Trust Region Algorithms For Nonlinear Constrained Optimization Problems

Trust region methods are efficient for solving unconstraint optimization problems. The basic idea of these methods is to approximate the optimization problem by a sequence of quadratic minimization problems subject to some trust region. An attractive property of trust region methods lies in their numerical stability and robustness. They can be applied to solve ill-conditioned problems. Under certain conditions, trust region methods are globally and super linearly convergent. However, the traditional trust region methods may have limits. The Hessian matrix of the trust region subproblem may not be positive definite. In this case, the trust region subproblem is relatively difficult to solve, the super convergence rate is hard to retain.Wu Q J(2004) proposed a modified BFGS update formula, which was applied to a general nonlinear unconstraint optimization problem. Hence.he obtained a modified BFGS trust region method for unconstrained optimization problem. This method has the property that the generated matrixes are positive definite. Under certain conditions, the method is feasible.In this paper, we succeed to generalize the modified BFGS update formula to constrained optimization problem. For inequality constrained optimization problem, we use the modified BFGS update formula construct a new trust region sub-problem, and then obtain a modified BFGS update trust region algorithm for inequality constrained optimization problem. Under certain conditions, we prove the method is feasible. For general nonlinear constrained optimization problem, we use penalty function make the constrained optimization problem to unconstrained optimization problem, and then we apply the modified BFGS update formula in to it. Hence, we obtain a BFGS update trust region algorithm for general nonlinear constrained optimization problem. The same as chapter three, we prove the method is feasible and its global convergence. At last, by numerical results are given which show the effectiveness of the proposed method.