##  [Subgradient](/subgradient-0) 

 Definition

A vector that generalizes the gradient for a convex (possibly nondifferentiable) function at a point: a subgradient g at x satisfies f(y) ≥ f(x) + g·(y−x) for all y in the domain; the set of all such g is the subdifferential.

 

 

 

 

 

 





## Principle

Principle

Subgradients define supporting hyperplanes to the epigraph of a convex function and provide first-order optimality conditions and descent directions when classical derivatives do not exist.

 

 

 

 

 





## Demonstration

Demonstration

For f(t)=|t| at t=0 the subdifferential is the interval [−1,1]; any g in [−1,1] satisfies |y| ≥ 0 + g·(y−0) for all y, illustrating nondifferentiability handled by a set of valid subgradients.

 

 

 

 

## Misapplication

Misapplication

Using subgradient-based algorithms without convexity assumptions or interpreting any selection of a one-sided slope at a nondifferentiable point as a valid subgradient for nonconvex functions leads to incorrect descent claims.

 

 

 

 

 





## Consequence

Consequence

When correctly applied to convex optimization, subgradients yield necessary and sufficient optimality criteria (0 in subdifferential) and support iterative methods (subgradient descent) that converge under proper step-size rules.

 

 

 

 

## Reversal

Reversal

The classical gradient is the unique subgradient at a point when the function is differentiable there; conversely, a singleton subdifferential implies differentiability of a convex function at that point.

 

 

 

 

 





## Boundary

Boundary

Primarily formulated for convex functions on convex domains; generalized notions (Clarke, limiting subgradients) extend to certain nonconvex settings but have different calculus rules and interpretations.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between the convex subgradient and generalized gradients for nonconvex functions (Clarke, Mordukhovich): they agree on convex cases but differ in set structure and calculus properties in nonconvex analysis.

 

 

 

 

 





## Synthesis

Synthesis

A subgradient is a supporting vector that replaces the gradient for convex nondifferentiable functions, forming a subdifferential set that furnishes optimality conditions and first-order methods for nonsmooth convex optimization.