Convex Functions

#Math
A function where any local minimum is guaranteed to be a global minimum

$\displaystyle f(\lambda a+(1-\lambda)b)\leq \lambda f(a)+(1-\lambda)f(b),\lambda \in [0,1]\rightarrow f(x)$ is convex

• If every point on a line between two points on our function lie above the function, the function's convex

$\displaystyle f(x)$ is convex $\displaystyle \rightarrow f''(x)>0~\forall~x$

$\displaystyle \mathbf{H}_{f}\succeq 0\rightarrow f$ is convex

• $\displaystyle \mathbf{H}_{f}$ is the Hessian of $\displaystyle f$ and must be a positive semidefinite matrix
• For multi-variate $\displaystyle f$