example (convexity and differentiabilityif is continuously differentiable multivariate functionthen is convex if and only if the univariate function ( : ( ) [ is convex function for any and in the interior of the domain of this property provides an alternative definition for convexity of multivariate and differentiable function to see why it is truefirst assume that is convex and [ thenusing the subgradient definition of convexity in ( )we have (af ( ( ) for some subgradient substituting with and dwe obtain ( ( ( vd for any two points [ thereforeg is convexbecause we have identified the existence of subgradient vd for each converselyassume that is convex for [ since is differentiablethen so is thenthe convexity of implies that there is subgradient at such thatg(tg( for all [ rearrangingg(tg( vt and taking the right limit as we obtain ( df (xthereforeg(tg( ( df (xand substituting yieldsf ( df (xdf ( )so that there exists subgradient vectornamely ( )for each hencef is convex by the definition in ( an optimization program of the form ( is said to be convex programming problem if the objective is convex function the inequality constraint functions {gi are convex the equality constraint functions {hi are affinethat isof the form > bi this is equivalent to both hi and -hi being convex for all table summarizes some commonly encountered problemsall of which are convexwith the exception of the quadratic programs with convex programming problem