subgradient the vector in ( may not be unique and is referred to as subgradient of one of the crucial properties of convex function is that jensen' inequality holds (see exercise in ) (xf (ex)for any random vector directional derivative example (convexity and directional derivativethe directional derivative of multivariate function at in the direction is defined as the right derivative of ( : ( dat ( df (xlim ( /tf ( )tt| lim this right derivative may not always exist howeverif is convex functionthen the directional derivative of at in the interior of its domain always exists (in any direction dto see thislet by jensen' inequality we have for any and in the interior of the domaint ( (xf making the substitution and rearranging the last equation yieldsf ( df (xf ( df (xt in other wordsthe function ( df ( ))/ is increasing for and therefore the directional derivative satisfiesf ( df (xf ( df (xinf lim > | henceto show existence it is enough to show that ( df ( ))/ is bounded from below since lies in the interior of the domain of we can choose small enough so that also lies in the interior thereforethe convexity of implies that there exists subgradient vector such that ( df (xv( din other wordsf ( df (xvd provides lower bound for all and the directional derivative of at an interior always exists (in any directionconcave function function satisfying ( with strict inequality is said to be strictly convex it is said to be (strictlyconcave function if is (strictlyconvex assuming that is an open setconvexity for is equivalent to (yf ( ( ) ( for all xy moreoverfor strict convexity is equivalent to the hessian matrix being positive definite for all xand convexity isequivalent to the hessian matrix being positive semidefinite for all xthat isy (xy for all and recall that we write to indicate that matrix is positive semidefinite