Abstract
Multivalued functions satisfying a general convexity condition are examined in the first section. The second section establishes a general transposition theorem for such functions and develops an abstract multiplier principle for them. In particular both convex inequality and linear equality constraints are seen to satisfy the same generalized constraint qualification. The final section examines quasi-convex programmes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.A. Abrams and A. Ben-Israel, “Optimality conditions and recession cones”,Operations Research 23 (1975) 549–553.
C. Berge,Espace topologique, fonction multivoque (Dunod, Paris, 1959).
B.D. Craven and B. Mond, “Transposition theorems for cone-convex functions”,SIAM Journal of Applied Mathematics 24 (1973) 603–612.
J.L. Kelley and I. Namioka,Linear topological spaces (Van Nostrand, New York, 1963).
R. Lehmann and W. Oettli, “The theorem of the alternative, the key theorem, and the vector maximization problem”,Mathematical Programming 8 (1975) 332–344.
D.G. Luenberger,Optimization by vector space methods (J. Wiley, New York, 1969).
D.G. Luenberger, “Quasi-convex programming”,SIAM Journal of Applied Mathematics 16 (1968) 1090–1095.
O.L. Mangasarian,Nonlinear programming (McGraw-Hill, New York, 1969).
T.S. Motzkin, “Beiträge zur Theorie der linearen ungleichungen”, Dissertation, University of Basel (Jerusalem, 1936).
J. Ponstein, “Seven kinds of convexity”,SIAM Review 2 (1967) 115–119.
A.P. Robertson and W.J. Robertson,Topological vector spaces (Cambridge University Press, Cambridge 1964).
S.M. Robinson, “Regularity and stability for convex multivalued functions”,Mathematics of Operations Research 1 (1976) 130–143.
R.T. Rockafellar,Convex analysis (Princeton University Press, 1970).
J. Stoer and C. Witzgall,Convexity and optimization in finite dimensions, I (Springer, Berlin, 1970).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Borwein, J. Multivalued convexity and optimization: A unified approach to inequality and equality constraints. Mathematical Programming 13, 183–199 (1977). https://doi.org/10.1007/BF01584336
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01584336