On the Convergence of Constrained Optimization Methods with Accurate Hessian Information on a Subspace

This paper analyzes methods of the type proposed by Coleman and Conn for nonlinearly constrained optimization. It is shown that if the reduced Hessian approximation is sufficiently accurate, then the method generates a sequence of iterates that converges one-step superlinearly. This result applies to a quasi-Newton implementation. If the reduced exact Hessian is used, the method has an R-order equal to that of the secant method. A similar result for a modified version of successive quadratic programming is also proved. Finally, some parallels between convergence results for methods that approximate the reduced Hessian method, and multiplier methods that use the reduced Hessian inverse, are pointed out.