K-convex function

K-convex functions, first introduced by Scarf,^[1] 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, $S \leq s$, 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.

Definition

Two equivalent definitions are as follows:

Definition 1 (The original definition)

A function $g: \mathbb{R}\rightarrow\mathbb{R}$ is K-convex if

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

for any $u, z\geq 0,$ and $b>0$.

Definition 2 (Definition with geometric interpretation)

A function $g: \mathbb{R}\rightarrow\mathbb{R}$ is K-convex if

$g(\lambda x+\bar{\lambda} y) \leq \lambda g(x) + \bar{\lambda} [g(y)+K]$

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

This definition admits a simple geometric interpretation related to the concept of visibility.^[2] Let $a \geq 0$. A point $(x,f(x))$ is said to be visible from $(y,f(y)+a)$ if all intermediate points $(\lambda x+\bar{\lambda} y, f(\lambda x+\bar{\lambda} y)), 0\leq \lambda \leq 1$ 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 $y\geq x$.

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

$\lambda = z/(b+z),\quad x=u-b,\quad y=u+z.$

Properties

Property 1

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

Property 2

If $g_1$ is K-convex and $g_2$ is L-convex, then for $\alpha \geq 0, \beta \geq 0,\; g=\alpha g_1 +\beta g_2$ is $(\alpha K+\beta L)$-convex.

Property 3

If $g$ is K-convex and $\xi$ is a random variable such that $E|g(x-\xi)|<\infty$ for all $x$, then $Eg(x-\xi)$ is also K-convex.

Property 4

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

• $g(S)\leq g(y)$, for all $y\in \mathbb{R}$;
• $g(S)+K=g(s)<g(y)$, for all $y<s$;
• $g(y)$ is a decreasing function on $(-\infty, s)$;
• $g(y)\leq g(z)+K$ for all $y, z$ with $s\leq y\leq z$.

References

1. Scarf, H. (1960). The Optimality of (S, s) Policies in the Dynamic Inventory Problem. Stanford, CA: Stanford University Press. p. Chapter 13.
2. Kolmogorov, A. N.; Fomin, S. V. (1970). Introduction to Real Analysis. New York: Dover Publications Inc.

External Links

• Gallego, Guillermo; Sethi, Suresh (16 September 2004). "K-CONVEXITY IN ℜⁿ" (PDF): 21. Retrieved January 21, 2016.