# zbMATH — the first resource for mathematics

Semicontinuity and quasiconvex functions. (English) Zbl 0892.90145
Summary: Criteria are derived for quasiconvex functions under lower semicontinuity and upper semicontinuity conditions. The results thus obtained generalize earlier results for convex functions. We also give new conditions under which a given function is $$r$$-convex in the sense given by Avriel.

##### MSC:
 90C25 Convex programming 26B25 Convexity of real functions of several variables, generalizations
Full Text:
##### References:
 [1] YANG, X. M., Convexity of Semicontinuous Functions, Opsearch, Vol. 31, pp. 309–317, 1994. [2] AVRIEL, M., r-Convex Functions, Discussion Paper 7106, Center for Operations Research and Econometrics, Université Catholique de Louvain, 1971. [3] JEYAKUMAR, V., and WOLKOWICZ, H., Zero Duality Gaps in Infinite-Dimensional Programming, Journal of Optimization Theory and Applications, Vol. 67, pp. 87–108, 1990. · Zbl 0687.90077 · doi:10.1007/BF00939737 [4] MANGASARIAN, O. L., Nonlinear Programming, McGraw-Hill, New York, New York, 1969. [5] ILLES, T., and KASSAY, G., Farkas Type Theorems for Generalized Convexities, Report 94–23, Faculty of Technical Mathematics and Informatics, Delft University of Technology, 1994. · Zbl 0821.90091 [6] MANGASARIAN, O. L., Pseudoconvex Functions, SIAM Journal on Control, Vol. 118, 1993. [7] NG, G. T., and NIKODEM, K., On Approximately Convex Functions, Proceedings of the American Mathematical Society, Vol. 118, pp. 103–108, 1993. · Zbl 0823.26006 · doi:10.1090/S0002-9939-1993-1159176-X [8] CRAVEN, B. D., and GLOVER, B. M., On Nondifferentiable Convex Functions, Optimization, Vol. 16, pp. 3–6, 1985. · Zbl 0567.46019 · doi:10.1080/02331938508842981 [9] ROCKAFELLAR, R. J., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1972.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.