Abstract
In this paper, a functional inequality constrained optimization problem is studied using a discretization method and an adaptive scheme. The problem is discretized by partitioning the interval of the independent parameter. Two methods are investigated as to how to treat the discretized optimization problem. The discretization problem is firstly converted into an optimization problem with a single nonsmooth equality constraint. Since the obtained equality constraint is nonsmooth and does not satisfy the usual constraint qualification condition, relaxation and smoothing techniques are used to approximate the equality constraint via a smooth inequality constraint. This leads to a sequence of approximate smooth optimization problems with one constraint. An adaptive scheme is incorporated into the method to facilitate the computation of the sum in the inequality constraint. The second method is to apply an adaptive scheme directly to the discretization problem. Thus a sequence of optimization problems with a small number of inequality constraints are obtained. Convergence analysis for both methods is established. Numerical examples show that each of the two proposed methods has its own advantages and disadvantages over the other.
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References
R.K. Brayton, G.D. Hachtel and A.L. Sangiovanni-Vincentlli, A survey of optimization techniques for integrated circuit design, Proc. IEEE 69 (1981) 1334–1362.
I.D. Coope and G.A. Watson, A projected Lagrangian algorithm for semi-infinite programming, Math. Prog. 32 (1985) 337–356.
I.D. Coope and C.J. Price, Exact penalty function methods for nonlinear semi-infinite programming, in: Semi-Infinite Programming, eds. R. Reemtsen and J.-J. Rückmann (Kluwer, 1998).
G. Gonzaga, E. Polak and R. Trahan, An improved algorithm for optimization problems with functional inequality constraints, IEEE Transactions on Automatic Control AC-25 (1980) 49–54.
A. Grace, Optimization Toolbox for Use with MATLAB User Guide(The Maths Works, South Natick, MA, 1990).
R.D. Grigoreff and R.M. Reemtsen, Discrete approximation of minimization problems. I. Theory, Numer. Funct. Anal. Optimiz. 11 (1990) 701–719.
R. Hettich, An implementation of a discretization methods for semi-infinite programming, Math. Prog. 34 (1986) 354–361.
R. Hettich and G. Gramlich, A note on an implementation of a method for quadratic semi-infinite programming, Math. Prog. 46 (1990) 249–254.
R. Hettich and K.O. Kortanek, Semi-infinite programming: theory, methods, and applications, SIAM Review 35 (1993) 380–429.
A. Kaplan and R. Tichatschke, Stable Methods for Ill-Posed Variational Problems(Akademie Verlag, Berlin, 1994).
L.S. Jennings and K.L. Teo, A computational algorithm for functional inequality constrained optimization problems, Automatica 26 (1990) 371–375.
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 and D.Q. Mayne, An algorithm for optimization problems with functional inequality constraints, IEEE Transactions on Automatic Control AC-21 (1976) 184–193.
E. Polak, D.Q. Mayne and D.M. Stimler, Control system design via semi-infinite optimization: a review, Proc. IEEE 72 (1984) 1777–1794.
E. Polak and Y. Wardi, Nondifferentiable optimization algorithm for designing control systems having singular value inequalities, Automatica 18 (1982) 267–283.
R. Reemtsen, Some outer approximation methods for semi-infinite optimization problems, Journal of Computational and Applied Mathematics 53 (1994) 87–108.
R. Reemtsen, Semi-infinite programming: discretization methods, in: Encyclopedia of Optimization, eds. C.A. Floudas and P.M. Pardalos (Kluwer, Boston, 1998) pp. 1–8.
R. Reemtsen and J.J. Rückmann, eds., Semi-Infinite Programming(Kluwer, Dordrecht, 1998).
J. Stoer and C.H. Witzgall, Convexity and Optimization in Finite Dimension I(Springer, Berlin, 1998).
Y. Tanaka, M. Fukushima and T. Ibaraki, A comparative study of several semi-infinite nonlinear programming algorithms, European Journal of Operational Research 36 (1988) 92–100.
K.L. Teo and C.J. Goh, A simple computational procedure for optimization problems with functional inequality constraints, IEEE Transactions on Automatic Control AD-32 (1987) 940–941.
K.L. Teo, V. Rehbock and L.S. Jennings, A new computational algorithm for functional inequality constrained optimization problems, Automatica 29 (1993) 789–792.
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Teo, K., Yang, X. & Jennings, L. Computational Discretization Algorithms for Functional Inequality Constrained Optimization. Annals of Operations Research 98, 215–234 (2000). https://doi.org/10.1023/A:1019260508329
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DOI: https://doi.org/10.1023/A:1019260508329