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 and , $S \geq s$, such that when the inventory level falls below level , an order is issued for a quantity that brings the inventory up to level , and nothing is ordered otherwise. Gallego and Sethi Gallego, G. and Sethi, S. P. (2005). ''K''-convexity in ℜⁿ. ''Journal of Optimization Theory & Applications,'' 127(1):71-88. 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 $g: \mathbb\rightarrow\mathbb$ is ''K''-convex if
:g(u)+z\left frac\right\leq g(u+z) + K
for any $u, z\geq 0,$ and $b>0$.

Definition 2 (Definition with geometric interpretation)

A function $g: \mathbb\rightarrow\mathbb$ is ''K''-convex if
:g(\lambda x+\bar y) \leq \lambda g(x) + \bar (y)+K/math>
for all x\leq y, \lambda \in ,1/math>, where $\bar=1-\lambda$.

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 $(\lambda x+\bar y, f(\lambda x+\bar y)), 0\leq \lambda \leq 1$ 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

Sethi S P, Cheng F. Optimality of (s, S) Policies in Inventory Models with Markovian Demand. INFORMS, 1997.

Property 1

If $g: \mathbb\rightarrow\mathbb$ 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 $g: \mathbb\rightarrow\mathbb$ is ''K''-convex, restriction of $g$ on any convex set $\mathbb\subset\mathbb$ is ''K''-convex.

Property 5

If $g: \mathbb\rightarrow\mathbb$ 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 $y\in \mathbb$;
* g(S)+K=g(s), for all y;
* $g(y)$ is a decreasing function on $(-\infty, s)$;
* $g(y)\leq g(z)+K$ for all $y, z$ with $s\leq y\leq z$.

References

Further reading

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{{Convex analysis and variational analysis

Convex analysis
Types of functions