Differential Geometric Methods for Solving Nonlinear Constrained Optimization Problems and a Related System of Nonlinear Equations: Global Analysis and Implementation

摘要

Robust method that can produce convergence from a very poor initial estimate of the optimal solution for larger problems are discussed. A differential geometric method is developed specifically to obtain robust algorithms without resorting to the penalty-type approach. In particular, a generic class of feasibility-improving gradient acute projection (FIGAP) methods and their Levenberg-Marquardt-type modifications is developed for solving the general nonlinear constrained minimization problems. Each method in this class is an amalgamation of a generalized gradient projection method and a generalized Newton-Raphson method which, respectively, take care of reducing the value of the objective function and satisfying constraint equations at the same time. The class of FIGAP methods contains various new methods as well as many of the existing methods. A unified theory is developed for the methods by using extensively the concept of various generalized inverses and related projectors, which facilitates geometric interpretation of the FIGAP methods. Analysis is given to the continuous analogs of the methods to obtain robust algorithms, which also gives insight into the global behavior of the related algorithms. Various new algorithms are derived from the general theory that use the QR decomposition, the SVD decomposition and other decompositions of the Jacobian matrix of the constraint functions. Quasi-Newton algorithms which estimate projected Hessian matrix and require in some cases only approximations of nonnegative definite matrix of size n-m are developed to enhance the local convergence, where n and m are numbers of variables and constraint equations, respectively. 1 figure. (ERA citation 05:003938)