Research On Qp-free And Generalized Gradient Projection Algorithms For Nonlinear Optimization Problems

This dissertation considers the nonlinear inequality constrained optimization and minimax optimization problems. Optimization is an important branch of operations research and cybernetics, which is widely used in the important areas of national economic planning, production management, engineering design, transportation and national defense construction. The core problem of optimization is the study of all kinds of optimization model, and the corresponding of theory, fast and effective numerical algorithms, which is very active in home and abroad. Nonlinear minimax optimization problem is a very important special class of nonlinear programming, on the one hand, there are many applications of the minimax problems in nonlinear programming and other mathematical problems; On the other hand, minimax optimization is widely used in many practical application problems, such as engineering design, optimal control, financial management, energy and environment, etc. With the rapid development of modern science and technology, and the arrival of the age for the big data, the scale of the corresponding problems will be more and more. Therefore, efficient and stable algorithms for the large scale minimax optimization problems have important theoretical significance and practical application value.The research works of this dissertation contain four parts as follows:In Chapter 2, a feasible QP-free algorithm for nonlinear inequality constrained optimization is proposed. At each iteration, the proposed algorithm solves only two systems of linear equations with a same coefficient matrix to obtain the feasible descent direction, the submatrix in the lower right corner of the coefficient matrix is a zero matrix, so the sparsity of the coefficient matrix is good. Under some necessary assumptions, the algorithm possesses global and strong convergence. Finally, some preliminary numerical experiments are reported to show that the algorithm is effective.In Chapter 3, combining the method of strongly sub-feasible directions and the working set technique, a new QP-free algorithm with an arbitrary initial iteration point for solving inequality constrained optimization is proposed. At each iteration, the algorithm solves only two systems of linear equations with a same uniformly nonsingular coefficient matrix to obtain the search direction, then combining two directions to yield the main search direction, the submatrix in the lower right corner of the coefficient matrix is a nonzero matrix. Furthermore,the positive definiteness assumption on the Hessian estimate is relaxed. Under some necessary assumptions, the algorithm not only possesses global and strong convergence, but also ensures that the iteration points can get into the feasible set after finite iterations. Finally, a series of preliminary numerical results are reported to show that the algorithm is promising.In Chapter 4, combining the techniques of generalized gradient projection and -working set, we present an algorithm for the nonlinear unconstrained minimax problems. Based on the stationary conditions, a new optimal identification function is given. We construct a new descent direction, which is generated by an-generalized gradient projection explicit formula. Under some mild assumptions,the algorithm possesses global and strong convergence. Finally, some preliminary numerical results show that the proposed algorithm performs efficiently.In Chapter 5, we consider the nonlinear minimax problems with inequality constraints. Based on the stationary conditions, with neither the exponential smooth function nor constrained smooth transformation, we propose a feasible QP-free algorithm for the discussed problems. By means of a new and much tighter working set, we develop a new technique for constructing the sub-matrix in the lower right corner of the coefficient matrix, which can avoid the complex pivoting operation and such that the coefficient matrix possesses good sparsity.At each iteration, to obtain the search direction, two reduced systems of linear equations with a same coefficient are solved. Under mild conditions, the proposed algorithm has global and strong convergence. Finally, some preliminary numericalexperiments are reported to show that the algorithm is promising.In Chapter 6, we summarize the main research work and achievements of this dissertation, and point out the further research works in the future.