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 have generalized the concept of K-convexity to higher dimensional Euclidean spaces.

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 have generalized the concept of K-convexity to higher dimensional Euclidean spaces. 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 . \lambda = z/(b+z),\quad x=u-b,\quad y=u+z. 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. 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 .

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Further reading

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