a sum is the sum of the subdifferentials." The subgradient The concepts of subderivative and subdifferential can be generalized to functions of several variables. If is a real-valued convex function defined on a convex open set in the Euclidean space , a vector in that space is called a subgradient at if for any one has that where the dot denotes the dot product. The set of all subgradients at is called the subdifferential at x0 and is denoted . The subdifferential is always a nonempty convex compact set. These concepts generalize further to convex functions on a convex set in a locally convex space . A functional in the dual space is called the subgradient at in if for all , The set of all subgradients at is called the subdifferential at and is again denoted . The subdifferential is always a convex closed set. It can be an empty set; consider for example an unbounded operator, which is convex, but has no subgradient. If is continuous, the subdifferential is nonempty. History The subdifferential on convex functions was introduced by Jean Jacques Moreau and R. Tyrrell Rockafellar in the early 1960s. The generalized subdifferential for nonconvex functions was introduced by F.H. Clarke and R.T. Rockafellar in the early 1980s. See also Weak derivative Subgradient method References External links Generalizations of the derivative Convex optimization Variational analysis