A variable metric algorithm for constrained minimization based on an augmented Lagrangian

This paper concerns the implementation of a recent idea, attributed to Hestenes and Powell, based on solving the equality constrained finite dimensional minimization problem\documentclass{article}\pagestyle{empty}\begin{document}$$ \min \,\{ \,f(x)|(x)\, = \,0,\,x \in \,R^n \} $$\end{document}via the unconstrained problem\documentclass{article}\pagestyle{empty}\begin{document}$$ \min \,\{ f\,(x)\, + \, < g(x),\,\lambda > + 0.5 < g(x),\,Kg(x) > \,|\,x\, \in \,R^n,\,\lambda \, \in \,R^p $$\end{document}where ƒ is a non‐linear functional, g is a non‐linear mapping into R p , K is a prescribed matrix of penalty constants and λ is the Lagrange multiplier. The computational algorithm is based on restoring active constraints to first order and adjusting x in the remaining necessary conditions by gradient projection. The minimization is performed by the variable metric rank‐two BGFS update with linear search by cubic interpolation. Computational results using the algorithm include two problems of minimum fuel trajectory optimization—two impulse rendezvous with Comet Encke and three impulse constrained positioning of a geostationary satellite.