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A trust region algorithm for nonlinearly constrained optimization. (English) Zbl 0631.65068

The authors describe a method using trust regions to solve the general nonlinearly equality constrained optimization problem ${}_{x\in {R}^{n}}f\left(x\right)$ subject to $c\left(x\right)=0$. The method works by iteratively minimizing a quadratic model of the Lagrangian subject to a possibly relaxed linearization of the problem constraints and a trust region constraint. It is shown that this method is globally convergent even if singular or indefinite Hessian approximations are made.

A second order correction step that brings the iterates closer to the feasible set is described. If sufficiently precise Hessian information is used, the correction step allows one to prove that the method is also locally quadratically convergent, and that the limit satisfies the second order necessary conditions for constrained optimization. An example is given to show that, without this correction, a situation similar to the Maratos effect may occur where the iteration is unable to move away from a saddle point.

##### MSC:
 65K05 Mathematical programming (numerical methods) 90C30 Nonlinear programming