I tried to inspect the attached file getPDF.jsp(nonsmooth).pdf, but its text was not readable/searchable in the tool, so the summary below is based on standard theory and the provided literature sources.

1. Basic idea of nonsmooth analysis

• 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.

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

2. Clarke generalized directional derivative

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

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.
• It is positively homogeneous and subadditive in v in the standard Clarke theory.
• It is usually larger than, or equal to, the ordinary directional derivative when both are considered in comparable settings.
• It captures worst-case local directional growth near x.

3. Clarke subdifferential

The Clarke subdifferential is commonly defined by:

∂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.

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

∂C f(x) = {∇f(x)}.

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

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.

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.

Sum rule

∂C(f + g)(x) ⊂ ∂C f(x) + ∂C g(x).

Under additional regularity conditions, equality may hold.

Scalar multiplication

For scalar a,

∂C(a f)(x) = a ∂C f(x).

If a < 0, the set is reflected.

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.

Quotient rule

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

∂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:

∂C(φ ∘ F)(x) ⊂ DF(x)^T ∂C φ(F(x)).

If φ is regular in Clarke’s sense, stronger forms are available.

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:

∂C f(x) ⊂ co ⋃_{i in I(x)} ∂C fi(x).

If the fi are smooth, this becomes:

∂C f(x) ⊂ co { ∇fi(x): i in I(x) }.

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

Fermat rule for nonsmooth optimization

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

0 ∈ ∂C f(x).

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

5. Classical papers and books

• F. H. Clarke’s work on generalized gradients is a foundational source for Clarke generalized derivatives in nonsmooth analysis.

• Clarke generalized directional derivatives and generalized gradients are discussed in the finite-dimensional nonsmooth optimization literature, including work by Hiriart-Urruty.

• Rockafellar’s work on generalized subgradients in mathematical programming is another foundational line; the cited paper outlines fundamentals of generalized directional derivatives and subgradients.

• Hiriart-Urruty’s finite-dimensional work discusses Clarke directional derivatives, Clarke generalized gradients, calculus rules, and applications to nonsmooth optimization.

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.

• 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.

• Recent Rockafellar publications indicate continuing work in set-valued and nonsmooth analysis, including 2025 work connected with the calculus of variations.

7. Good reading path

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

A practical first reading list would be:

• Clarke’s work on generalized gradients and Clarke generalized derivatives.
• Hiriart-Urruty’s work on generalized derivatives and nonsmooth optimization.
• Rockafellar’s “Generalized Subgradients in Mathematical Programming.”
• Recent papers using Goldstein or Clarke-type subdifferentials in nonsmooth optimization algorithms and optimality conditions.

I tried to inspect the attached file getPDF.jsp(nonsmooth).pdf, but its text was not readable/searchable in the tool, so the summary below is based on standard theory and the provided literature sources.

1. Basic idea of nonsmooth analysis

• 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.

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

2. Clarke generalized directional derivative

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

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.
• It is positively homogeneous and subadditive in v in the standard Clarke theory.
• It is usually larger than, or equal to, the ordinary directional derivative when both are considered in comparable settings.
• It captures worst-case local directional growth near x.

3. Clarke subdifferential

The Clarke subdifferential is commonly defined by:

∂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.

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

∂C f(x) = {∇f(x)}.

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

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.

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.

Sum rule

∂C(f + g)(x) ⊂ ∂C f(x) + ∂C g(x).

Under additional regularity conditions, equality may hold.

Scalar multiplication

For scalar a,

∂C(a f)(x) = a ∂C f(x).

If a < 0, the set is reflected.

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.

Quotient rule

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

∂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:

∂C(φ ∘ F)(x) ⊂ DF(x)^T ∂C φ(F(x)).

If φ is regular in Clarke’s sense, stronger forms are available.

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:

∂C f(x) ⊂ co ⋃_{i in I(x)} ∂C fi(x).

If the fi are smooth, this becomes:

∂C f(x) ⊂ co { ∇fi(x): i in I(x) }.

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

Fermat rule for nonsmooth optimization

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

0 ∈ ∂C f(x).

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

5. Classical papers and books

• F. H. Clarke’s work on generalized gradients is a foundational source for Clarke generalized derivatives in nonsmooth analysis.

• Clarke generalized directional derivatives and generalized gradients are discussed in the finite-dimensional nonsmooth optimization literature, including work by Hiriart-Urruty.

• Rockafellar’s work on generalized subgradients in mathematical programming is another foundational line; the cited paper outlines fundamentals of generalized directional derivatives and subgradients.

• Hiriart-Urruty’s finite-dimensional work discusses Clarke directional derivatives, Clarke generalized gradients, calculus rules, and applications to nonsmooth optimization.

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.

• 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.

• Recent Rockafellar publications indicate continuing work in set-valued and nonsmooth analysis, including 2025 work connected with the calculus of variations.

7. Good reading path

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

A practical first reading list would be:

• Clarke’s work on generalized gradients and Clarke generalized derivatives.
• Hiriart-Urruty’s work on generalized derivatives and nonsmooth optimization.
• Rockafellar’s “Generalized Subgradients in Mathematical Programming.”
• Recent papers using Goldstein or Clarke-type subdifferentials in nonsmooth optimization algorithms and optimality conditions.