Two-Sided Quadratic Functional Growth

• Two-Sided QFG is a property that defines both lower and upper quadratic bounds for functionals relative to the distance from optimal solutions, ensuring stability in various mathematical models.
• It connects functional growth with metric subregularity and error bounds, serving as a first-order certificate for the robustness of convergence in optimization and regularity analysis.
• QFG underlies advanced algorithms by enabling linear convergence rates and robust performance in convex, nonconvex, saddle point, and stochastic systems.

Two-Sided Quadratic Functional Growth (QFG) is a property describing how functionals, generators, or solution gaps exhibit at least quadratic behavior (from both upper and lower directions) relative to their arguments or the distance to the solution set. This concept generalizes classical quadratic growth and error bounds in optimization, stochastic analysis, and PDEs, providing stability guarantees, convergence rates, and well-posedness results under relaxed conditions. QFG is instrumental in modern convex, nonconvex, and saddle point optimization, stochastic differential equations, empirical process theory, and regularity analysis.

1. Definition and Mathematical Formulation

The core of QFG is the requirement that the deviation of a function or an optimality metric is bounded below (and sometimes also above) by a quadratic function of the distance between the point under consideration and the set of solutions. For minimization problems, the standard QFG condition is

$g(x) - g^* \geq \kappa_g D_X(x, P_{X^*}(x))$

where $g$ is the objective function, $g^*$ its minimum, $P_{X^*}(x)$ the projection of $x$ onto the minimizer set $X^*$, and $D_X(\cdot,\cdot)$ is a (possibly Bregman) distance metric. For saddle point problems, QFG is two-sided, involving both primal and dual variables:

$$f(x, \bar{y}) - f(\bar{x}, y) \geq D_{Z}^M(z, \bar{z})$$

where $z = [x^\top, y^\top]^\top$, $\bar{z} = P_{Z^*}(z)$ (projection onto solution set $Z^*$), and $M$ is a diagonal matrix of positive weights. This inequality ensures the "functional gap" grows at least quadratically with respect to the proximity to $Z^*$. Two-sided means both directions are controlled, guaranteeing robustness and stability in practice (Melcher et al., 13 Oct 2025).

Quadratic growth conditions also appear in PDE and stochastic analysis, e.g., in 2BSDEs with generators satisfying

$$F_t(\omega, y, z, a) \leq a + B|y| + C|a^{1/2} z|^2$$

and in matrix optimization with spectral functions via

$f(X) \geq f(X^*) + (\kappa/2)\|X - X^*\|^2$

for $X$ in a neighborhood of $X^*$ (Cui et al., 2017).

2. Stability, Subdifferential, and Metric Subregularity Connections

QFG is intimately linked to critical point stability and error bounds, especially via the metric subregularity of the subdifferential mapping. For lsc semi-algebraic functions, quadratic functional growth

$$f(x) \geq f(\bar{x}) + \frac{\alpha}{2}\|x-\bar{x}\|^2 \quad \forall x \text{ near } \bar{x}$$

is equivalent to strong metric subregularity:

$$\|x - \bar{x}\| \leq \kappa\, d(0; \partial f(x)) \quad \forall x \text{ near } \bar{x}$$

where $d(0; \partial f(x))$ denotes the minimal norm from $0$ to the subdifferential $\partial f(x)$ (Drusvyatskiy et al., 2013). For convex matrix optimization, metric subregularity of the subdifferential similarly underpins QFG in spaces of matrices (Cui et al., 2017). This equivalence provides a "first-order certificate" for quadratic growth, allowing stability and convergence analysis without resorting to explicit second-order derivatives.

3. QFG in Stochastic Differential Equations, BSDEs, and PDEs

QFG arises naturally in stochastic differential equations with generators that grow quadratically in their control components. In second-order BSDEs (2BSDEs), generators exhibit two-sided quadratic growth in $z$, necessitating analysis in BMO spaces and energy inequalities to establish well-posedness. The representation formula for the solution is given by a supremum over standard BSDEs:

$$Y_{t_1} = \operatorname{ess}\sup_{P' \in \mathcal{P}_H(t_1+, P)} y^{P'}_{t_1}(t_2, Y_{t_2})$$

ensuring uniqueness and enabling stochastic control theory connections (Possamaï et al., 2012).

In backward stochastic partial differential equations (BSPDEs), QFG controls both the gradient (spatial derivatives) and martingale components, ensuring existence and uniqueness of weak solutions. For instance, the BSPDE

$$du = -\Big\{ [(a^{ij} u_{x^j} + \sigma^{ik} q^k)_{x^i} + f(t,x,u,u_x,q)] dt \Big\} + q^k\, dW^k$$

with $f$ quadratic in $u_x$ and $q$ illustrates two-sided QFG in infinite-dimensional settings (Du et al., 2012).

In PDE analysis, quasilinear parabolic systems with drivers growing quadratically in the gradient, e.g.,

$$|f^i(t, x, u, p)| \leq C_f [1 + |p^i||p| + \sum_{j < i} |p^j|^2 + |p|^{2-\epsilon}]$$

are handled via QFG and BMO martingale techniques, yielding regularity and existence of classical solutions (Jackson, 2022).

4. QFG in Optimization and Saddle Point Algorithms

QFG generalizes the strong convexity requirement in optimization. For convex–concave saddle point problems

$\min_{x \in X} \max_{y \in Y} f(x, y)$

the two-sided QFG (or quadratic gradient growth, QGG) replaces strong convexity–strong concavity assumptions. This relaxation extends the set of problems solvable with linear convergence guarantees. The generalized accelerated primal–dual (GAPD) algorithm exploits two-sided QFG to achieve linear convergence under the contraction

$$D_{Z}^{A_k - \Gamma B_k}(\bar{z}_k, z_k) \leq \rho^k D_{Z}^{A_0}(\bar{z}_0, z_0)$$

where $A_k$, $B_k$ are method-specific parameters and $\rho \in (0, 1)$ depends on QFG constants (Melcher et al., 13 Oct 2025). The algorithm tackles structured saddle point problems, e.g.,

$$\min_{x \in X} \max_{y \in Y} [ h(C_1x) + \langle Ax, y \rangle - g(C_2y) ]$$

where $h$ and $g$ are strongly convex, and $C_1$, $C_2$, $A$ are general linear operators, provided appropriate structural conditions yield two-sided QFG.

Restart schemes for accelerated first-order methods use QFG to guarantee linear convergence in non-strongly convex settings, enabling robust performance for practical optimization tasks such as model predictive control, Lasso regression, and more (Alamo et al., 2021).

5. Applications and Regularity Implications

QFG underpins robust control, finance, and empirical process theory. In robust risk-sensitive control, the entropic risk measure for 2BSDEs with quadratic generators is defined via exponential transformation, and value processes are characterized by QFG properties (Possamaï et al., 2012). BSPDEs with quadratic nonhomogeneous terms appear in non-Markovian stochastic control, leading to stochastic HJB equations with two-sided QFG (Du et al., 2012).

In empirical process theory, quadratic forms for the two-sample testing problem exploit QFG, as boundedness under the null and quadratic growth under alternatives yield asymptotic chi-square distributions and sensitivity to functional differences:

$$Q_n = \eta(m, n)^\top [\tilde{C}(g)]^{-1} \eta(m, n), \quad Q_n \xrightarrow{d} \chi^2_k$$

(Bárcenas et al., 2015).

Uniform ellipticity with $p$–$q$ growth is achieved in variational calculus by constructing integrands $f(z) = g(|z|)$ with controlled oscillating exponents, ensuring the Hessian satisfies uniform bounds and thus regularity properties such as local Lipschitz continuity (Filippis et al., 2020).

Quadratic mean-field BSDEs and their mean-field reflected or anticipated extensions are solved via QFG, BMO martingale theory, and fixed-point contraction, yielding existence, uniqueness, comparison, and stochastic representations for fully nonlinear and nonlocal PDEs (Hao et al., 2022, Hu et al., 2022, Hu et al., 2019).

6. Significance and Extensions

The two-sided QFG condition is central to the analysis and performance of numerical algorithms, stability theory, and regularity results across stochastic analysis, optimization, and PDE theory. The relaxation from strong convexity–strong concavity to QFG broadens the class of models for which robust convergence and stability can be established. QFG provides computable criteria for quadratic error bounds, supports superior rates in augmented Lagrangian methods (Cui et al., 2017), and enables advanced saddle point algorithms to operate efficiently in resource allocation, game theory, and adversarial learning contexts (Melcher et al., 13 Oct 2025).

The equivalence between QFG and metric subregularity (as elucidated for semi-algebraic and prox-regular functions) forms a foundation for error bounds and the justification of linear convergence in nonconvex and composite settings (Drusvyatskiy et al., 2013, Chieu et al., 2021). QFG appears naturally in rich mathematical structures—from geometric growth of binary quadratic forms on trees (Spalding et al., 2017) to quantum stochastic system performance criteria via quadratic-exponential functionals (Vladimirov et al., 2022).

Future research directions involve extending QFG analysis to fully nonlinear stochastic systems, higher-degree growth phenomena, complex interaction models in mean-field games, and the refinement of numerical schemes that robustly exploit two-sided quadratic growth for high-dimensional problems.