A Class Of Nonlinear Lagrange Methods For Nonlinear Optimization Problems

Classic Lagrangians in which the multiplier vectors and the constraint mappings are invloved in linear ways, play an important role in studies on the duality theories of convex programmings, especially that of linear programmings and quadratic programmings which should be express through Classic Lagrangians. But in nonconvex programmings, the primal problems and the duality problems which are based on Classic Lagrangians have duality gaps. So many sholars are become more and more interested in studying on the varies of Classic Larangians. Nonlinear Lagrangians are varies of Classic Larangians, in which the multiplier vectors and the constraint mappings are invloved in nonlinear ways. We employ dual methods based on nonlinear Lagrangians to solve optimization problems, which is called Nonlinear Lagrangian Methods. This dissertation studies mainly theories and according numerical implementation of a class of Lagrangian methods for nonlinear optimization problems. The main results, obtained in this dissertation, may be summarized as follws:1. Chapter 2 constructs two nonlinear Lagrangians for solving nonlinear optimization problems with inequality constraints and establishes the theoretical framework of corresponding dual algorithms. We prove, under some mild assumptions, the local convergence theorems for the two dual algrithms and present the error bounds for approximate solutions. We also prove that the two algorithms are global convergence when growth condition is fulfilled.2. Chapter 3 studies corresponding dual problems and related dual theories and saddle point theories. We prove, under some mild conditions, the values of the objective functions of the primal and dual problems coincide in their optimal points. We also point that the optimal solutions to the primal and the dual problems have no duality gap at the saddle points.3. Chapter 4 reports numerical experiments for the dual algorithms in Chapter 2. We apply these dual algorithms to solve some nonlinear optimization problems with relative small scale, including inequality constrained optimization problems and unconstrained minimax problems. The numerical results demonstrate that the dual algorithms are effective.