On Theory Of A Class Of Dual Algorithms In Nonlinear Optimization

This dissertation studies mainly theories and according numerical implementation of a class of dual algorithms for nonlinear optimization problems, including unconstrained minimax problems and constrained nonlinear programming problems. The main results, obtained in this dissertation, may be summarized as follows:1. Chapter 2 establishes the theoretical framework of a class of dual algorithms for solving nonlinear optimization problems with inequality constraints. We prove, under some mild assumptions, the local convergence theorem for this class of dual algorithms and present the error bound for approximate solutions. The modified barrier function methods of Polyak (1992) and the augmented Lagrange function method of Bertsekas (1982) are verified to be the special cases of the class of dual algorithms. A dual algorithm, based on an exponential potential function, is proven to be included in this class of dual algorithms. Detailed convergence results for the dual algorithm are given and the estimate of the condition number of the potential function's Hessian is established, which depends on the penalty parameter.2. Chapter 3 is devoted to the study of the convergence theory of a dual algorithm for unconstrained minimax problems. A dual algorithm for solving unconstrained minimax problems, based on the penalty function of Bertsekas (1982), is presented. We prove that there exits a threshold of the penalty parameter satisfying that the sequences generated by the dual algorithm converge locally to the Kuhn-Tuker point of the unconstrained minimax problems when the penalty parameter is less than the threshold. And we establish the error bound of the solutionswith the penalty parameter. Furthermore, the condition number of potential function's Hessian is estimated, which depends on the penalty parameter.3. Chapter 4 studies the techniques for numerical realization of the dual algorithms and generalization of the dual algorithms to general unconstrained nonlinear programming. At first, we construct modified dual algorithms to overcome the drawback that it needs to resolve a sequential unconstrained minimization problems exactly in the step 2 of the dual algorithms in Chapter 2 and Chapter 3. It is proven that these modified dual algorithms still have the same convergence results as those of the conceptional dual algorithms in Chapter 2 and Chapter 3. Secondly, a dual algorithm is constructed for general constrained nonlinear programming problems and the local convergence theorem is established accordingly .The condition number of modified Lagrange function's Hessian is estimated, which also depends on the penalty parameter.4. Chapter 5 reports numerical experiments for the dual algorithms in Chapter 2, Chapter 3 and Chapter 4. We apply these dual algorithms to solve a large number of nonlinear optimization problems with relative small scale, including inequality constrained optimization problems, unconstrained minimax problems and general constrained optimization problems. The numerical results demonstrate that the dual algorithms are effective and the listed tables on the numerical results confirm the correctness of some theoretical results obtained in the previous chapters.