ИСТИНА

In this study we obtain sharp local convergence results for the augmented Lagrangian method and for the linearly constrained Lagrangian method. Both of these two methods gave rise to successful software. The augmented Lagrangian method is used by LANCELOT [7], ALGENCAN [1], and PENNON [9] software packages; while the linearly constrained Lagrangian method is the base of MINOS [8] and lterSD [4]. Nevertheless, the local convergence theory of these methods has had several drawbacks that we remedy in the study. Specically, we extend the sharpest known theorems about local convergence of the two methods [2, 6] to the case of problems with Lipschitzian derivatives dropping the assumption of twice dierentiability. This extension has been highly desirable since none of the methods in question involve second derivatives in their design. In addition, this extension makes it possible to apply the augmented Lagrangian method and the linearly constrained Lagrangian method to the lifted reformulations of MPCC [5], which find several applications in engineering including optimal design of plastic tructures [3]. Concerning the linearly constrained Lagrangian method, we also improve the result from [6] with respect to the rate of convergence, replacing the superlinear convergence rate estimate by the quadratic one. In our analysis, the augmented Lagrangian method and the linearly constrained Lagrangian method are viewed as special cases of an abstract Newtonian iterative framework that we develop and study first. We believe that this framework may serve as a convenient tool for local convergence analysis of many other algorithms as well. References [1] ALGENCAN. http://www.ime.usp.br/egbirgin/tango/. [2] D. Fernandez and M.V. Solodov. Local convergence of exact and inexact augmented Lagrangian methods under the second-order sucient optimality condition. SIAM J. Optim., 22:384-407, 2012. [3] M.C. Ferris and F. Tin-Loi. On the solution of a minimum weight elastoplastic problem involving displacement and complementarity constraints. Comput. Methods Appl. Mech. Engrg., 174(1-2):108-120, 1999. [4] R. Fletcher. A sequential linear constraint programming algorithm for NLP. SIAM J. Optim., 22(3):772-794, 2012. [5] A.F. Izmailov, A.L. Pogosyan, and M.V. Solodov. Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints. Comput. Optim. and Appl., 51:199-221, 2012. [6] A.F. Izmailov and M.V. Solodov. Inexact Josephy{Newton framework for generalized equations and its applications to local analysis of Newtonian methods for constrained optimization. Comput. Optim. Appl., 46:347-368, 2010. [7] LANCELOT. http://www.cse.scitech.ac.uk/nag/lancelot/lancelot.shtml. [8] B.A. Murtagh and M.A. Saunders. MINOS 5.0 user's guide. Technical Report SOL 83.20, Stanford University, 1983. [9] PENNON. http://www.penopt.com/.