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 $(s,S)$ policy in inventory control theory. The policy is characterized by two numbers s and S, $S\ge s$, 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 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 $g:ℝ\to ℝ$ is K-convex if

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

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

Definition 2 (Definition with geometric interpretation)

A function $g:ℝ\to ℝ$ is K-convex if

$g(\lambda x+\overline{\lambda }y)\le \lambda g(x)+\overline{\lambda }[g(y)+K]$

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

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

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

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

Properties

Property 1

If $g:ℝ\to ℝ$ is K-convex, then it is L-convex for any $L\ge K$. In particular, if $g$ is convex, then it is also K-convex for any $K\ge 0$.

Property 2

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

Property 3

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

Property 4

If $g:ℝ\to ℝ$ is K-convex, restriction of $g$ on any convex set $𝔻\subset ℝ$ is K-convex.

Property 5

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

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