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This is a generalization of the concept of strongly convex function; by taking () = we recover the definition of strong convexity. It is worth noting that some authors require the modulus ϕ {\displaystyle \phi } to be an increasing function, [ 17 ] but this condition is not required by all authors.
The policy is characterized by two numbers s and S, , such that when the inventory level falls below level s, an order is issued for a quantity that brings the inventory up to level S, and nothing is ordered otherwise. Gallego and Sethi [2] have generalized the concept of K-convexity to higher dimensional Euclidean spaces.
When the points are interpreted as probability distributions – notably as either values of the parameter of a parametric model or as a data set of observed values – the resulting distance is a statistical distance. The most basic Bregman divergence is the squared Euclidean distance.
then is called strictly convex. [1]Convex functions are related to convex sets. Specifically, the function is convex if and only if its epigraph. A function (in black) is convex if and only if its epigraph, which is the region above its graph (in green), is a convex set.
Quasiconvexity is a generalisation of convexity for functions defined on matrices, to see this let and ((,),) with (,) =. The Riesz-Markov-Kakutani representation theorem states that the dual space of C 0 ( R m × d ) {\displaystyle C_{0}(\mathbb {R} ^{m\times d})} can be identified with the space of signed, finite Radon measures on it.
If : is a continuous function and is closed, then is closed.; If : is a continuous function and is open, then is closed if and only if it converges to along every sequence converging to a boundary point of .
In complex geometry and analysis, the notion of convexity and its generalizations play an important role in understanding function behavior. Examples of classes of functions with a rich structure are, in addition to the convex functions, the subharmonic functions and the plurisubharmonic functions.
The concept of K-convexity generalizes K-convexity introduced by Scarf (1960) [2] to higher dimensional spaces and is useful in multiproduct inventory problems with fixed setup costs. Scarf used K-convexity to prove the optimality of the (s, S) policy in the single product case. Several papers are devoted to obtaining optimal policies for ...