Abstract
In this paper we propose a primal-dual algorithm for the solution of general nonlinear programming problems. The core of the method is a local algorithm which relies on a truncated procedure for the computation of a search direction, and is thus suitable for large scale problems. The truncated direction produces a sequence of points which locally converges to a KKT pair with superlinear convergence rate.
The local algorithm is globalized by means of a suitable merit function which is able to measure and to enforce progress of the iterates towards a KKT pair, without deteriorating the local efficiency. In particular, we adopt the exact augmented Lagrangian function introduced in Pillo and Lucidi (SIAM J. Optim. 12:376–406, 2001), which allows us to guarantee the boundedness of the sequence produced by the algorithm and which has strong connections with the above mentioned truncated direction.
The resulting overall algorithm is globally and superlinearly convergent under mild assumptions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bertsekas, D.P.: Constrained Optimization and Lagrange Multipliers Methods. Academic Press, San Diego (1982)
Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1995)
Byrd, R.H., Gilbert, J.C., Nocedal, J.: A trust region method based on interior point techniques for nonlinear programming. Math. Program. Ser. A 89, 149–185 (2000)
Byrd, R.H., Hribar, M.E., Nocedal, J.: An interior point algorithm for large-scale nonlinear programming. SIAM J. Optim. 9, 877–900 (1999)
Byrd, R.H., Schnabel, R.B., Shultz, G.A.: A trust region algorithm for nonlinearly constrained optimization. SIAM J. Numer. Anal. 24, 1152–1170 (1987)
Di Pillo, G., Grippo, L.: A class of continuously differentiable exact penalty function algorithms for nonlinear programming problems. In: Toft-Christensen, E. (ed.) System Modelling and Optimization. Springer, Berlin (1984)
Di Pillo, G., Grippo, L.: An exact penalty method with global convergence properties. Math. Program. 36, 1–18 (1986)
Di Pillo, G., Grippo, L.: Exact penalty functions in constrained optimization. SIAM J. Control Optim. 27, 1333–1360 (1989)
Di Pillo, G., Lucidi, S.: An augmented Lagrangian function with improved exactness properties. SIAM J. Optim. 12, 376–406 (2001)
Di Pillo, G., Liuzzi, G., Lucidi, S., Palagi, L.: A truncated Newton method in an augmented Lagrangian framework for nonlinear programming. Technical Report 09-07, Department of Computer and System Sciences, University of Rome “La Sapienza”, Rome, Italy (2007). Available for download at URL http://www.dis.uniroma1.it/~liuzzi/papers/TR09_07.pdf
Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. Ser. A 91, 201–213 (2002)
Facchinei, F.: Minimization of SC1 functions and the Maratos effect. Oper. Res. Lett. 17, 131–137 (1995)
Facchinei, F., Lucidi, S.: Quadratically and superlinearly convergent algorithms for the solution of inequality constrained minimization problems. J. Optim. Theory Appl. 85, 265–289 (1995)
Fletcher, R., Gould, N.I.M., Leyffer, S., Toint, Ph.L., Wachter, A.: Global convergence of trust-region SQP-filter algorithms for general nonlinear programming. SIAM J. Optim. 13, 635–659 (2002)
Fletcher, R., Leyffer, S.: Nonlinear programming without a penalty function. Math. Program. 91, 239–270 (2002)
Forsgren, A., Gill, P.E.: Primal-dual interior methods for nonconvex nonlinear programming. SIAM J. Optim. 8, 1132–1152 (1998)
Gay, D.M., Overton, M.L., Wright, M.H.: A primal-dual interior method for nonconvex nonlinear programming. In: Yuan, Y. (ed.) Advances in Nonlinear Programming. Kluwer Academic, Dordrecht (1998)
Gill, P.E., Murray, W., Saunders, M.A.: \(\mathsf{SNOPT}\) : an SQP algorithm for large-scale constrained optimization. SIAM J. Optim. 9, 979–1006 (2001)
Glad, T., Polak, E.: A multiplier method with automatic limitation of penalty growth. Math. Program. 17, 140–155 (1979)
Gould, N.I.M., Orban, D., Toint, Ph.L.: CUTEr and SifDec: a constrained and unconstrained testing environment, revisited. ACM Trans. Math. Softw. 29, 373–394 (2003)
Grippo, L., Lampariello, F., Lucidi, S.: A truncated Newton method with nonmonotone line search for unconstrained optimization. J. Optim. Theory Appl. 60, 401–419 (1989)
Grippo, L., Lampariello, F., Lucidi, S.: A class of nonmonotone stabilization methods in unconstrained optimization. Numer. Math. 59, 779–805 (1991)
Guddat, J., Guerra Vasquez, F., Jongen, H.Th.: Parametric Optimization: Singularities, Pathfollowing and Jumps. Wiley, New York (1990)
Lucidi, S.: New results on a continuously differentiable exact penalty function. SIAM J. Optim. 2, 558–574 (1992)
Orthega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, San Diego (1970)
Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)
Shanno, D.F., Vanderbei, R.J.: An interior point algorithm for nonconvex nonlinear programming. Comput. Optim. Appl. 13, 231–252 (1999)
Wachter, A., Biegler, L.T.: Global and local convergence of line search filter methods for nonlinear programming. Technical Report B-01-09, Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA (2001)
Yabe, H., Yamashita, H.: Superlinear and quadratic convergence of some primal-dual interior point methods for constrained optimization. Math. Program. 75, 377–397 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work has been supported by MIUR-PRIN 2005 Research Program on New Problems and Innovative Methods in Nonlinear Optimization.
Rights and permissions
About this article
Cite this article
Di Pillo, G., Liuzzi, G., Lucidi, S. et al. A truncated Newton method in an augmented Lagrangian framework for nonlinear programming. Comput Optim Appl 45, 311–352 (2010). https://doi.org/10.1007/s10589-008-9216-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-008-9216-3