Abstract
In their proof of Gilbert–Pollak conjecture on Steiner ratio, Du and Hwang (Proceedings 31th FOCS, pp. 76–85 (1990); Algorithmica 7:121–135, 1992) used a result about localization of the minimum points of functions of the type max y∈Y f(·, y). In this paper, we present a generalization of such a localization in terms of generalized vertices, when we minimize over a compact polyhedron, and Y is a compact set. This is also a strengthening of a result of Du and Pardalos (J. Global Optim. 5:127–129, 1994). We give also a random version of our generalization.
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References
Du, D.Z., Hwang, F.K.: An approach for proving lower bounds: solution of Gilbert–Pollak’s conjecture on Steiner ration. Proceedings 31th FOCS, pp. 76–85 (1990)
Du D.Z. and Hwang F.K. (1992). A proof of the Gilbert–Pollak conjecture on the Steiner ratio. Algorithmica 7: 121–135
Du, D.Z., Pardalos, P.M., Wu, W.: Mathematical Theory of Optimization. Kluwer Academic Publishers (2001)
Du D.Z. and Pardalos P.M. (1994). A continuous version of a result of Du and Hwang. J. Global Optim. 5: 127–129
Du, D.Z.: Minimax and its applications. In: Horst, R., Pardalos, P. (eds.) Handbook of Global Optimization. Kluwer (1995)
Himmelberg C.J. (1975). Measurable relations. Fund. Math. 87: 53–72
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Georgiev, P.G., Pardalos, P.M. & Chinchuluun, A. Localization of minimax points. J Glob Optim 40, 489–494 (2008). https://doi.org/10.1007/s10898-007-9250-1
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DOI: https://doi.org/10.1007/s10898-007-9250-1