An Augmented Penalization Algorithm for the Equality Constrained Minimization Problem



In this contribution an iterative method for solving the nonlinear minimization problem with equality constraints is presented. The method is based on the sequential minimization of the differentiable penalization function known as augmented Lagrangian. Each unconstrained minimization subproblem is solved by using a conjugate-gradient technique combined with a trust-region strategy of globalization, which is especially efficient for large-scale problems. The update of the multipliers and the penalty parameter is done by using standard schemes. The theoretical properties and the behavior of the algorithm are discussed. Details of the implementation are presented, the algorithm is tested with a set of classic problems and with a minimax formulation to the problems which belong to the well known family of Hard-Sphere Problems.


[1] D.P. Bertsekas, “Constrained Optimization and Lagrange Multiplier Methods”, Athenas Scientific, Belmont, MA, 1996.

A.R. Conn, N.I.M. Gould and Ph.L. Toint, A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds, SIAM Journal on Numer. Analysis, 28, No.3 (1991), 545-572.

J.H. Conway and N.J.C. Sloane, “Sphere Packing, Lattices and Groups”, Springer-Verlag, New York, 1988.

J.P. Dussault, Augmented penalty algorithms, IMA Journal of Numerical Analysis, 18 (1998), 355-372.

N.I.M. Gould, On the convergence of a sequential penalty function method for constrained minimization, SIAM Journal on Numer. Analysis, 26 (1989), 107-128.

M.R. Hestenes, Multiplier and gradient methods, Journal of Optimization Theory and Applications, 4 (1969), 303-320.

W. Hock and K. Schittkowski, Test examples for nonlinear programming codes, in “Lecture Notes in Economics and Mathematical Systems” (M. Beckmann and H.P. K¨unzi, eds.), 187, Springer-Verlag, Berlin, 1981.

N. Kreji´c, J.M. Martínez, M. Mello and E.A. Pilotta, Validation of an augmented Lagrangian algorithm with a Gauss-Newton Hessian approximation using a set of Hard-Spheres problems, Computational Optimization and Applications, 16 (2000), 247-263.

M.J.D. Powell, A method for nonlinear constraints in minimization problems, in “Optimization” (R. Fletcher, ed.), pp. 283-298, Academic Press, London, 1969.

K. Schittkowski, More test examples for nonlinear programming codes, in “Lecturs Notes in Economics and Mathematical Systems” (M. Beckmann and W. Krelle, eds.), Springer-Verlag, Berlin, 1987.

G.N. Sottosanto, “Un algoritmo de penalización aumentada para el problema de minimización con restricciones”, Master’s thesis, Departamento de Matemática, Universidad Nacional del Sur, Bahía Blanca, Argentina, 2001.

T. Steihaug, The conjugate-gradient method and trust-regions in large-scale ptimization, SIAM Journal on Numer. Analysis, 20(3) (1983), 626-637.


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Trends in Computational and Applied Mathematics

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