Abstract
In this paper, we consider a generalized semi-infinite optimization problem where the index set of the corresponding inequality constraints depends on the decision variables and the involved functions are assumed to be continuously differentiable. We derive first-order necessary optimality conditions for such problems by using bounds for the upper and lower directional derivatives of the corresponding optimal value function. In the case where the optimal value function is directly differentiable, we present first-order conditions based on the linearization of the given problem. Finally, we investigate necessary and sufficient first-order conditions by using the calculus of quasidifferentiable functions.
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Graettinger, T. J., and Krogh, B. H., The Acceleration Radius: A Global Performance Measure for Robotic Manipulators, IEEE Journal of Robotics and Automation, Vol. 4, pp. 60–69, 1988.
Hettich, R., and Still, G., Semi-Infinite Programming Models in Robotics, Parametric Optimization and Related Topics II, Edited by J. Guddat, H. T. Jongen, B. Kummer, and F. Nožička, Mathematical Research, Akademie Verlag, Berlin, Germany, Vol. 62, pp. 112–118, 1991.
Krabs, W., On Time-Minimal Heating or Cooling of a Ball, International Series of Numerical Mathematics, Birkhäuser, Basel, Switzerland, Vol. 81, pp. 121–131, 1987.
Hoffmann, A., and Reinhardt, R., On Reverse Chebyshev Approximation Problems, Preprint M 08/94, Faculty of Mathematics and Natural Sciences, Technical University of Ilmenau, 1994.
Kaplan, A., and Tichatschke, R., On a Class of Terminal Variational Problems, Parametric Optimization and Related Topics IV, Edited by J. Guddat, H. T. Jongen, F. Nožička, G. Still, and F. Twilt, Peter Lang, Frankfurt, Germany, pp. 185–199, 1997.
Jongen, H. T., RÜckmann, J. J., and Stein, O., Generalized Semi-Infinite Optimization: A First-Order Optimality Condition and Examples, Mathematical Programming, Vol. 83, pp. 145–158, 1998.
Hettich, R., and Still, G., Second-Order Optimality Conditions for Generalized Semi-Infinite Programming Problems, Optimization, Vol. 34, pp. 195–211, 1995.
Weber, G. W., Generalized Semi-Infinite Optimization: On Some Foundations, Preprint 1825, Department of Mathematics, Darmstadt University of Technology, 1996.
Polak, E., On the Mathematical Foundations of Nondifferentiable Optimization in Engineering Design, SIAM Review, Vol. 29, pp. 21–89, 1987.
Hettich, R., and Kortanek, K. O., Semi-Infinite Programming: Theory, Methods, and Applications, SIAM Review, Vol. 35, pp. 380–429, 1993.
Reemtsen, R., and RÜckmann, J. J., Editors, Semi-Infinite Programming, Kluwer Academic Publishers, Boston, Massachusetts, 1998.
Hettich, R., Jongen, H. T., and Stein, O., On Continuous Deformations of Semi-Infinite Optimization Problems, Approximation and Optimization in the Carribean II, Edited by M. Florenzano, J. Guddat, M. Jimenez, H. T. Jongen, G. L. Lagomasino, and F. Marcellan, Peter Lang, Frankfurt, Germany, pp. 406–424, 1995.
Jongen, H. T., and RÜckmann, J. J., One-Parameter Families of Feasible Sets in Semi-Infinite Optimization, Global Optimization.
Jongen, H. T., and Stein, O., On Generic One-Parametric Semi-Infinite Optimization, SIAM Journal of Optimization, Vol. 7, pp. 1103–1137, 1997.
Fiacco, A. V., Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, New York, New York, 1983.
Bonnans, J. F., and Shapiro, A., Optimization Problems with Perturbations, a Guided Tour, SIAM Review.
Mangasarian, O. L., and Fromovitz, S., The Fritz John Necessary Optimality Conditions in the Presence of Equality and Inequality Constraints, Journal of Mathematical Analysis and Applications, Vol. 17, pp. 37–47, 1967.
Gauvin, J., A Necessary and Sufficient Regularity Condition to Have Bounded Multipliers in Nonconcave Programming, Mathematical Programming, Vol. 12, pp. 136–138, 1977.
Kurcyusz, S., On the Existence and Nonexistence of Lagrange Multipliers in Banach Spaces, Journal of Optimization Theory and Applications, Vol. 20, pp. 81–110, 1976.
Gauvin, J., and Dubeau, F., Differential Properties of the Marginal Function in Mathematical Programming, Mathematical Programming Study, Vol. 19, pp. 101–119, 1982.
Lempio, F., and Maurer, H., Differential Stability in Infinite-Dimensional Nonlinear Programming, Applied Mathematics and Optimization, Vol. 6, pp. 139–152, 1980.
Levitin, E. S., On Differential Properties of the Optimal Value of Parametric Problems of Mathematical Programming, Doklady Akademii Nauk SSSR, Vol. 224, pp. 1354–1358, 1975.
Gauvin, J., and Tolle, J. W., Differential Stability in Nonlinear Programming, SIAM Journal on Control and Optimization, Vol. 15, pp. 294–311, 1977.
John, F., Extremum Problems with Inequalities as Subsidiary Conditions, Studies and Essays, R. Courant Anniversary Volume, Wiley-Interscience, New York, New York, pp. 187–204, 1948.
Golshtein, E. G., Theory of Convex Programming, Translations of Mathematical Monographs, American Mathematical Society, Providence, Rhode Island, Vol. 36, 1972.
Bonnans, J. F., Ioffe, A. D., and Shapiro, A., Expansion of Exact and Approximate Solutions in Nonlinear Programming, Proceedings of the French-German Conference on Optimization, Edited by D. Pallaschke, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, pp. 103–117, 1992.
Gauvin, J., and Janin, R., Directional Behavior of Optimal Solutions in Nonlinear Mathematical Programming, Mathematics of Operations Research, Vol. 13, pp. 629–649, 1988.
Jongen, H. T., RÜckmann, J. J., and Stein, O., Disjunctive Optimization: Critical Point Theory, Journal of Optimization Theory and Applications, Vol. 93, pp. 321–336, 1997.
Demyanov, V. F., and Vasilev, L. V., Nondifferentiable Optimization, Optimization Software, Publications Division, New York, New York, 1985.
Shapiro, A., On Optimality Conditions in Quasidifferentiable Optimization, SIAM Journal on Control and Optimization, Vol. 22, pp. 610–617, 1984.
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Rückmann, J.J., Shapiro, A. First-Order Optimality Conditions in Generalized Semi-Infinite Programming. Journal of Optimization Theory and Applications 101, 677–691 (1999). https://doi.org/10.1023/A:1021746305759
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DOI: https://doi.org/10.1023/A:1021746305759