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.

1 Definition
2 Properties
3 References
4 External Links

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 P. (2014). "K-convexity in $\mathbb{R}^n$ ". External link in |title= (help)

fonte: https://web.archive.org/web/20160221211028/https://en.wikipedia.org/wiki/K-convex_function

Conheça Walt Disney World