F-convex functions: Properties and applications. (English) Zbl 0535.90074

Generalized concavity in optimization and economics, Proc. NATO Adv. Study Inst., Vancouver/Can. 1980, 301-334 (1981).
Summary: [For the entire collection see Zbl 0534.00022.]
For a given family of functions F, a function is F-convex if its epigraph is supported at each point by a member of F. We introduce notions such as F-subgradients and F-conjugate functions, and generalized results associated with their classical counterparts. In particular we derive monotonicity results for the F-subgradients, and unimodality results for differences of F-convex and F-concave functions. A Fenchel-type duality theory for primal problems involving such differences is given. Other augmented Lagrangean type duals, involving closely related notions of generalized convexity, are briefly sketched. We also outline a numerical method for a root-finding problem, which is based on ”F-approximation” rather than ”linearization” as in Newton’s method.


90C25 Convex programming
26B25 Convexity of real functions of several variables, generalizations
90C55 Methods of successive quadratic programming type
49N15 Duality theory (optimization)


Zbl 0534.00022