K-convex function
None
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 policy in inventory control theory . The policy is characterized by two numbers s and 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 [2] 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 is K -convex if
for any and .
Definition 2 (Definition with geometric interpretation)
A function is K -convex if
for all , where .
This definition admits a simple geometric interpretation related to the concept of visibility. [3] Let . A point is said to be visible from if all intermediate points lie below the line segment joining these two points. Then the geometric characterization of K -convexity can be obtain as:
- A function is K -convex if and only if is visible from for all .
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
Properties
Property 1
If is K -convex, then it is L -convex for any . In particular, if is convex, then it is also K -convex for any .
Property 2
If is K -convex and is L -convex, then for is -convex.
Property 3
If is K -convex and is a random variable such that for all , then is also K -convex.
Property 4
If is K -convex, restriction of on any convex set is K -convex.
Property 5
If is a continuous K -convex function and as , then there exit scalars and with such that
- , for all ;
- , for all ;
- is a decreasing function on ;
- for all with .
References
- ↑ Scarf, H. (1960). The Optimality of (S, s) Policies in the Dynamic Inventory Problem . Stanford, CA: Stanford University Press. p. Chapter 13.
- ↑ Gallego, G. and Sethi, S. P. (2005). K -convexity in ℜ n . Journal of Optimization Theory & Applications, 127(1):71-88.
- ↑ Kolmogorov, A. N.; Fomin, S. V. (1970). Introduction to Real Analysis . New York: Dover Publications Inc.
- ↑ Sethi S P, Cheng F. Optimality of (s, S) Policies in Inventory Models with Markovian Demand. INFORMS, 1997.
External links
-
Gallego, Guillermo; Sethi, Suresh (16 September 2004).
"K-CONVEXITY IN ℜ
n
"
(PDF)
: 21
. Retrieved
January 21,
2016
.
{{ cite journal }}
: Cite journal requires| journal=
( help )
Basic concepts | |
---|---|
Topics (list) | |
Maps | |
Main results (list) | |
Sets | |
Series | |
Duality | |
Applications and related |