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K-convex function

Two equivalent definitions are as follows:

Definition 1 (The original definition)[edit]

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

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

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

Definition 2 (Definition with geometric interpretation)[edit]

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

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

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

This definition admits a simple geometric interpretation related to the concept of visibility.^[3] Let ${\displaystyle a\geq 0}$. A point ${\displaystyle (x,f($x))} is said to be visible from ${\displaystyle (y,f($y)+a)} if all intermediate points ${\displaystyle (\lambda x+{\bar {\lambda }}y,f(\lambda x+{\bar {\lambda }}y)),0\leq \lambda \leq 1}$ lie below the line segment joining these two points. Then the geometric characterization of K-convexity can be obtain as:

A function ${\displaystyle g}$isK-convex if and only if ${\displaystyle (x,g($x))} is visible from ${\displaystyle (y,g($y)+K)} for all ${\displaystyle y\geq x}$.

Proof of Equivalence[edit]

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

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

Properties[edit]

^[4]

Property 1[edit]

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

Property 2[edit]

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

Property 3[edit]

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

Property 4[edit]

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

Property 5[edit]

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

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

References[edit]

Further reading[edit]

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