2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76285We introduce real vector spaces composed of set-valued maps on an open set. They are also complete metric spaces, lattices, commutative rings. The set of differentiable functions is a dense subset of these spaces and the classical gradient may be extended in these spaces as a closed operator. If a function f belongs to the domain of such extension, then f is locally lipschitzian and the values of extended gradient coincide with the values of Clarke's gradient. However, unlike Clarke's gradient, our generalized gradient is a linear operator.Optimization and Control49K99text