×

Posynomial geometric programming as a special case of semi-infinite linear programming. (English) Zbl 0682.90078

This paper develops a wholly linear formulation of the posynomial geometric programming problem. It is shown that the primal geometric programming problem is equivalent to a semi-infinite linear program, and the dual problem is equivalent to a generalized linear program. Furthermore, the duality results that are available for the traditionally defined primal-dual pair are readily obtained from the duality theory for semi-infinite linear programs. It is also shown that two efficient algorithms (one primal based and the other dual based) for geometric programming actually operate on the semi-infinite linear program and its dual.
Reviewer: J.Rajgopal

MSC:

90C30 Nonlinear programming
90C34 Semi-infinite programming
90C05 Linear programming
49N15 Duality theory (optimization)
90-08 Computational methods for problems pertaining to operations research and mathematical programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kelley, J. E.,The Cutting Plane Method for Solving Convex Programs, SIAM Review, Vol. 8, pp. 703-712, 1960. · Zbl 0098.12104
[2] Gustafson, S.-Å., andKortanek, K. O.,Semi-Infinite Programming and Applications, Mathematical Programming: The State of the Art, Edited by A. Bachem, M. Grostchel, and B. Korte, Springer-Verlag, New York, New York, 1983.
[3] Avriel, M., Dembo, R. S., andPassy, U.,Solution of Generalized Geometric Programs, International Journal of Numerical Methods in Engineering, Vol. 9, pp. 149-168, 1975. · Zbl 0299.65035 · doi:10.1002/nme.1620090112
[4] Bricker, D. L., andRajgopal, J.,Yet Another Geometric Programming Dual Algorithm, Operations Research Letters, Vol. 2, pp. 177-180, 1982. · Zbl 0525.90079 · doi:10.1016/0167-6377(83)90051-2
[5] Duffin, R. J., Peterson, E. L., andZener, C.,Geometric Programming: Theory and Applications, John Wiley and Sons, New York, New York, 1967.
[6] Duffin, R. J.,Linearizing Geometric Programs, SIAM Review, Vol. 12, pp. 211-227, 1970. · Zbl 0229.90047 · doi:10.1137/1012043
[7] Dinkel, J. J., Elliott, W. H., andKochenberger, G. A.,A Linear Programming Approach to Geometric Programs, Naval Research Logistics Quarterly, Vol. 25, pp. 39-53, 1978. · Zbl 0399.90090 · doi:10.1002/nav.3800250105
[8] Glashoff, K., andGustafson, S.-Å.,Linear Optimization and Approximation, Springer-Verlag, New York, New York, 1983. · Zbl 0526.90058
[9] Glashoff, K.,Duality Theory of Semi-Infinite Programming, Semi-Infinite Programming: Proceedings of a Workshop, Bad Honnef, Germany, 1978; Edited by R. Heffich, Springer-Verlag, New York, New York, 1979. · Zbl 0417.90073
[10] Haar, A., Über Linear Ungleichungen, Acta Mathematica, Vol. 2, pp. 1-14, 1924. · JFM 50.0699.02
[11] Duffin, R. J., andKarlovitz, L. A.,An Infinite Linear Program with Duality Gap, Management Science, Vol. 12, pp. 122-134, 1965. · Zbl 0133.42604 · doi:10.1287/mnsc.12.1.122
[12] Gochet, W., Smeers, Y., andKortanek, K. O.,Using Semi-Infinite Programming in Geometric Programming, Proceedings of the 20th International Meeting of TIMS, Tel Aviv, Israel, 1973; Edited by E. Skifler, Academic Press, New York, New York, Vol. 2, pp. 430-438, 1973.
[13] Dantzig, G. B.,Linear Programming and Extensions, Princeton University Press, Princeton, New Jersey, 1963. · Zbl 0108.33103
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.