QUADRATICALLY AND SUPERLINEARLY CONVERGENT ALGORITHMS FOR THE SOLUTION OF INEQUALITY CONSTRAINED MINIMIZATION PROBLEMS

In this paper, some Newton and quasi-Newton algorithms for the solutio n of inequality constrained minimization problems are considered. All the algorithms described produce sequences {x(k)} converging q-superli nearly to the solution. Furthermore, under mild assumptions, a q-quadr atic convergence rate in x is also attained. Other features of these a lgorithms are that only the solution of linear systems of equations is required at each iteration and that the strict complementarity assump tion is never invoked. First, the superlinear or quadratic convergence rate of a Newton-like algorithm is proved. Then, a simpler version of this algorithm is studied, and it is shown that it is superlinearly c onvergent. Finally, quasi-Newton versions of the previous algorithms a re considered and, provided the sequence defined by the algorithms con verges, a characterization of superlinear convergence extending the re sult of Boggs, Tolle, and Wang is given.