Abstract
In this study we introduce the notion of supporting functions and exploit some of their important properties. We make use of these functions to develop generalized optimality conditions of a mixed stationary-saddle type, where the supporting functions play either the role of the gradients/directional derivatives and/or the role of the original functions. These conditions may be applied to certain problems involving nondifferentiable and/or nonconvex functions. Classical, as well as new stationary and saddle optimality conditions follow from our approach in a natural fashion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Abadie, “On the Kuhn—Tucker Theorem”, in:Nonlinear programming, Ed. J. Abadie (North-Holland, Amsterdam, 1967).
M.S. Bazaraa, “A theorem of the alternative with application to convex programming: optimality, duality, and stability”,Journal of Mathematical Analysis and Applications 41 (1973).
M.S. Bazaraa and J.J. Goode, “Necessary optimality criteria in mathematical programming in the presence of differentiability”,Journal of Mathematical Analysis and Applications 40 (1972).
M.S. Bazaraa, J.J. Goode and M.Z. Nashed, “On the cones of tangents with application to mathematical programming”,Journal of Optimization Theory and Applications, to appear.
M.S. Bazaraa, J.J. Goode and C.M. Shetty, “Optimality criteria in nonlinear programming without differentiability”,Operations Research 19 (1971).
J. Bram, “The Lagrange Multiplier Theorem for max—min with several constraints”,SIAM Journal on Applied Mathematics 14 (1966).
M.D. Canon, C.D. Cullum, Jr. and E. Polak,Theory of optimal control and mathematical programming (McGraw-Hill, New York, 1970).
J. Danskin, “On the theory of min—max”,SIAM Journal on Applied Mathematics 14 (1966).
V.F. Demyanov, “On the solution of certain minimax problems, I”,Cybernetics 2 (1966).
V.F. Demyanov, “Algorithms for some minimax problems”,Journal of Computer and Systems Sciences 2 (1968).
V.F. Demyanov and A.M. Rubinov,Approximate methods in optimization problems (Elsevier, Amsterdam, 1970).
J.E. Falk and R.M. Soland, “An algorithm for separable nonconvex programming problems”,Management Science 15 (1969).
A.M. Geoffrion, “Generalized benders decomposition”,Journal of Optimization Theory and Applications 10 (1972).
W.W. Hogan, “Directional derivatives for extremal value functions with applications to the completely convex case”,Operations Research 21 (1973).
F. John, “Extremum problems with inequalities as subsidiary conditions”, in:Studies and essays: Courant anniversary volume, Eds. K.O. Friedrichs and O.E. Neugebauer (Interscience, New York, 1948).
H.W. Kuhn and A.W. Tucker, “Nonlinear programming”, in:Proceedings of the second Berkeley symposium on mathematical statistics and probability, Ed. J. Neyman (University of California, Berkeley, Calif., 1951).
O.L. Mangasarian,Nonlinear programming (McGraw-Hill, New York, 1969).
R.T. Rockafellar, “Duality in nonlinear programming”, in:Mathematics of the decision sciences, Eds. G.B. Dantzig and A.F. Veinott (American Mathematical Society, Providence, R.I., 1969).
R.T. Rockafellar,Convex analysis (Princeton Univ. Press, Princeton, N.J., 1970).
W.I. Zangwill,Nonlinear programming: a unified approach (Prentice-Hall, Englewood Cliffs, N.J., 1969).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bazaraa, M.S., Goode, J.J. Extension of optimality conditions via supporting functions. Mathematical Programming 5, 267–285 (1973). https://doi.org/10.1007/BF01580133
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01580133