Convex Functions

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

$f(\lambda a+(1-\lambda )b)\le \lambda f(a)+(1-\lambda )f(b),\lambda \in [0,1]\to f(x)$ is convex

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

$f(x)$ is convex $\to f^{\prime \prime }(x)>0\forall x$

${\mathbf{H}}_f⪰0\to f$ is convex

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