function is extremely complex, the number of unknown variables and constrained conditions are very large. How to find more rapid and efficient optimization algorithms for solving large-scale optimization problems is an urgent work. The main work of this paper is to study the theory and algorithms for solving nonlinear optimization problems, and to verify the effectiveness of these algorithms through a large number of numerical experiments. There are eight chapters in the thesis.Chapter I is introductory. The purpose of this research, significance, the status of this thesis and the main research contents are discussed. In Chapter II, the overview, progress and challenges about the method of moving asymptotes model are introduced.The method of solving large–scale nonlinear optimization is discussed from Chapters III to V. In Chapter III, a new method of moving asymptotes is proposed for solving large-scale unconstrained optimization problems. In this method, we established new nonquadratic model - new subproblems of moving asymptote, discussed the convex separable properties of new subproblem, and obtained a descent direction by solving a subproblem of moving asymptotes in each iteration. New rules for controlling the asymptotes parameters are designed by using the trust region radius and some approximation properties such that the global convergence of this method is obtained. In Chapter IV, a new method of moving asymptotes for large-scale optimization problems subject to linear equality constraints is discussed. Linear equality constraints are deleted with null space technique and original constrained optimization problems is equivalent converted into unconstrained optimization problems in this method. Finally, a new moving asymptotes trust region algorithm for solving large-scale linear equality constrained optimization problems is established. In Chapter V, a convex approximation-dual method for large scale optimization problems subject to linear inequality constraints is discussed. In each step of the iterative process, according to the properties of new subprblem, a descent direction is obtained by solving a convex separable subproblem with dual technique. Finally, a new moving asymptotes dual-trust region algorithm for solving large-scale linear inequality constrained optimization problems is established and the global convergence of the new method is provedIn Chapter VI, the one-dimensional case of moving asymptotes model is studied. Based on the one-dimensional moving asymptotes function (a fractional approximation function), a class of new parametric Secant methods were established to solve nonlinear, univariate and unconstrained optimization problems. Convergence analysis shows the proposed methods to be ( 2+ 1)-order convergent. Finally, under different conditions, the selection of parameters is discussed and corresponding algorithms are proposed. In Chapter VII, some applications of new algorithms in financial decision-making optimization problem are given. Some models of portfolio optimization in financial decision-making are discussed in detail. Numerical experiments and analysis are carried out for some practical problems in various portfolio optimization models. The numerical results show that our proposed algorithms are able to deal with these problems.The global convergence of all proposed algorithms are established and proved under some reasonable conditions in the thesis. Numerical experiments for all these methods are widely carried out. Results show that these methods are very significant and worth to study further. In the last, all the algorithms developed in the thesis are summarized and some problems that are worthwhile for further research are proposed.