Abstract
The continuity of the optimal value function of a parametric convex semi-infinite program is secured by a weak regularity condition that also implies the convergence of certain discretization methods for semi-infinite problems. Since each discretization level yields a parametric program, a sequence of optimal value functions occurs. The regularity condition implies that, with increasing refinement of the discretization, this sequence converges uniformly with respect to the parameter to the optimal value function corresponding to the original semi-infinite problem. Our result is applicable to the convergence analysis of numerical algorithms based on parametric programming, for example, rational approximation and computation of the eigenvalues of the Laplacian.
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References
Hettich, R., An Implementation of a Discretization Method for Semi-Infinite Programming, Mathematical Programming, Vol. 34, pp. 354–361, 1986.
Reemtsen, R., Discretization Methods for the Solution of Semi-Infinite Programming Problems, Journal of Optimization Theory and Applications, Vol. 71, pp. 85–103, 1991.
Reemtsen, R., Some Outer Approximation Methods for Semi-Infinite Optimization Problems, Journal of Computational and Applied Mathematics, Vol. 53, pp. 87–108, 1994.
Polak, E., Optimization: Algorithms and Consistent Approximations, Springer, Berlin, Germany, 1997.
Hettich, R., and Kortanek, K. O., Semi-Infinite Programming: Theory, Methods, and Applications, SIAM Review, Vol. 35,No. 3, 1993.
Bonnans, J. F., and Shapiro, A., Optimization Problems with Perturbations: A Guided Tour, SIAM Review, Vol. 40,No. 2, 1998.
Brosowski, B., Parametric Semi-Infinite Linear Programming, I: Continuity of the Feasible Set and the Optimal Value, Mathematical Programming Study, Vol. 21, pp. 18–42, 1984.
Zencke, P., and Hettich, R., Directional Derivatives for the Value Function in Semi-Infinite Programming, Mathematical Programming, Vol. 38, pp. 323–340, 1987.
Gugat, M., One-Sided Derivatives for the Value Function in Convex Parametric Programming, Optimization, Vol. 28, pp. 301–314, 1994.
Hettich, R., and Zencke, P., An Algorithm for General Restricted Rational Chebyshev Approximation, SIAM Journal on Numerical Analysis, Vol. 27, pp. 1024–1033, 1990.
Gugat, M., The Newton Differential Correction Algorithm for Rational Chebyshev Approximation with Constrained Denominators, Numerical Algorithms, Vol. 13, pp. 107–122, 1996.
Hettich, R., and Gugat, M., Optimization under Functional Constraints (Semi-Infinite Programming) and Applications, Modern Methods of Optimization, Edited by W. Krabs and J. Zowe, Springer, Berlin, Germany, 1992.
Hettich, R., Haaren, E., Ries, M., and Still, G., Accurate Numerical Approximations of Eigenfrequencies and Eigenfunctions of Elliptic Membranes, Zamm, Vol. 67, pp. 589–597, 1987.
Ekeland, I., and Temam, R., Convex Analysis and Variational Problems, North-Holland, Amsterdam, Holland, 1976.
Apostol, T., Mathematical Analysis, Addison-Wesley, Reading, Massachusetts, 1979.
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Gugat, M. Convex Semi-Infinite Parametric Programming: Uniform Convergence of the Optimal Value Functions of Discretized Problems. Journal of Optimization Theory and Applications 101, 191–201 (1999). https://doi.org/10.1023/A:1021779213028
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DOI: https://doi.org/10.1023/A:1021779213028