A Tutorial on Hadamard Semidifferentials

The Hadamard semidifferential is more general than the Fréchet differential now dominant in undergraduate mathematics education. By slightly changing the definition of the forward directional derivative, the Hadamard semidifferential rescues the chain rule, enforces continuity, and permits differentiation across maxima and minima. It also plays well with convex analysis and naturally extends differentiation to smooth embedded submanifolds, topological vector spaces, and metric spaces of shapes and geometries. The current elementary exposition focuses on the more familiar territory of analysis in Euclidean spaces and applies the semidifferential to some representative problems in optimization and statistics. These include algorithms for proximal gradient descent, steepest descent in matrix completion, and variance components models.