K-convex function

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 [math]\displaystyle{ (s,S) }[/math] policy in inventory control theory. The policy is characterized by two numbers s and S, [math]\displaystyle{ S \geq s }[/math], 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 have generalized the concept of K-convexity to higher dimensional Euclidean spaces.

Definition

Two equivalent definitions are as follows:

Definition 1 (The original definition)

Let K be a non-negative real number. A function [math]\displaystyle{ g: \mathbb{R}\rightarrow\mathbb{R} }[/math] is K-convex if

[math]\displaystyle{ g(u)+z\left[\frac{g(u)-g(u-b)}{b}\right] \leq g(u+z) + K }[/math]

for any [math]\displaystyle{ u, z\geq 0, }[/math] and [math]\displaystyle{ b\gt 0 }[/math].

Definition 2 (Definition with geometric interpretation)

A function [math]\displaystyle{ g: \mathbb{R}\rightarrow\mathbb{R} }[/math] is K-convex if

[math]\displaystyle{ g(\lambda x+\bar{\lambda} y) \leq \lambda g(x) + \bar{\lambda} [g(y)+K] }[/math]

for all [math]\displaystyle{ x\leq y, \lambda \in [0,1] }[/math], where [math]\displaystyle{ \bar{\lambda}=1-\lambda }[/math].

This definition admits a simple geometric interpretation related to the concept of visibility. Let [math]\displaystyle{ a \geq 0 }[/math]. A point [math]\displaystyle{ (x,f(x)) }[/math] is said to be visible from [math]\displaystyle{ (y,f(y)+a) }[/math] if all intermediate points [math]\displaystyle{ (\lambda x+\bar{\lambda} y, f(\lambda x+\bar{\lambda} y)), 0\leq \lambda \leq 1 }[/math] lie below the line segment joining these two points. Then the geometric characterization of K-convexity can be obtain as:

A function [math]\displaystyle{ g }[/math] is K-convex if and only if [math]\displaystyle{ (x,g(x)) }[/math] is visible from [math]\displaystyle{ (y,g(y)+K) }[/math] for all [math]\displaystyle{ y\geq x }[/math].

Proof of Equivalence

It is sufficient to prove that the above definitions can be transformed to each other. This can be seen by using the transformation

[math]\displaystyle{ \lambda = z/(b+z),\quad x=u-b,\quad y=u+z. }[/math]

Properties

Property 1

If [math]\displaystyle{ g: \mathbb{R}\rightarrow\mathbb{R} }[/math] is K-convex, then it is L-convex for any [math]\displaystyle{ L\geq K }[/math]. In particular, if [math]\displaystyle{ g }[/math] is convex, then it is also K-convex for any [math]\displaystyle{ K\geq 0 }[/math].

Property 2

If [math]\displaystyle{ g_1 }[/math] is K-convex and [math]\displaystyle{ g_2 }[/math] is L-convex, then for [math]\displaystyle{ \alpha \geq 0, \beta \geq 0,\; g=\alpha g_1 +\beta g_2 }[/math] is [math]\displaystyle{ (\alpha K+\beta L) }[/math]-convex.

Property 3

If [math]\displaystyle{ g }[/math] is K-convex and [math]\displaystyle{ \xi }[/math] is a random variable such that [math]\displaystyle{ E|g(x-\xi)|\lt \infty }[/math] for all [math]\displaystyle{ x }[/math], then [math]\displaystyle{ Eg(x-\xi) }[/math] is also K-convex.

Property 4

If [math]\displaystyle{ g: \mathbb{R}\rightarrow\mathbb{R} }[/math] is K-convex, restriction of [math]\displaystyle{ g }[/math] on any convex set [math]\displaystyle{ \mathbb{D}\subset\mathbb{R} }[/math] is K-convex.

Property 5

If [math]\displaystyle{ g: \mathbb{R}\rightarrow\mathbb{R} }[/math] is a continuous K-convex function and [math]\displaystyle{ g(y)\rightarrow \infty }[/math] as [math]\displaystyle{ |y|\rightarrow \infty }[/math], then there exit scalars [math]\displaystyle{ s }[/math] and [math]\displaystyle{ S }[/math] with [math]\displaystyle{ s\leq S }[/math] such that

• [math]\displaystyle{ g(S)\leq g(y) }[/math], for all [math]\displaystyle{ y\in \mathbb{R} }[/math];
• [math]\displaystyle{ g(S)+K=g(s)\lt g(y) }[/math], for all [math]\displaystyle{ y\lt s }[/math];
• [math]\displaystyle{ g(y) }[/math] is a decreasing function on [math]\displaystyle{ (-\infty, s) }[/math];
• [math]\displaystyle{ g(y)\leq g(z)+K }[/math] for all [math]\displaystyle{ y, z }[/math] with [math]\displaystyle{ s\leq y\leq z }[/math].

External links

• Gallego, Guillermo; Sethi, Suresh (16 September 2004) (PDF). K-CONVEXITY IN ℜⁿ. pp. 21. https://www.utdallas.edu/~sethi/Postscript/Kconvexity091504.pdf. Retrieved January 21, 2016.