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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.

MSC:

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

Citations:

Zbl 0534.00022