gradiensets

Gradiensets are mathematical objects that collect gradient vectors of a scalar function across a domain. They are used to capture local directional information that informs optimization analysis and the study of gradient dynamics.

Formally, let f: R^n -> R be differentiable on a domain D ⊆ R^n. The gradienset of f over D is defined as G_f(D) = { ∇f(x) : x ∈ D }. The gradienset is a subset of R^n whose shape and size reflect features of f, such as curvature and how rapidly the gradient changes across the domain. In convex settings, gradients satisfy monotonicity properties: for x, y ∈ D, ⟨∇f(x) − ∇f(y), x − y⟩ ≥ 0.

Variants and related concepts include the subgradient set for non-differentiable functions, denoted ∂f(x), and the Clarke generalized gradient, which extend the idea of a gradient to broader classes of functions. While gradiensets emphasize the collection of gradients across a region, subgradients provide a pointwise, set-valued alternative that remains well-defined under non-smoothness.

Structure and interpretation: the gradienset is usually a geometric object in R^n. Its geometry can reveal function features such as linear regions in piecewise-affine models, sensitivity to perturbations, and potential directions for descent in optimization algorithms.

Examples: for a quadratic function f(x) = 1/2 x^T Q x with Q symmetric, ∇f(x) = Qx and G_f(D) = { Qx : x ∈ D }. If D is a ball, G_f(D) is the image of that ball under the linear map Q, typically an ellipsoid or related shape.

Applications include analyzing gradient flow, convergence behavior of algorithms, and studying how objective values respond to changes in input. The term gradienset is a portmanteau of gradient and set, used in some optimization discussions to emphasize the ecosystem of gradient information across a domain. See also gradient, subgradient, Jacobian, Hessian, and convex analysis.