K-convex functions, first introduced by Scarf, are a special weakening of the concept of convex function which is crucial in the proof of the optimality of the ( s , S ) policy in inventory control theory. The policy is characterized by two numbers s and S, S ≥ 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.
Two equivalent definitions are as follows:
A function g : R → R is K-convex if
g ( u ) + z [ g ( u ) − g ( u − b ) b ] ≤ g ( u + z ) + K for any u , z ≥ 0 , and b > 0 .
A function g : R → R is K-convex if
g ( λ x + λ ¯ y ) ≤ λ g ( x ) + λ ¯ [ g ( y ) + K ] for all x ≤ y , λ ∈ [ 0 , 1 ] , where λ ¯ = 1 − λ .
This definition admits a simple geometric interpretation related to the concept of visibility. Let a ≥ 0 . A point ( x , f ( x ) ) is said to be visible from ( y , f ( y ) + a ) if all intermediate points ( λ x + λ ¯ y , f ( λ x + λ ¯ y ) ) , 0 ≤ λ ≤ 1 lie below the line segment joining these two points. Then the geometric characterization of K-convexity can be obtain as:
A function
g is
K-convex if and only if
( x , g ( x ) ) is visible from
( y , g ( y ) + K ) for all
y ≥ x .
It is sufficient to prove that the above definitions can be transformed to each other. This can be seen by using the transformation
λ = z / ( b + z ) , x = u − b , y = u + z . If g : R → R is K-convex, then it is L-convex for any L ≥ K . In particular, if g is convex, then it is also K-convex for any K ≥ 0 .
If g 1 is K-convex and g 2 is L-convex, then for α ≥ 0 , β ≥ 0 , g = α g 1 + β g 2 is ( α K + β L ) -convex.
If g is K-convex and ξ is a random variable such that E | g ( x − ξ ) | < ∞ for all x , then E g ( x − ξ ) is also K-convex.
If g : R → R is a continuous K-convex function and g ( y ) → ∞ as | y | → ∞ , then there exit scalars s and S with s ≤ S such that
g ( S ) ≤ g ( y ) , for all y ∈ R ; g ( S ) + K = g ( s ) < g ( y ) , for all y < s ; g ( y ) is a decreasing function on ( − ∞ , s ) ; g ( y ) ≤ g ( z ) + K for all y , z with s ≤ y ≤ z .