Abstract
A first order criterion for pseudo-convexity and second order criteria for quasi-convexity and pseudo-convexity are given for twice differentiable functions on open convex sets. The relationships between these second order criteria and other known criteria are also analysed. Finally, the numbers of operations required to verify these criteria are calculated and compared.
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References
K.J. Arrow and A.C. Enthoven, “Quasi-concave programming”,Econometrica 29 (1961) 779–800.
M. Avriel and S. Schaible, “Second order characterization of pseudo-convex functions”,Mathematical Programming 14 (1978) 170–185.
J.R. Bunch and B.N. Parlett, “Direct methods for solving symmetric indefinite systems of linear equations”,SIAM Journal on Numerical Analysis 8 (1971) 639–655.
R.W. Cottle, “Manifestations of the Schur complement”,Linear algebra and its Applications 8 (1974) 189–211.
J.P. Crouzeix, “Contributions à l'étude des fonctions quasi-convexes”, Thèse de doctorat, U.E.R. des Sciences Exactes et Naturelles, Université de Clermont-Ferrand II, France (1977).
J.P. Crouzeix, “A second order condition for quasi-convexity”,Mathematical Programming 18 (1980) 349–352.
G. Debreu, “Definite and semidefinite quadratic forms”,Econometrica 20 (1952) 295–300.
W.E. Diewert, M. Avriel and I. Zang, “Nine kinds of quasi-convexity and concavity”,Journal of Economic Theory (to appear).
J.A. Ferland, “Matrix-theoretic criteria for the quasi-convexity of twice continuously differentiable functions”,Linear Algebra and its Applications 38 (1981) 51–63.
J.A. Ferland, “Mathematical programming problems with quasi-convex objective functions”,Mathematical Programming 3 (1972) 296–301.
J.A. Ferland, “Quasi-convex and pseudo-convex functions on solid convex sets”, Technical Report 71-4, Department of Operations Research, Stanford University, Stanford, CA (April 1971).
D. Gale,The theory of linear economic models (McGraw-Hill, New York, 1960).
D.W. Katzner,Static demand theory (The MacMillen Company, New York, 1970).
L.J. Lau, “Testing and imposing monotonicity, convexity and quasi-convexity constraints”, Reprint no 268, The Stanford Institute for Mathematical Studies in the Social Sciences (1978).
O.L. Mangasarian,Non-linear programming (McGraw-Hill, New York, 1969).
P. Mereau and J.G. Paquet, “Second order conditions for pseudo-convex functions”,SIAM Journal on Applied Mathematics 27 (1974) 131–137.
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This research was supported by the Natural Sciences and Engineering Research Council Canada, Grant no. A8312, and by NATO Research Grant no. 1934.
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Crouzeix, JP., Ferland, J.A. Criteria for quasi-convexity and pseudo-convexity: Relationships and comparisons. Mathematical Programming 23, 193–205 (1982). https://doi.org/10.1007/BF01583788
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DOI: https://doi.org/10.1007/BF01583788