r-convex functions

A Classroom Note on Twice Continuously Differentiable Strictly Convex and Strongly Quasiconvex Functions

“…See also Avriel (1972), Kemp and Kimura (1978) and, for characterizations of twice continuously differentiable quasiconvex functions, Crouzeix (1980), Crouzeix and Ferland (1982), Diewert and others (1981). Needless to say, condition (1) is trivially satisfied for r = 1.…”

Generalized convexity in mathematical programming

“…f is said to be a convex function if for all x, y € C , 0 < A 5 1 , If f is di'fferentiable, an alternate and equivalent definition is (2) f(y) -fix) 2 (y-x where V/(x) denotes the gradient (column) vector of p a r t i a l derivatives of / with respect to x . Whereas (1) says that linear approximation over-estimates the value of the function, (2) says that a tangent to / at x l i e s on or below / .…”

“…Following the terminology of [ 2 ] , we now extend the definition of convexity in (l) as follows. A function / is said to be r convex if i t satisfies, for a l l x, y in the domain of / , Recall now that we restricted our definition of Mif; X) to fiy) and fix) positive.…”

“…This was done so that M (f; X) will be defined for a l l r . in order to allow zero and negative values of f for a l l r , Avriel [2] and Martos [77] independently define r-convex functions as follows: / i s said to be r-convex if for a l l r , X , -°° 5 r 2 °° , 0 5 X 5 1 , i t satisfies…”

“…Avriel [2] calls functions that satisfy (33) for r < 0 , superaonvex and, for r > 0 , suboonvex. Thus superconvexity implies convexity which implies subconvexity which in turn, implies quasi-convexity.…”