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Second order characterizations of pseudoconvex functions

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Abstract

Second order characterizations for (strictly) pseudoconvex functions are derived in terms of extended Hessians and bordered determinants. Additional results are presented for quadratic functions.

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References

  1. K.J. Arrow and A.D. Enthoven, “Quasi-concave programming”,Econometrica 29 (1961) 779–800.

    Google Scholar 

  2. M. Avriel, “r-convex functions”,Mathematical Programming 2 (1972) 309–323.

    Google Scholar 

  3. M. Avriel, “Solution of certain nonlinear programs involvingr-convex functions”,Journal of Optimization Theory and Applications 11 (1973) 159–174.

    Google Scholar 

  4. M. Avriel,Nonlinear programming: analysis and methods”, (Prentice Hall, Englewood Cliffs, NJ, 1976).

    Google Scholar 

  5. M. Avriel and I. Zang, “Generalized convex functions with applications to nonlinear programming”, in: P. van Moeseke, ed.,Mathematical programs for activity analysis (North-Holland, Amsterdam, 1974) pp. 23–33.

    Google Scholar 

  6. V.J. Bowman and T.C. Gleason, “A note on second order conditions for pseudo-convexity”, Working paper 44-74-75, Graduate School of Industrial Administration, Carnegie-Mellon University (May 1975).

  7. R.W. Cottle, “Manifestation of the Schur complement”,Linear Algebra and its Applications 8 (1974) 189–211.

    Google Scholar 

  8. R.W. Cottle and J.A. Ferland, “On pseudo-convex functions of nonnegative variables”,Mathematical Programming 1 (1971) 95–101.

    Google Scholar 

  9. R.W. Cottle and J.A. Ferland, “Matrix-theoretic criteria for the quasi-convexity and pseudoconvexity of quadratic functions”,Linear Algebra and its Applications 5 (1972) 123–136.

    Google Scholar 

  10. J.A. Ferland, “Quasi-convex and pseudo-convex functions on solid convex sets”, Tech. Rept. 71-4, Operations Research House, Stanford University (April 1971).

  11. J.A. Ferland, “Maximal domains of quasi-convexity and pseudoconvexity for quadratic functions”,Mathematical Programming 3 (1972) 178–192.

    Google Scholar 

  12. J.A. Ferland, “Mathematical programming problems with quasi-convex objective functions”,Mathematical Programming 3 (1972) 296–301.

    Google Scholar 

  13. A.V. Fiacco, “Second order sufficient conditions for weak and strict constrained minima”,SIAM Journal on Applied Mathematics 16 (1968) 105–108.

    Google Scholar 

  14. F.R. Gantmacher,The Theory of Matrices, Vol. I (Chelsea, New York, 1960).

  15. O.L. Mangasarian, “Pseudo-convex functions”,SIAM Journal on Control 3 (1965) 281–290.

    Google Scholar 

  16. O.L. Mangasarian,Nonlinear programming, (McGraw-Hill, New York, 1969).

    Google Scholar 

  17. B. Martos, “Nem-lineáris programozási módszerek hatóköre” (Power of nonlinear programming methods), A Magyar Tudományos Akadémia Közgazdaságtudományi Intézetének Közleményei, 20, Budapest (1966).

  18. B. Martos, “Subdefinite matrices and quadratic forms”,SIAM Journal on Applied Mathematics 17 (1969) 1215–1223.

    Google Scholar 

  19. B. Martos, “Quadratic programming with a quasi-convex objective function”,Operations Research 19 (1971) 82–97.

    Google Scholar 

  20. B. Martos,Nonlinear programming: theory and methods, (North-Holland, Amsterdam, 1975).

    Google Scholar 

  21. P. Mereau and J.G. Paquet, “Second order conditions for pseudoconvex functions”,SIAM Journal on Applied Mathematics 27 (1974) 131–137.

    Google Scholar 

  22. J. Ponstein, “Seven kinds of convexity”,SIAM Review 9 (1967) 115–119.

    Google Scholar 

  23. S. Schaible, “Beiträge zur quasikonvexen Programmierung”, Ph.D. Dissertation, Universität Köln (1971).

  24. S. Schaible, “Quasi-convex optimization in general real linear spaces”,Zeitschrift für Operations Research 16 (1972) 205–213.

    Google Scholar 

  25. S. Schaible, “Quasi-concave, strictly quasi-concave and pseudo-concave functions”, in: R. Henn, H.P. Künzi and H. Schubert, eds.Methods of operations research 17 (Anton Hain, Meisenheim, 1973) pp. 308–316.

    Google Scholar 

  26. S. Schaible, “Quasi-convexity and pseudo-convexity of cubic functions”,Mathematical Programming 5 (1973) 243–247.

    Google Scholar 

  27. S. Schaible, “Second order characterizations of pseudo-convex quadratic functions”,Journal of Optimization Theory and Applications 21 (1977) 15–26.

    Google Scholar 

  28. J. Stoer and C. Witzgall, “Convexity and optimization in finite dimensions I”, (Springer, Berlin, 1970).

    Google Scholar 

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Avriel, M., Schaible, S. Second order characterizations of pseudoconvex functions. Mathematical Programming 14, 170–185 (1978). https://doi.org/10.1007/BF01588964

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  • DOI: https://doi.org/10.1007/BF01588964

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