gradientem

Gradientem is a term occasionally encountered in mathematical discussions of gradients and constrained optimization. It is not a standard, widely adopted concept, and its exact meaning varies by author. In general, gradientem refers to the component of the gradient that lies in a specified subspace of directions, such as a constraint tangent space.

Definition and interpretation: Let f be differentiable on a manifold M, and let S_p be a subspace of the tangent space T_pM at point p. The gradientem of f at p with respect to S_p is the projection of the Riemannian gradient ∇f(p) onto S_p. In Euclidean space with the usual metric, grad_S f(p) = P_S(∇f(p)), where P_S is the orthogonal projection to S_p. If S is the tangent space to a constraint at p, grad_S f provides the descent directions that respect the constraint.

Relationship to optimization: Gradientem is described as the constrained-descent component of the gradient and is related to projected gradients and Lagrange multiplier methods. When constraints are inactive, gradientem reduces to the ordinary gradient.

Example: Minimize f(x,y)=x^2+y^2 subject to x+y=1. The unconstrained gradient is ∇f=(2x,2y). The gradientem relative to the constraint’s tangent space (direction (1,-1)) is the projection of ∇f onto that direction: grad_S f = ((2x-2y)/2)(1,-1) = (x−y, y−x).

Status and usage: The term gradientem is not standard in mainstream references and is mainly used in didactic or exploratory contexts to illustrate the idea of gradient projection under constraints.