Abstract
In this paper a nonlinear penalty method via a nonlinear Lagrangian function is introduced for semi-infinite programs. A convergence result is established which shows that the sequence of optimal values of nonlinear penalty problems converges to that of semi-infinite programs. Moreover a conceptual convergence result of a discretization method with an adaptive scheme for solving semi-infinite programs is established. Preliminary numerical experiments show that better optimal values for some nonlinear semi-infinite programs can be obtained using the nonlinear penalty method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.W. Adams, FIR digital filters with least-squares stopbands subject to peak-gain constraints, IEEE Transactions on Circuits and Systems 39 (1991) 376–388.
A. Charnes, W.W. Cooper and K.O. Kortanek, Duality, Harr programs, and finite sequence spaces, Proceedings of the National Academy of Sciences 48 (1962) 782–786.
D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods (Athena Scientific, Belmont, MA, 1996).
A.V. Fiacco and K.O. Kortanek, eds., Semi-Infinite Programming and Application (Springer, 1983).
C.J. Goh and X.Q.Yang, A sufficient and necessary condition for nonconvex constrained optimization, Applied Mathematics Letters 10 (1997) 9–12.
R. Hettich and G. Gramlich, A note on an implementation of a method for quadratic semi-infinite programming, Mathematical Programming 46 (1990) 249–254.
R. Hettich and K.O. Kortanek, Semi-infinite programming: theory, methods and applications, SIAM Review 35 (1993) 380–429.
A.D. Ioffe, Second-order conditions in nonlinear nonsmooth problems of semi-infinite programming, in: [4] (1979) pp. 262-280.
V. Jeyakumar and H. Wolkowicz, Zero duality gaps in infinite programming, J. Optim. Theory Appl. 67 (1990) 87–108.
E.R. Panier and A.L. Tits, A globally convergent algorithm with adaptively refined discretization for semi-infinite optimization problems arising in engineering design, IEEE Transactions on Automatic Control 34 (1989) 903–908.
E. Polak, On the mathematical foundations of nondifferentiable optimization in engineering design, SIAM Review 21 (1987) 21–89.
E. Polak, Optimization: Algorithms and Consistent Approximations (Springer, New York, 1997).
R. Reemstem and S. Gorner, Numerical methods for semi-infinite programming: A survey, in: [14] (1998) pp. 195-276.
R. Reemtsen and J.-J. Ruckmann, eds., Semi-Infinite Programming (Kluwer Academic, Boston, 1998).
A.M. Rubinov, B.M. Glover and X.Q. Yang, Decreasing functions with applications to penalization, SIAM Journal on Optimization 10 (1999) 289–313.
K.L. Teo, V. Rehbock and L.S. Jennings, A new computational algorithm for functional inequality constrained optimization problems, Automatica 29 (1993) 789–792.
K.L. Teo, X.Q. Yang and L.S. Jennings, Computational discretization algorithms for functional inequality constrained optimization, Annals of Operations Research 98 (2000) 215–234.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Yang, X., Teo, K. Nonlinear Lagrangian Functions and Applications to Semi-Infinite Programs. Annals of Operations Research 103, 235–250 (2001). https://doi.org/10.1023/A:1012911307208
Issue Date:
DOI: https://doi.org/10.1023/A:1012911307208