Semigradable

Semigradable is a term encountered in some branches of mathematical optimization and analysis to describe a property of a function that admits a semigradient, a generalized first-order derivative used for non-smooth or non-differentiable settings. The concept is not universally standardized, and its precise use can vary between authors.

Informally, a function f: R^n -> R is semigradable at a point x if there exists a vector g in R^n that serves as a semigradient: g provides a first-order linear approximation to f near x. In practical terms, this means that near x the function can be bounded or approximated by the linear model f(x) + g^T (y - x) plus higher-order terms. The semigradient plays a role similar to a gradient in smooth problems, offering a directionally informative linear reference.

Relationship to other concepts is straightforward in common cases. If f is differentiable at x, the gradient ∇f(x) acts as a semigradient. If f is convex, any subgradient g in the subdifferential ∂f(x) serves as a semigradient, aligning semigradability with well-known nonsmooth analysis. For some nonconvex or piecewise-smooth functions, a suitable semigradient may or may not exist at a given point, depending on the local geometry of f.

Applications typically involve iterative optimization methods that rely on a first-order model of the objective, even when full differentiability is unavailable. Semigradability thus provides a framework for extending gradient-based ideas to a broader class of functions.

See also: gradient, subgradient, generalized gradient, nonsmooth optimization.