Nonlinear Lagrange Methods For Solving Nonlinear Optimization Problems

Nonlinear Lagrangians are variants of the classical Lagrangian, in which the multiplier vectors or constraint functions are involved in nonlinear ways. Nonlinear Lagrange methods are dual methods based on nonlinear Lagrangians for solving optimization problems. As dual methods usually require no restrictions on the feasibility of primal variables, nonlinear Lagrange methods are playing important roles in solving constrained optimization problems. This dissertation is devoted to studying two families of nonlinear Lagrangians for solving nonlinear programming problems with inequality constraints and the convergence theory for their associated dual algorithms. We establish theory frameworks of the two classes of nonlinear Lagrange methods and find a new nonlinear Lagrangian produced by an NCP function in the second class, which is superior over other known nonlinear Lagrangians in both theory analysis and numerical performance. The main results obtained in this dissertation may be summarized as follows:1. Chapter 2 establishes a theory framework for a class of nonlinear Lagrangians, which are linear with respect to the multiplier variables. Firstly, a set of assumptions are proposed to guarantee the convergence of the nonlinear Lagrangian algorithms, to analyze the condition numbers of Lagrange Hessians as well as to develop dual approaches. The convergence theorem shows that the dual algorithm based on any of nonlinear Lagrangians in the class is locally convergent when the penalty parameter k is greater than a threshold and the error bound of primal-dual solutions is proportional to k^-1. An analysis reveals that the condition number of Lagrange Hessian at an optimal solution is proportional to the penalty parameter k. Secondly, the duality theory based on the proposed nonlinear Lagrangians is established, including the duality theorem, the second order sufficiency optimality conditions for the dual problems, the saddle point theorem, and a sufficient condition for the existence of saddle points characterized by the perturbation function. After that, the convergence for the second-order multiplier scheme is discussed, which proves that if the Hessians of problem functions are Lipschitz continuous, then the sequence generated by the second-order multiplier scheme converges quadratically. Finally, numerical results are given to verify the validity of the dual algorithms based on various nonlinear Lagrangians.2. Chapter 3 establishes a theory framework for another class of nonlinear Lagrangians, which are nonlinear with respect to the multiplier variables. Firstly, a set of assumptions are proposed to guarantee the convergence of the nonlinear Lagrangian algorithms, to analyze the...