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 ${\displaystyle (s,S)}${\displaystyle (s,S)} policy in inventory control theory. The policy is characterized by two numbers s and S, ${\displaystyle S\geq s}${\displaystyle 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 ^[2] have generalized the concept of K-convexity to higher dimensional Euclidean spaces.

Two equivalent definitions are as follows:

Let K be a non-negative real number. A function ${\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }${\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} } is K-convex if

${\displaystyle g(u)+z\left[{\frac {g(u)-g(u-b)}{b}}\right]\leq g(u+z)+K}${\displaystyle g(u)+z\left[{\frac {g(u)-g(u-b)}{b}}\right]\leq g(u+z)+K}

for any ${\displaystyle u,z\geq 0,}${\displaystyle u,z\geq 0,} and ${\displaystyle b>0}${\displaystyle b>0}.

A function ${\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }${\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} } is K-convex if

${\displaystyle g(\lambda x+{\bar {\lambda }}y)\leq \lambda g(x)+{\bar {\lambda }}[g(y)+K]}${\displaystyle g(\lambda x+{\bar {\lambda }}y)\leq \lambda g(x)+{\bar {\lambda }}[g(y)+K]}

for all ${\displaystyle x\leq y,\lambda \in [0,1]}${\displaystyle x\leq y,\lambda \in [0,1]}, where ${\displaystyle {\bar {\lambda }}=1-\lambda }${\displaystyle {\bar {\lambda }}=1-\lambda }.

This definition admits a simple geometric interpretation related to the concept of visibility.^[3] Let ${\displaystyle a\geq 0}${\displaystyle a\geq 0}. A point ${\displaystyle (x,f(x))}${\displaystyle (x,f(x))} is said to be visible from ${\displaystyle (y,f(y)+a)}${\displaystyle (y,f(y)+a)} if all intermediate points ${\displaystyle (\lambda x+{\bar {\lambda }}y,f(\lambda x+{\bar {\lambda }}y)),0\leq \lambda \leq 1}${\displaystyle (\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 ${\displaystyle g}${\displaystyle g} is K-convex if and only if ${\displaystyle (x,g(x))}${\displaystyle (x,g(x))} is visible from ${\displaystyle (y,g(y)+K)}${\displaystyle (y,g(y)+K)} for all ${\displaystyle y\geq x}${\displaystyle y\geq x}.

It is sufficient to prove that the above definitions can be transformed to each other. This can be seen by using the transformation

${\displaystyle \lambda =z/(b+z),\quad x=u-b,\quad y=u+z.}${\displaystyle \lambda =z/(b+z),\quad x=u-b,\quad y=u+z.}

^[4]

If ${\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }${\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} } is K-convex, then it is L-convex for any ${\displaystyle L\geq K}${\displaystyle L\geq K}. In particular, if ${\displaystyle g}${\displaystyle g} is convex, then it is also K-convex for any ${\displaystyle K\geq 0}${\displaystyle K\geq 0}.

If ${\displaystyle g_{1}}${\displaystyle g_{1}} is K-convex and ${\displaystyle g_{2}}${\displaystyle g_{2}} is L-convex, then for ${\displaystyle \alpha \geq 0,\beta \geq 0,\;g=\alpha g_{1}+\beta g_{2}}${\displaystyle \alpha \geq 0,\beta \geq 0,\;g=\alpha g_{1}+\beta g_{2}} is ${\displaystyle (\alpha K+\beta L)}${\displaystyle (\alpha K+\beta L)}-convex.

If ${\displaystyle g}${\displaystyle g} is K-convex and ${\displaystyle \xi }${\displaystyle \xi } is a random variable such that ${\displaystyle E|g(x-\xi )|<\infty }${\displaystyle E|g(x-\xi )|<\infty } for all ${\displaystyle x}${\displaystyle x}, then ${\displaystyle Eg(x-\xi )}${\displaystyle Eg(x-\xi )} is also K-convex.

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

If ${\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }${\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} } is a continuous K-convex function and ${\displaystyle g(y)\rightarrow \infty }${\displaystyle g(y)\rightarrow \infty } as ${\displaystyle |y|\rightarrow \infty }${\displaystyle |y|\rightarrow \infty }, then there exit scalars ${\displaystyle s}${\displaystyle s} and ${\displaystyle S}${\displaystyle S} with ${\displaystyle s\leq S}${\displaystyle s\leq S} such that

• ${\displaystyle g(S)\leq g(y)}${\displaystyle g(S)\leq g(y)}, for all ${\displaystyle y\in \mathbb {R} }${\displaystyle y\in \mathbb {R} };
• ${\displaystyle g(S)+K=g(s)<g(y)}${\displaystyle g(S)+K=g(s)<g(y)}, for all ${\displaystyle y<s}${\displaystyle y<s};
• ${\displaystyle g(y)}${\displaystyle g(y)} is a decreasing function on ${\displaystyle (-\infty ,s)}${\displaystyle (-\infty ,s)};
• ${\displaystyle g(y)\leq g(z)+K}${\displaystyle g(y)\leq g(z)+K} for all ${\displaystyle y,z}${\displaystyle y,z} with ${\displaystyle s\leq y\leq z}${\displaystyle s\leq y\leq z}.

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

• Convex analysis
• Types of functions

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