Subgradient Growth Condition

• Subgradient Growth Condition is a property defining how the objective function grows near a minimizer by linking value gaps to the subdifferential structure.
• It encompasses forms like uniform quadratic growth, tilt stability, and strong metric regularity, which are critical for guaranteeing robust local minimality.
• By establishing second-order error bounds and stability measures, this condition directly informs the design and convergence analysis of nonsmooth optimization algorithms.

The subgradient growth condition is a central analytical property in nonsmooth optimization that quantifies the local or global relationship between the function value gap and the geometry of the subdifferential mapping. It formalizes how the objective function "grows" away from a minimizer through second-order behavior or error bounds involving subgradients, providing a bridge between stability, local regularity, and algorithmic convergence rates.

1. Definitions and Foundational Frameworks

The subgradient growth condition manifests in several closely related forms, each capturing a different aspect of how the local geometry of a function f: ℝⁿ → ℝ ∪ {+∞} controls the function’s behavior near a minimizer x̄:

• Uniform Quadratic Growth:

There exists κ > 0 and neighborhood U of x̄ such that

$$f(x) \geq f(x̄) + κ \|x - x̄\|^2, \quad \forall x \in U$$

Equivalently, this asserts a strong local minimum in the sense of second-order variational analysis (Drusvyatskiy et al., 2012).

• Tilt Stability:

The perturbed minimization mapping

$$M(v) = \operatorname{argmin}_{\|x - x̄\| \leq ε} \{ f(x) - \langle v, x\rangle \}$$

is single-valued and Lipschitz in v near v = 0. Tilt stability ensures that the minimizers respond in a stable (Lipschitz) way to linear perturbations, unifying notions of strong minimizers with subdifferential regularity (Drusvyatskiy et al., 2012).

• Strong Metric Regularity of the Limiting Subdifferential:

The limiting subdifferential ∂f is said to be strongly metrically regular at (x̄, 0) if, locally, there is a unique solution x(v) to v ∈ ∂f(x), and x(·) is Lipschitz in v. This ensures locally robust invertibility and Lipschitz continuity of the inverse subdifferential mapping (Drusvyatskiy et al., 2012).

These properties link to classical and extended second-order concepts through the quadratic growth condition and its generalizations across both smooth and nonsmooth analysis.

2. Equivalence of Subgradient Growth, Tilt Stability, and Metric Regularity

The foundational result, formalized in [(Drusvyatskiy et al., 2012), Theorem 3.5], is the equivalence of:

1. Uniform quadratic growth (stable strong local minimality).
2. Tilt stability.
3. Strong metric regularity of the limiting subdifferential.

The main technical route involves:

• Reduction to Convexity:

Using local convexification techniques, the nonsmooth function is approximated by a convexified envelope with matching subdifferential structure locally.

• Linking Quadratic Growth to Metric Regularity:

For convex functions, strong metric regularity of the subdifferential ([∂f]⁻¹ is single-valued and Lipschitz) is equivalent to the quadratic growth property

$$f(x) \geq f(x̄) + \langle v, x-x̄\rangle + κ\|x-x̄\|^2, \quad \forall x \in U$$

for some v ∈ ∂f(x̄) (Drusvyatskiy et al., 2012, Bauschke et al., 2014).

• Implications for Nonsmooth Analysis:

With prox-regularity and subdifferential continuity, the same regularity properties (growth, stability, and invertibility) transfer to generally nonsmooth, lower-semicontinuous functions (Drusvyatskiy et al., 2012).

The table below summarizes the equivalence under standard regularity:

Property	Equivalent Condition (under regularity)	Main Paper
Uniform Quadratic Growth	f(x) ≥ f(x̄) + κ ‖x−x̄‖²	(Drusvyatskiy et al., 2012)
Tilt Stability at x̄	M(v) Lipschitz, single-valued near v=0	(Drusvyatskiy et al., 2012)
Strong Metric Regularity of ∂f at (x̄, 0)	(∂f)⁻¹ Lipschitz, single-valued near 0	(Drusvyatskiy et al., 2012)

These equivalences provide a rigorous theoretical bridge between local growth properties and the analytic structure of the subdifferential, crucial for both sensitivity analysis and numerical algorithms.

3. The Subgradient Graphical Derivative and Second-Order Sufficient/Necessary Conditions

The subgradient graphical derivative provides a variational-analytic generalization of the Hessian in nonsmooth settings (Chieu et al., 2019):

• Definition:

For a function f at (x, v), the graphical derivative of ∂f is

$$D(\partial f)(x|v)(w) = \{ z \in ℝ^n \mid (w, z) \in T_{\mathrm{gph}∂f}(x, v) \}$$

This set-valued map generalizes the action of ∇²f(x)w for smooth f.

• Positive Definiteness:

Uniform quadratic growth is equivalent (under subdifferential continuity, prox-regularity, or twice epi-differentiability) to the positive definiteness of D(∂f)(x̄|0):

$$\forall\, w \neq 0,\, z \in D(\partial f)(x̄|0)(w) \implies \langle z, w\rangle \geq c\|w\|^2$$

This second-order condition is both sufficient and necessary for quadratic growth and thus for strong local minimality (Chieu et al., 2019).

• Metric Subregularity Connection:

Strong metric subregularity of ∂f is equivalent to positive definiteness of D(∂f)(x̄|0), further reinforcing the connection between second-order geometry and invertibility properties of the generalized gradient (Chieu et al., 2019, Chieu et al., 2021).

4. Algorithmic and Practical Implications

Subgradient growth conditions are directly linked with the local convergence rates and robustness of first-order optimization algorithms:

• Quadratic Growth and Local Convergence:

Strong local quadratic growth guarantees local linear or quadratic convergence rates for subgradient, bundle, and majorization-minimization algorithms (Drusvyatskiy et al., 2012, Yang et al., 2015).

• Bundle Methods and Error Bounds:

Growth conditions (including Hölderian and sharp error bounds) inform step size policies and restarting schedules, yielding accelerated convergence and iteration complexity improvements (Johnstone et al., 2017, Freund et al., 2015, Yang et al., 2015).

• Sensitivity and Stability:

The equivalence to strong metric regularity implies that minimizers and KKT points respond stably (Lipschitz) to perturbations in problem data; thus, the subgradient growth condition underpins robust parametric sensitivity (Drusvyatskiy et al., 2012, Hang et al., 2023).

• Breakdown Without Regularity:

Violations of prox-regularity or subdifferential continuity (for instance, in

$$f(x) = \begin{cases} 1 + x^4, & x < 0 \ x^2, & x \geq 0 \end{cases}$$

) can lead to failure of these equivalence results, highlighting the necessity of the stated regularity prerequisites (Drusvyatskiy et al., 2012).

5. Error Bound Conditions and Extensions

The subgradient growth condition is naturally related to more general error bound properties:

• Growth Error Bounds:

Inequalities of the form

$d(x, S) \leq \varphi(f(x) - f(x̄))$

Where S is the set of minimizers, and φ is a desingularizing function, generalize the subgradient growth condition to a broader class of convex and nonconvex functions (Jin, 2023).

• Kurdyka–Łojasiewicz (KL) Inequality:

The KL condition,

$φ'(f(x) - f(x̄))⋅d(0, ∂f(x)) ≥ 1$

is a generalized subgradient growth property, ensuring that function value gaps force the subdifferential norm to be large. In convex cases, KL and growth error bounds are essentially equivalent (Jin, 2023).

• Sufficient/Necessary Conditions for Linear Convergence:

The positive definiteness of the subgradient graphical derivative and strong metric (sub)regularity conditions guarantee linear rates when plugged into algorithmic frameworks (Chieu et al., 2019, Chieu et al., 2021).

6. Applications in Control, Variational Inequalities, and Machine Learning

Subgradient growth and its variants appear broadly:

• Optimization Stability:

Error bounds and growth conditions play a crucial role in stability and perturbation analysis, especially in variational inequalities and optimal control.

• Iterative Optimization:

Ensuring a subgradient growth property (or verifying quadratic growth) is essential for guaranteeing practical performance and rate certificates in modern nonsmooth optimization algorithms, especially those based on projection or bundle methods (Bauschke et al., 2014, Freund et al., 2015).

• Robust Recovery and Learning:

In problems such as robust low-rank recovery and deep learning with nonsmooth activations, subgradient growth conditions underpin the avoidance of trap points and spurious minima (Josz et al., 2022, Ding et al., 2021).

7. Limitations, Regularity Assumptions, and Counterexamples

The equivalence among subgradient growth, tilt stability, and strong metric regularity is contingent upon strong regularity assumptions, notably:

• Prox-Regularity: The function must not deviate sharply from a smooth/convex template.
• Subdifferential Continuity: The epigraphical geometry must not contain "kinks" that disconnect subgradient behavior from local minimizers.

Counterexamples demonstrate sharp failure of the theory when these constraints are not met, emphasizing the necessity for careful verification in any application (Drusvyatskiy et al., 2012).

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In conclusion, the subgradient growth condition provides a unified geometric and variational analytic framework for understanding local (and global) stability, convergence, and robustness in nonsmooth and variational optimization. Its deep equivalence with tilt stability and metric regularity underpins much of modern variational analysis, algorithm design, and sensitivity theory, binding together function growth, second-order structure, and solution mapping stability.