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, $S\geq s$, 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 $g:\mathbb {R} \rightarrow \mathbb {R}$ 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. 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 K-convex, restriction of $g$ on any convex set $\mathbb {D} \subset \mathbb {R}$ is K-convex.

Property 5

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 $s\leq y\leq z$.

Further reading

• Gallego, G.; Sethi, S. P. (2005). "${\mathcal {K}}$-convexity in ${\mathfrak {R}}^{n}$" (PDF). Journal of Optimization Theory and Applications. 127 (1): 71–88. doi:10.1007/s10957-005-6393-4. MR 2174750.