This post summarizes convex sets and convex functions, and their properties.

Convex Sets

Convex Functions

(Convexity) A function $f$ is convex if $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y),$$ for all $x,y\in dom(f)$ and $t\in(0,1)$.

(Strict Convexity) A function $f$ is strictly convex if $$f(tx+(1-t)y)< tf(x)+(1-t)f(y),$$ for all $x,y\in dom(f)$ and $t\in(0,1)$.

(Strong Convexity) A function $f$ is $m$-strongly convex if $f(x)-\frac{m}{2}\|x\|_2^2$ is convex.

Strong convexity implies strict convexity, which implies convexity.

• If a convex function has a local minimum $x^{\ast}$, then $x^{\ast}$ is also a global minimum.
• If a strictly convex function has a local/global minimum $x^{\ast}$, then $x^{\ast}$ is the unique global minimum.

A proper convex function is an extended real-valued convex function with a non-empty domain, that never takes on the value $-\infty$ and also is not identically equal to $+\infty $.

Convex Functions over $K$