Trust region algorithms for optimization with nonlinear equality and inequality constraints.

摘要

We consider the general nonlinear optimization problem defined as, minimize a nonlinear real-valued function of several variables, subject to a set of nonlinear equality and inequality constraints. This class of problems arise in many real life applications, for example in engineering design, chemical equilibrium, simulation and data fitting. In this research, we present algorithms that use the trust region technique to solve these problems. First, we develop an algorithm for solving the nonlinear equality constrained optimization, then we generalize the algorithm to handle the inclusion of nonlinear inequality constraints in the problem. The algorithms use the successive quadratic programming (SQP) approach and trust region technique. We define a model subproblem which minimizes a quadratic approximation of the Lagrangian subject to modified relaxed linearizations of the problem nonlinear constraints and a trust region constraint. Inequality constraints are handled by a compromise between an active set strategy and IQP subproblem solution technique. An analysis which describes the local convergence properties of our algorithms is presented. The algorithms are implemented and the model minimization is done approximately by using the dogleg approach. Numerical results are presented and compared with the results of a popular line search method. Some examples are presented in which the ability of our method to use directions of negative curvature results in greater reliability. Results of the numerical experiments indicate that our method is very robust and reasonably efficient.