What Is Nonsmooth Analysis? Understanding Clarke's Subdifferential and Its Calculus Rules

• Classical calculus works well for smooth functions, but many important functions are nonsmooth, such as:

  • f(x)=|x|
  • f(x)=max_i f_i(x)
  • distance functions
  • value functions in optimization/control
  • ReLU-type piecewise-linear losses

• Instead of asking for one derivative, nonsmooth analysis asks for a set of generalized derivatives.^EE

• For a locally Lipschitz function f: R^n -> R, Clarke introduced generalized directional derivatives and generalized gradients as tools for nonsmooth optimization.^LE

2. Clarke Generalized Directional Derivative

For locally Lipschitz f, the Clarke directional derivative at x in direction v is commonly written as:^E

f°(x; v) = limsup_{y -> x, t ↓ 0} [f(y + t v) - f(y)] / t.

Key points:

• It is designed for locally Lipschitz nonsmooth functions.^EE
• It is positively homogeneous and subadditive in v in the standard Clarke theory.^LE
• It is usually larger than, or equal to, the ordinary directional derivative when both are considered in comparable settings.^E
• It captures worst-case local directional growth near x.^EE

3. The Clarke Subdifferential

The Clarke subdifferential is commonly defined by:^E

∂C f(x) = { ξ in R^n: f°(x; v) >= <ξ, v> for all v in R^n }.

An equivalent standard description for locally Lipschitz f is that the Clarke subdifferential is the convex hull of limits of nearby classical gradients where the function is differentiable.^EE

∂C f(x) = co { limits of ∇f(x_k): x_k -> x, f differentiable at x_k }.

Here co means convex hull.

Key properties:

• ∂C f(x) is a set-valued generalized derivative for locally Lipschitz functions.^EE
• In the standard finite-dimensional Clarke theory, ∂C f(x) is nonempty, compact, and convex for locally Lipschitz f.^EE
• If f is continuously differentiable near x, then the Clarke subdifferential reduces to the ordinary gradient.^EE

∂C f(x) = {∇f(x)}.

• If f is convex, the Clarke subdifferential agrees with the usual convex-analysis subdifferential in standard settings.^LE

Example:

f(x) = |x|.

Then

∂C f(x) =
  {-1},        x < 0
  [-1, 1],     x = 0
  {1},         x > 0.

This is the standard basic example, showing how a corner is represented by a whole interval of possible slopes.^EE

4. Calculus Rules for the Clarke Subdifferential

Let f, g: R^n -> R be locally Lipschitz near x; Clarke calculus rules are a central part of nonsmooth optimization theory.^LEE

Sum Rule

∂C(f + g)(x) ⊂ ∂C f(x) + ∂C g(x).

Under additional regularity conditions, equality may hold.^LE

Scalar Multiplication

For scalar a,

∂C(a f)(x) = a ∂C f(x).

If a < 0, the set is reflected.^E

Product Rule

∂C(fg)(x) ⊂ f(x) ∂C g(x) + g(x) ∂C f(x).

This is one of the standard inclusion-type calculus rules in Clarke subdifferential calculus.^LE

Quotient Rule

If g(x) ≠ 0 and g is bounded away from zero near x, the standard quotient rule has the form:^E

∂C(f/g)(x) ⊂ [g(x) ∂C f(x) - f(x) ∂C g(x)] / g(x)^2.

Chain Rule

If F: R^n -> R^m is strictly differentiable at x and φ: R^m -> R is locally Lipschitz near F(x), the Clarke chain rule is commonly stated as an inclusion of the following type:^LE

∂C(φ ∘ F)(x) ⊂ DF(x)^T ∂C φ(F(x)).

If φ is regular in Clarke's sense, stronger forms are available.^LE

Max Rule

If

f(x) = max { f1(x),..., fm(x) },

where each fi is locally Lipschitz, define the active index set

I(x) = { i: fi(x) = f(x) }.

Then the standard Clarke max rule gives an inclusion of the following form:^LE

∂C f(x) ⊂ co ⋃_{i in I(x)} ∂C fi(x).

If the fi are smooth, this becomes:^E

∂C f(x) ⊂ co { ∇fi(x): i in I(x) }.

For many standard max-functions, equality holds under suitable regularity assumptions.^LE

Fermat Rule for Nonsmooth Optimization

If x is a local minimizer of locally Lipschitz f, then the nonsmooth Fermat condition is:^LE

0 ∈ ∂C f(x).

This is the nonsmooth analogue of ∇f(x)=0.^LE

5. Classical Papers and Books

• F. H. Clarke's work on generalized gradients is a foundational source for Clarke generalized derivatives in nonsmooth analysis.^E
• Clarke generalized directional derivatives and generalized gradients are discussed in the finite-dimensional nonsmooth optimization literature, including work by Hiriart-Urruty.^L
• Rockafellar's work on generalized subgradients in mathematical programming is another foundational line; the cited paper outlines fundamentals of generalized directional derivatives and subgradients.^S
• Hiriart-Urruty's finite-dimensional work discusses Clarke directional derivatives, Clarke generalized gradients, calculus rules, and applications to nonsmooth optimization.^L

6. Recent Literature Directions

• Recent nonsmooth optimization research continues to use Clarke-type objects and related relaxations; for example, a 2025 paper studies convergence speed using the Goldstein subdifferential, described as a relaxed version of the Clarke subdifferential used in several algorithms.^L
• Clarke generalized directional derivatives continue to appear in optimality conditions for set optimization problems; a 2025 paper studies approximate weak minimal solutions using a new Clarke-type generalized derivative.^A
• Recent Rockafellar publications indicate continuing work in set-valued and nonsmooth analysis, including 2025 work connected with the calculus of variations.^S

7. Good Reading Path

1. Start with Clarke's generalized directional derivative and subdifferential.^LEE
2. Learn the core calculus rules: sum, product, chain, max, and Fermat rules.^LE
3. Study convex subdifferentials and compare them with Clarke subdifferentials.^LS
4. Move to variational analysis: normal cones, coderivatives, and broader generalized differentiation frameworks.^SS
5. Read recent optimization papers using Clarke or Goldstein subdifferentials in algorithms.^L

A practical first reading list would be:

• Clarke's work on generalized gradients and Clarke generalized derivatives.^E
• Hiriart-Urruty's work on generalized derivatives and nonsmooth optimization.^L
• Rockafellar's "Generalized Subgradients in Mathematical Programming."^S
• Recent papers using Goldstein or Clarke-type subdifferentials in nonsmooth optimization algorithms and optimality conditions.^LA