Karamardian, S.; Schaible, S.; Crouzeix, J. P. Characterization of generalized monotone maps. (English) Zbl 0792.90070 J. Optimization Theory Appl. 76, No. 3, 399-413 (1993). Summary: This paper is a sequel to a paper of the first and the second author [J. Optimization Theory Appl. 66, No. 1, 37-46 (1990; Zbl 0697.90067)] in which several kinds of generalized monotonicity were introduced for maps. They were related to generalized convexity properties of functions in the case of gradient maps. In the present paper, we derive first-order characterizations of generalized monotone maps based on a geometrical analysis of generalized monotonicity. These conditions are both necessary and sufficient for generalized monotonicity. Specialized results are obtained for the affine case. Cited in 3 ReviewsCited in 57 Documents MSC: 90C30 Nonlinear programming 26B25 Convexity of real functions of several variables, generalizations Keywords:one-dimensional maps; first-order characterizations; affine maps; generalized monotonicity Citations:Zbl 0679.90055; Zbl 0697.90067 PDFBibTeX XMLCite \textit{S. Karamardian} et al., J. Optim. Theory Appl. 76, No. 3, 399--413 (1993; Zbl 0792.90070) Full Text: DOI References: [1] Karamardian, S., andSchaible, S.,Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37-46, 1990. · Zbl 0679.90055 [2] Avriel, M., Diewert, W. E., Schaible, S., andZang, I.,Generalized Concavity, Plenum Publishing Corporation, New York, New York, 1988. [3] Karamardian, S.,Complementarity over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445-454, 1976. · Zbl 0304.49026 [4] Schaible, S.,Generalized Monotone Maps and Variational Inequalities, Proceedings of the 14th Annual Conference of Associazione per la Matematica Applicata alle Scienze Economiche e Sociali, Edited by A. M. Cerquetti, Pescara, Italy, September 1990, pp. 597-607, 1990. [5] Arrow, K. J., andEnthoven, A. C.,Quasiconcave Programming, Econometrica, Vol. 29, pp. 779-800, 1961. · Zbl 0104.14302 [6] Avriel, M.,r-Convex Functions, Mathematical Programming, Vol. 2, pp. 309-323, 1972. · Zbl 0249.90063 [7] Schaible, S.,Second-Order Characterizations of Pseudoconvex Quadratic Functions, Journal of Optimization Theory and Applications, Vol. 21, pp. 15-26, 1977. · Zbl 0336.26006 [8] Avriel, M., andSchaible, S.,Second-Order Characterizations of Pseudoconvex Functions, Mathematical Programming, Vol. 14, pp. 170-185, 1978. · Zbl 0382.90071 [9] Schaible, S.,Quasiconvex, Pseudoconvex, and Strictly Pseudoconvex Quadratic Functions, Journal of Optimization Theory and Applications, Vol. 35, pp. 303-338, 1981. · Zbl 0446.90071 [10] Crouzeix, J. P., andFerland, J. A.,Criteria for Quasiconvexity and Pseudoconvexity: Relationships and Comparisons, Mathematical Programming, Vol. 23, pp. 193-205, 1982. · Zbl 0479.90067 [11] Mazzoleni, P.,Monotonicity Properties and Generalized Concavity, Paper Presented at the International Workshop on Generalized Convexity and Fractional Programming, University of California at Riverside, 1989. [12] Castagnoli, E.,On Order-Preserving Functions, Proceedings of the 8th Italian-Polish Symposium on System Analysis and Decision Support in Economics and Technology, Levico Terme, September 1989, Edited by M. Fedrizzi and J. Kacprzyk, Omnitech Press, Warszawa, pp. 151-165, 1990. [13] Castagnoli, E., andMazzoleni, P.,Order-Preserving Functions and Generalized Convexity, Rivista di Matematica per le Scienze Economiche e Sociali, Vol. 14, pp. 33-45, 1991. · Zbl 0760.26013 [14] Castagnoli, E., andMazzoleni, P.,Generalized Monotonicity and Convexity for Poor Vector-Order Relations, Paper Presented at the International Workshop on Generalized Convexity and Fractional Programming, University of California at Riverside, 1989. 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.