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

↑ Scarf, H. (1960). The Optimality of (S, s) Policies in the Dynamic Inventory Problem. Stanford, CA: Stanford University Press. p. Chapter 13.
↑ 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.

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