Augmented Lagrangian method with nonmonotone penalty parameters for constrained optimization
- 362 Downloads
At each outer iteration of standard Augmented Lagrangian methods one tries to solve a box-constrained optimization problem with some prescribed tolerance. In the continuous world, using exact arithmetic, this subproblem is always solvable. Therefore, the possibility of finishing the subproblem resolution without satisfying the theoretical stopping conditions is not contemplated in usual convergence theories. However, in practice, one might not be able to solve the subproblem up to the required precision. This may be due to different reasons. One of them is that the presence of an excessively large penalty parameter could impair the performance of the box-constraint optimization solver. In this paper a practical strategy for decreasing the penalty parameter in situations like the one mentioned above is proposed. More generally, the different decisions that may be taken when, in practice, one is not able to solve the Augmented Lagrangian subproblem will be discussed. As a result, an improved Augmented Lagrangian method is presented, which takes into account numerical difficulties in a satisfactory way, preserving suitable convergence theory. Numerical experiments are presented involving all the CUTEr collection test problems.
KeywordsNonlinear programming Augmented Lagrangian methods Penalty parameters Numerical experiments
Unable to display preview. Download preview PDF.
- 1.Andreani, R., Haeser, G., Martínez, J.M.: On sequential optimality conditions for smooth constrained optimization. Optimization (to appear) Google Scholar
- 5.Andreani, R., Martínez, J.M., Svaiter, B.F.: A new sequential optimality condition for constrained optimization and algorithmic consequences. SIAM J. Optim. (to appear) Google Scholar
- 7.Birgin, E.G., Fernández, D., Martínez, J.M.: On the boundedness of penalty parameters in an Augmented Lagrangian method with lower level constraints. Technical Report, Department of Applied Mathematics, State University of Campinas, Brazil Google Scholar
- 10.Buys, J.D.: Dual algorithms for constrained optimization problems. Doctoral Dissertation, University of Leiden, Leiden, the Netherlands (1972) Google Scholar
- 20.Conn, A.R., Gould, N.I.M., Toint, Ph.L.: Lancelot: A Fortran Package For farge Scale Nonlinear Optimization. Springer, Berlin (1992) Google Scholar
- 25.Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283–298. Academic Press, New York (1969) Google Scholar