Goldstein Subdifferential in Nonconvex Optimization

• Goldstein subdifferential is a neighborhood-enlarged generalized gradient for locally Lipschitz nonconvex functions, integrating Clarke subgradients over an ε-ball.
• It enables approximate stationarity and descent by using the convex hull of nearby gradients to reveal otherwise hidden descent directions.
• The formulation underpins robust optimization methods including gradient sampling, control, and stochastic optimization, offering quantifiable convergence guarantees.

The Goldstein subdifferential is a neighborhood-enlarged generalized gradient for locally Lipschitz, generally nonconvex functions. In the recent nonsmooth-optimization literature it is defined by replacing the pointwise Clarke subdifferential at $x$ with the convexified collection of Clarke subgradients from an $\varepsilon$-ball around $x$, thereby encoding nearby nonsmooth geometry rather than only first-order information at a single point. This enlargement is used to define approximate stationarity, to construct certified descent directions, and to analyze algorithmic convergence in settings ranging from gradient sampling and line-search methods to direct policy search and projected stochastic optimization (Maleknia et al., 2023).

1. Definition and basic construction

For a locally Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$, the underlying pointwise object is the Clarke subdifferential

$$\partial f(\mathbf x) := \operatorname{conv}\left\{\boldsymbol{\xi}\in\mathbb{R}^n:\exists\, \mathbf x_i\notin\Omega_f,\ \mathbf x_i\to \mathbf x,\ \nabla f(\mathbf x_i)\to \boldsymbol{\xi}\right\}.$$

The Goldstein $\varepsilon$-subdifferential is then defined by

$$\partial_\varepsilon f(\mathbf x) := \operatorname{cl\,conv}\{\partial f(\mathbf y): \mathbf y\in \mathcal B(\mathbf x,\varepsilon)\},$$

that is, the closed convex hull of all Clarke subdifferentials in the $\varepsilon$-ball around $\mathbf x$ (Maleknia et al., 2023). A closely related formulation used elsewhere is

$$\partial_{\varepsilon} f(x) := \operatorname{conv}\!\left(\bigcup_{y\in B_{\varepsilon}(x)} \partial f(y)\right), \qquad B_{\varepsilon}(x)=\{y:\|y-x\|\le \varepsilon\},$$

which emphasizes convexification over nearby points (Gebken, 2024).

Several elementary properties recur throughout the literature. The set $\varepsilon$0 is nonempty, convex, and compact; it satisfies $\varepsilon$1; and it is upper semicontinuous as a set-valued map in the formulations cited above (Maleknia et al., 2023). These properties explain why the construction is attractive algorithmically: it retains convexity and compactness while weakening exact pointwise stationarity.

The main motivation is geometric. In nonsmooth regions, and even at points where the function happens to be smooth, the Clarke subdifferential at the current iterate may be too local to reveal a useful descent direction. Enlarging the subdifferential to a neighborhood allows nearby kinks and active pieces to influence the descent model. This suggests that the Goldstein construction is best viewed as a local aggregation device: it regularizes nonsmooth first-order information without passing to a globally smoothed objective.

2. Stationarity and descent geometry

The Goldstein subdifferential is used primarily through its minimum-norm element. A standard descent model is

$\varepsilon$2

with normalized direction $\varepsilon$3 when $\varepsilon$4 (Maleknia et al., 2023). In this sense, Goldstein descent replaces “take a subgradient” with “project the origin onto a convexified neighborhood-gradient cloud.”

Approximate stationarity is then measured by the distance from $\varepsilon$5 to $\varepsilon$6. One formulation defines $\varepsilon$7-criticality by

$\varepsilon$8

meaning that there exists $\varepsilon$9 with $x$0 (Gebken, 2024). In robust control, an analogous notion is $x$1-stationarity: $x$2 for the policy-search objective $x$3 (Guo et al., 2022). Although the parameter order varies across papers, the principle is the same: Goldstein stationarity is approximate first-order criticality relative to a neighborhood radius and a norm tolerance.

The geometry of the minimum-norm element yields explicit descent inequalities. If $x$4 is the projection of $x$5 onto $x$6, then either $x$7 or

$x$8

which implies

$x$9

(Gebken, 2024). A control-theoretic variant states that if $f:\mathbb{R}^n\to\mathbb{R}$0 is the minimum-norm element of $f:\mathbb{R}^n\to\mathbb{R}$1, and the segment from $f:\mathbb{R}^n\to\mathbb{R}$2 in direction $f:\mathbb{R}^n\to\mathbb{R}$3 of length up to $f:\mathbb{R}^n\to\mathbb{R}$4 stays in the feasible set, then

$f:\mathbb{R}^n\to\mathbb{R}$5

(Guo et al., 2022). These estimates are the core reason Goldstein subgradients are useful in nonconvex nonsmooth descent: they produce quantitative decrease from an approximate generalized gradient.

3. Algorithmic approximation and implementable methods

In practice, $f:\mathbb{R}^n\to\mathbb{R}$6 is rarely available in closed form. Recent descent methods therefore build finite inner approximations from sampled subgradients. For a locally Lipschitz objective, one paper maintains a working set

$f:\mathbb{R}^n\to\mathbb{R}$7

uses $f:\mathbb{R}^n\to\mathbb{R}$8 as an inner approximation of the Goldstein subdifferential, and computes the search direction by solving

$f:\mathbb{R}^n\to\mathbb{R}$9

(Maleknia et al., 2023). If the resulting direction fails to give sufficient decrease, the approximation is refined by adding a new subgradient outside the current convex hull. A lemma used in that scheme states that if

$$\partial f(\mathbf x) := \operatorname{conv}\left\{\boldsymbol{\xi}\in\mathbb{R}^n:\exists\, \mathbf x_i\notin\Omega_f,\ \mathbf x_i\to \mathbf x,\ \nabla f(\mathbf x_i)\to \boldsymbol{\xi}\right\}.$$0

then $$\partial f(\mathbf x) := \operatorname{conv}\left\{\boldsymbol{\xi}\in\mathbb{R}^n:\exists\, \mathbf x_i\notin\Omega_f,\ \mathbf x_i\to \mathbf x,\ \nabla f(\mathbf x_i)\to \boldsymbol{\xi}\right\}.$$1, so the working model becomes strictly richer (Maleknia et al., 2023).

The corresponding line-search mechanism is a two-point variant of Mifflin’s line search, denoted T-PLS, with two trial step lengths and three tests: a decrease test inside the $$\partial f(\mathbf x) := \operatorname{conv}\left\{\boldsymbol{\xi}\in\mathbb{R}^n:\exists\, \mathbf x_i\notin\Omega_f,\ \mathbf x_i\to \mathbf x,\ \nabla f(\mathbf x_i)\to \boldsymbol{\xi}\right\}.$$2-ball, a second decrease test for an actual descent step, and a subgradient test that returns a new Goldstein subgradient when a sampled Clarke subgradient is sufficiently aligned with the current direction. The paper emphasizes that these subgradients are arbitrary subgradients rather than quasi-secants, and proves finite termination under weak upper semismoothness (Maleknia et al., 2023).

A multiobjective extension uses the same architecture objective-wise. For

$$\partial f(\mathbf x) := \operatorname{conv}\left\{\boldsymbol{\xi}\in\mathbb{R}^n:\exists\, \mathbf x_i\notin\Omega_f,\ \mathbf x_i\to \mathbf x,\ \nabla f(\mathbf x_i)\to \boldsymbol{\xi}\right\}.$$3

finite sets $$\partial f(\mathbf x) := \operatorname{conv}\left\{\boldsymbol{\xi}\in\mathbb{R}^n:\exists\, \mathbf x_i\notin\Omega_f,\ \mathbf x_i\to \mathbf x,\ \nabla f(\mathbf x_i)\to \boldsymbol{\xi}\right\}.$$4 are maintained for each objective, and the search direction is generated from

$$\partial f(\mathbf x) := \operatorname{conv}\left\{\boldsymbol{\xi}\in\mathbb{R}^n:\exists\, \mathbf x_i\notin\Omega_f,\ \mathbf x_i\to \mathbf x,\ \nabla f(\mathbf x_i)\to \boldsymbol{\xi}\right\}.$$5

A new Mifflin-style “Finding an Effective Subgradient” routine enriches only those objectives whose current approximations are inadequate; under weak upper semismoothness, any accumulation point of the outer scheme is Clarke substationary: $$\partial f(\mathbf x) := \operatorname{conv}\left\{\boldsymbol{\xi}\in\mathbb{R}^n:\exists\, \mathbf x_i\notin\Omega_f,\ \mathbf x_i\to \mathbf x,\ \nabla f(\mathbf x_i)\to \boldsymbol{\xi}\right\}.$$6 (Maleknia et al., 2024).

Gradient-free methods use a different implementability route. For Lipschitz $$\partial f(\mathbf x) := \operatorname{conv}\left\{\boldsymbol{\xi}\in\mathbb{R}^n:\exists\, \mathbf x_i\notin\Omega_f,\ \mathbf x_i\to \mathbf x,\ \nabla f(\mathbf x_i)\to \boldsymbol{\xi}\right\}.$$7, a deterministic GFM and stochastic SGFM optimize via two-point zeroth-order estimators and aim directly at $$\partial f(\mathbf x) := \operatorname{conv}\left\{\boldsymbol{\xi}\in\mathbb{R}^n:\exists\, \mathbf x_i\notin\Omega_f,\ \mathbf x_i\to \mathbf x,\ \nabla f(\mathbf x_i)\to \boldsymbol{\xi}\right\}.$$8-Goldstein stationary points. The target criterion is

$$\partial f(\mathbf x) := \operatorname{conv}\left\{\boldsymbol{\xi}\in\mathbb{R}^n:\exists\, \mathbf x_i\notin\Omega_f,\ \mathbf x_i\to \mathbf x,\ \nabla f(\mathbf x_i)\to \boldsymbol{\xi}\right\}.$$9

and the expected complexity stated in the abstract is

$\varepsilon$0

for both deterministic and stochastic settings (Lin et al., 2022).

4. Smoothing, quantitative convergence, and the Goldstein modulus

A notable theoretical link connects Goldstein subdifferentials to uniform smoothing. For the smoothed function

$\varepsilon$1

where $\varepsilon$2 is the uniform distribution on the unit Euclidean ball, one paper proves

$\varepsilon$3

This result gives a direct bridge between zeroth-order optimization of a smoothed objective and Goldstein stationarity of the original nonsmooth objective (Lin et al., 2022). A plausible implication is that Goldstein subdifferentials furnish a first-order interpretation of ball smoothing rather than merely an auxiliary stationarity notion.

Recent work also analyzes how approximate Goldstein criticality controls actual convergence to a minimizer. If $\varepsilon$4 is a minimum of order $\varepsilon$5, meaning

$\varepsilon$6

and suitable higher-order semismoothness holds, then a sequence of $\varepsilon$7-critical points satisfies explicit distance estimates. For $\varepsilon$8, if $\varepsilon$9, then eventually

$$\partial_\varepsilon f(\mathbf x) := \operatorname{cl\,conv}\{\partial f(\mathbf y): \mathbf y\in \mathcal B(\mathbf x,\varepsilon)\},$$0

For $$\partial_\varepsilon f(\mathbf x) := \operatorname{cl\,conv}\{\partial f(\mathbf y): \mathbf y\in \mathcal B(\mathbf x,\varepsilon)\},$$1,

$$\partial_\varepsilon f(\mathbf x) := \operatorname{cl\,conv}\{\partial f(\mathbf y): \mathbf y\in \mathcal B(\mathbf x,\varepsilon)\},$$2

(Gebken, 2024). This shows that vanishing neighborhood radius and vanishing Goldstein-subgradient norm can be converted into a quantitative rate to the minimizer under local growth assumptions.

A complementary line of work introduces the Goldstein modulus

$$\partial_\varepsilon f(\mathbf x) := \operatorname{cl\,conv}\{\partial f(\mathbf y): \mathbf y\in \mathcal B(\mathbf x,\varepsilon)\},$$3

where $$\partial_\varepsilon f(\mathbf x) := \operatorname{cl\,conv}\{\partial f(\mathbf y): \mathbf y\in \mathcal B(\mathbf x,\varepsilon)\},$$4 is the shortest vector in $$\partial_\varepsilon f(\mathbf x) := \operatorname{cl\,conv}\{\partial f(\mathbf y): \mathbf y\in \mathcal B(\mathbf x,\varepsilon)\},$$5 (Kong et al., 2024). The modulus satisfies $$\partial_\varepsilon f(\mathbf x) := \operatorname{cl\,conv}\{\partial f(\mathbf y): \mathbf y\in \mathcal B(\mathbf x,\varepsilon)\},$$6 for an $$\partial_\varepsilon f(\mathbf x) := \operatorname{cl\,conv}\{\partial f(\mathbf y): \mathbf y\in \mathcal B(\mathbf x,\varepsilon)\},$$7-Lipschitz objective and $$\partial_\varepsilon f(\mathbf x) := \operatorname{cl\,conv}\{\partial f(\mathbf y): \mathbf y\in \mathcal B(\mathbf x,\varepsilon)\},$$8 if and only if $$\partial_\varepsilon f(\mathbf x) := \operatorname{cl\,conv}\{\partial f(\mathbf y): \mathbf y\in \mathcal B(\mathbf x,\varepsilon)\},$$9 is Clarke critical. It also upper-bounds the distance to any Clarke critical point $\varepsilon$0: $\varepsilon$1 The same paper relates near-linear convergence of Goldstein-style methods to linear growth of this modulus near minimizers,

$\varepsilon$2

and proves that, under that condition, an idealized Goldstein descent method has objective gap

$\varepsilon$3

after $\varepsilon$4 subgradient calls (Kong et al., 2024). This suggests a quantitative interpretation of the Goldstein radius as a robust local slope scale.

5. Applications in control and constrained stochastic optimization

The Goldstein subdifferential has proved especially effective when nonsmooth stationarity interacts with additional structure. In direct policy search for discrete-time $\varepsilon$5 state-feedback synthesis, the objective

$\varepsilon$6

is continuous, nonconvex, locally Lipschitz, and subdifferentially regular on the open nonconvex set of stabilizing controllers. The paper defines

$\varepsilon$7

proves that every Clarke stationary point is a global minimum, and uses Goldstein’s subgradient method

$\varepsilon$8

to obtain global convergence provided the initialization is stabilizing (Guo et al., 2022). Choosing

$\varepsilon$9

keeps all iterates in the stabilizing set, makes $\mathbf x$0 monotonically decreasing, and yields $\mathbf x$1. For constant $\mathbf x$2, the paper also gives the finite-time bound

$\mathbf x$3

hence $\mathbf x$4-stationarity in

$\mathbf x$5

iterations (Guo et al., 2022).

A different application arises in projected SGD over a compact convex set $\mathbf x$6. There the Goldstein subdifferential is generated by the indicator of the feasible set: $\mathbf x$7 where $\mathbf x$8 is the normal cone (Zheng et al., 3 Oct 2025). Stationarity is measured by

$\mathbf x$9

which reduces to the standard gradient norm in the unconstrained case. The paper proves asymptotic convergence under Robbins–Monro stepsizes, $$\partial_{\varepsilon} f(x) := \operatorname{conv}\!\left(\bigcup_{y\in B_{\varepsilon}(x)} \partial f(y)\right), \qquad B_{\varepsilon}(x)=\{y:\|y-x\|\le \varepsilon\},$$0 non-asymptotic bounds in expectation, and, for IID sub-Gaussian data, $$\partial_{\varepsilon} f(x) := \operatorname{conv}\!\left(\bigcup_{y\in B_{\varepsilon}(x)} \partial f(y)\right), \qquad B_{\varepsilon}(x)=\{y:\|y-x\|\le \varepsilon\},$$1 high-probability bounds (Zheng et al., 3 Oct 2025). In this setting the Goldstein construction acts as a constraint-aware relaxation of the exact normal-cone condition for local minimizers.

6. Relation to other subdifferentials and common misconceptions

The Goldstein subdifferential should not be conflated with the classical convex subdifferential or with other generalized subdifferentials that appear in Banach-space variational analysis. A short note on maximal monotonicity studies only the standard convex-analytic subdifferential of a proper convex lower semicontinuous function,

$$\partial_{\varepsilon} f(x) := \operatorname{conv}\!\left(\bigcup_{y\in B_{\varepsilon}(x)} \partial f(y)\right), \qquad B_{\varepsilon}(x)=\{y:\|y-x\|\le \varepsilon\},$$2

and explicitly states that it does not discuss Goldstein’s subdifferential, Clarke subdifferentials, or other approximate nonsmooth constructions (Răşcanu, 2019). Likewise, the “Subdifferential Test for Optimality” develops an abstract Banach-space subdifferential satisfying the Separation Principle and notes that Goldstein’s subdifferential is not mentioned and is not verified to satisfy that framework (Jules et al., 2012).

A second source of confusion concerns convex $$\partial_{\varepsilon} f(x) := \operatorname{conv}\!\left(\bigcup_{y\in B_{\varepsilon}(x)} \partial f(y)\right), \qquad B_{\varepsilon}(x)=\{y:\|y-x\|\le \varepsilon\},$$3-subdifferential calculus. Papers on the subdifferential of pointwise suprema use Fenchel $$\partial_{\varepsilon} f(x) := \operatorname{conv}\!\left(\bigcup_{y\in B_{\varepsilon}(x)} \partial f(y)\right), \qquad B_{\varepsilon}(x)=\{y:\|y-x\|\le \varepsilon\},$$4-subdifferentials such as

$$\partial_{\varepsilon} f(x) := \operatorname{conv}\!\left(\bigcup_{y\in B_{\varepsilon}(x)} \partial f(y)\right), \qquad B_{\varepsilon}(x)=\{y:\|y-x\|\le \varepsilon\},$$5

together with active-index sets, compactification of the index set, and Valadier-type formulas for exact convex subdifferentials (Correa et al., 2017). Although these constructions are “Goldstein-adjacent” in the sense that they involve approximation, convexification, and limiting procedures, they remain firmly within convex analysis and do not define the Goldstein subdifferential itself (Correa et al., 2020).

A third distinction concerns second-order variational analysis. Work on quadratic growth and strong metric subregularity for non-prox-regular functions is formulated in terms of proximal, regular, and limiting/Mordukhovich subdifferentials, together with subgradient graphical derivatives and second subderivatives; it is conceptually relevant but not Goldstein-specific (Chieu et al., 2021). The same applies to BV-based geometric analyses of convex one-homogeneous functionals and to reconstruction of polyhedral regular subdifferentials from directional derivatives: those papers study exact subdifferentials, not neighborhood-enlarged Goldstein objects (Chambolle et al., 2012).

In this sense, the Goldstein subdifferential occupies a specific niche. It is a locally aggregated, compact-convex, optimization-oriented enlargement of the Clarke subdifferential, designed for locally Lipschitz nonsmooth problems where exact pointwise stationarity is either too brittle or too difficult to exploit algorithmically. Its recent literature is therefore concentrated not in maximal-monotonicity theory or classical convex calculus, but in descent methods, quantitative stationarity analysis, robust control, gradient sampling, and stochastic or zeroth-order nonsmooth optimization.