Proximal Bundle Surrogate Framework

• The proximal bundle surrogate framework is a method for nonsmooth composite convex optimization that builds piecewise-linear surrogate models using subgradient cuts and a proximal regularization term.
• Each iteration solves a simple quadratic surrogate subproblem, ensuring robust stabilization and providing iteration-complexity guarantees across various convex and composite settings.
• Recent advances extend the framework to stochastic, weakly convex, and primal–dual formulations, employing active-cut and aggregate strategies for efficient bundle management.

The Proximal Bundle Surrogate Framework is a family of algorithmic devices for nonsmooth and composite convex optimization, distinguished by the use of piecewise-linear lower (minorant) surrogate models regularized by a strong convexity-inducing proximal term. Central to this framework is the construction of local approximations to a nonsmooth function by aggregating subgradient-based cuts obtained at previous iterates, forming a so-called “bundle.” Each algorithmic step then solves a structurally simple (typically quadratic or quadratic-programming) surrogate problem involving the bundle model plus a proximal term and, if present, smooth or “easy” convex regularization. The framework offers rigorous iteration-complexity guarantees, robust stabilization, and has been extended to a range of settings, including stochastic, composite, weakly convex, and constrained optimization.

1. General Design and Surrogate Model Construction

The framework addresses minimization problems of the form

$\min_{x\in\mathbb{R}^n} \,\phi(x) := f(x) + h(x),$

where $f$ is a (possibly nonsmooth) convex function and $h$ is convex (often with additional structure such as being smooth or proximable). At every iteration, a “bundle” $C_j$ consisting of selected past points $\{x^{i}\}$ and their subgradient information $\{f'(x^{i})\}$ is maintained. The lower cutting-plane model is defined as

$$f_j(u) := \max_{\xi\in C_j} \{ f(\xi) + \langle f'(\xi),\,u-\xi\rangle \},$$

ensuring $f_j(u) \leq f(u)$ for all $u$.

This model is regularized via a proximal quadratic centered at a reference (prox-center) $x^c_{j-1}$:

$$\Psi_j(u) := f_j(u) + h(u) + \frac{1}{2A}\|u-x^c_{j-1}\|^2,$$

where $A>0$ is a (constant or adaptive) proximal parameter. This construction yields a strongly convex surrogate that is tight at the bundle points, efficiently solvable, and stabilizes the otherwise ill-posed nonsmooth problem (Liang et al., 2020).

The surrogate model can be generalized to, for example, composite settings (with additional smooth or proximable components), weakly convex objectives via convexification, or expectation-valued functions in the stochastic regime (Liang et al., 2022, Liang et al., 2023).

2. Algorithmic Core: Prox-Bundle Subproblems and Iteration Structure

Each iteration solves the prox-bundle subproblem:

$$x^j = \mathop{\mathrm{argmin}}_{u\in\mathbb{R}^n} \left\{ f_j(u) + h(u) + \frac{1}{2A}\|u-x^c_{j-1}\|^2 \right\}.$$

This step yields a trial point $x^j$ and a model value $m_j = \Psi_j(x^j)$. To decide whether to accept $x^j$ as the new prox-center ("serious step") or to stay at $x^c_{j-1}$ and enrich the bundle ("null step"), a model gap test is used:

$t_j := \Psi^e_j(\tilde a_j) - m_j,$

where $$\Psi^e_j(u) = \Psi_j(u) + \frac{1}{2A}\|u-x^c_{j-1}\|^2$$ is an auxiliary model, and $$\tilde a_j = \operatorname{Argmin}\left\{\Psi^e_j(u)\ :\ u\in\{x^j,x^c_{j-1}\}\right\}$$. If $t_j\leq\delta$ for a fixed tolerance $\delta>0$, the iteration is serious; otherwise, it is null, and $C_{j+1}$ is augmented, often restricting to cuts active at $x^j$ to control bundle growth (Liang et al., 2020).

This two-phase (inner/outer) iteration structure is fundamental and underpins classical, relaxed, and stochastic variants (Liang et al., 2022, Ouorou, 2020). For weakly convex or hybrid objectives, convexification and alternative stationarity surrogates are integrated, but the basic loop—a bundle-regularized surrogate is refined until sufficient predicted progress is established—remains unchanged (Liang et al., 2023, Liang et al., 2021).

Table: Key Elements of Surrogate Construction

Component	Definition	Role
Bundle Model	$$f_j(u) = \max_{\xi\in C_j} \{f(\xi) + \langle f'(\xi),u-\xi\rangle\}$$	Lower model of $f$, powers cuts
Prox Term	$\frac{1}{2A}\|u-x^c\|^2$	Stabilization, strong convexity
Full Surrogate	$f_j(u) + h(u) + \frac{1}{2A}\|u-x^c\|^2$	Optimization model at each step
Test Quantity	$t_j=\Psi^e_j(\tilde a_j)-m_j$	Accept/reject rule

3. Convergence Theory and Complexity Guarantees

The framework achieves optimal or near-optimal iteration complexity for broad classes of problems. For convex and p-convex $h$, and fixed stepsize $A>0$, the number of serious steps needed to reach $\epsilon$-optimality scales as $\mathcal{O}(1/\epsilon^2)$—matching the best results for nonsmooth convex minimization (Liang et al., 2020, Liang et al., 2021).

In the stochastic case, with expectation-valued $f$ and using only a single cut, the stochastic composite proximal bundle method attains the optimal sample-complexity $\mathcal{O}(1/\epsilon^2)$ and is universal, requiring no problem parameters as input (Liang et al., 2022).

Extensions to weakly convex or hybrid settings preserve these guarantees up to logarithmic factors, provided appropriate convexification and regularized stationary-point conditions are used (Liang et al., 2023). For composite smooth+piecewise linear or strongly convex scenarios, accelerated rates become possible (e.g., $\mathcal{O}(1/\sqrt{\epsilon})\log(1/\epsilon)$ for certain smooth composite protocols) (Fersztand et al., 29 Apr 2025).

The convergence proofs typically exploit strong convexity induced by the prox term, telescoping Lyapunov arguments, and model gap recursion. Null-step iterations remain bounded—often logarithmic per serious step—even with fixed or absolute-accuracy null tests.

4. Extensions and Variants: Composite, Weakly Convex, and Primal–Dual Formulations

The proximal bundle surrogate framework is agnostic to the structure of $h$, supporting a range of composite optimization settings:

• Composite Proximal Bundle: When $h$ is simple (e.g., $\ell_1$-regularization or indicator function of a convex set), the subproblem is a composite quadratic program, efficiently solved using projected QP or dual techniques (Liang et al., 2021).
• Weakly Convex Extensions: For $f$ that is $m$-weakly convex (e.g., sum of smooth nonconvex and convex nonsmooth terms), local convexification via $(m/2)\|x-y\|^2$ allows the construction of a valid surrogate and convergence to $(\eta,\epsilon)$-Moreau stationarity with explicit bounds (Liang et al., 2023, Liao et al., 2 Sep 2025).
• Primal–Dual Bundle Surrogates: For linearly constrained convex programs, the bundle approach generalizes dual ascent and multiplier methods: at each update, a bundle surrogate is built for either the primal or dual subproblem (or both), yielding improved robustness and convergence rates compared to basic gradient and standard augmented Lagrangian methods (Zheng et al., 18 Nov 2025, Liao et al., 12 Feb 2025).

Numerical and theoretical work has shown that limited-memory bundle variants (e.g., single-cut, aggregate-cut), as well as adaptive strategies for prox parameter and bundle management, retain complexity bounds and practical efficacy (Liang et al., 2022, Fersztand et al., 24 Nov 2024).

5. Implementation and Practical Considerations

Efficient implementation maintains a modest, possibly dynamically pruned bundle; each subproblem is a quadratic (possibly composite) program solvable by modern QP solvers. Bundle updates are managed via active-cut retention (i.e., keeping only cuts active at the model minimizer), and in stochastic or large-scale scenarios, single-cut strategies with cut aggregation are effective (Liang et al., 2022, Fersztand et al., 24 Nov 2024). Parameters such as prox stepsize or accuracy tolerance can be fixed or adaptively tuned, with universal variants achieving optimal bounds without knowledge of smoothness or Lipschitz constants (Liang et al., 2 Apr 2024).

Variants with fixed absolute accuracy null-step rules (as opposed to classical relative rules) achieve improved iteration-complexity $O(\epsilon^{-4/5})$ and facilitate a Frank–Wolfe duality interpretation, further informing bundle management policies and complexity proofs (Fersztand et al., 24 Nov 2024). For smooth objectives, Nesterov-style acceleration via smooth lower surrogate models yields further improvements (Fersztand et al., 29 Apr 2025).

Implementation supports constrained, composite, stochastic, and weakly convex problems with minimal configuration, and bundle memory remains manageable due to active set strategies.

6. Significance, Applications, and Limitations

The proximal bundle surrogate framework has been established as a foundational approach for nonsmooth and composite convex optimization. Its rigorous complexity theory, generality, and robust numerical behavior have led to widespread adoption in large-scale, stochastic, robust, and hybrid settings, as well as in constrained and augmented Lagrangian schemes (Liang et al., 2020, Liao et al., 12 Feb 2025, Zheng et al., 18 Nov 2025).

Applications range from robust optimization, machine learning with composite penalties, and high-dimensional estimation to semidefinite, conic, and multistage adaptive robust optimization (Ning et al., 2018, Liao et al., 12 Feb 2025). In sampling, bundle-based oracles are now leveraged to control complexity for log-concave non-smooth distributions (Liang et al., 2021, Liang et al., 2 Apr 2024).

The framework's main limitation is the per-iteration cost of solving (possibly large) QPs as the bundle grows, though active-cut and aggregate-cut policies effectively control this. Extensions to nonconvex optimization require careful convexification and stationarity surrogate analysis; recent work demonstrates that the bundle principle extends to several weakly convex and composite nonconvex classes (Liao et al., 2 Sep 2025).

7. Recent Advances and Open Directions

The last five years have seen the development of optimal and universal variants for a range of regimes:

• Relaxed and parameter-robust variants with optimal complexity for large parameter ranges (Liang et al., 2020).
• Stochastic composite bundle surrogates with single-cut universality (Liang et al., 2022).
• Weakly convex and hybrid models with verifiable and optimal stationarity guarantees (Liang et al., 2023, Liao et al., 2 Sep 2025).
• Frank–Wolfe interpretations and improved bundle management for accelerated convergence (Fersztand et al., 24 Nov 2024).
• Accelerated protocols exploiting smoothness, achieving rates such as $O(\epsilon^{-1/2}\log(1/\epsilon))$ (Fersztand et al., 29 Apr 2025).
• Primal–dual and conic optimization with bundle surrogates in both primal and dual, leading to practical advances in robustness and flexibility (Zheng et al., 18 Nov 2025, Liao et al., 12 Feb 2025).
• Incorporation into sampling methods for non-smooth potentials via proximal MCMC (Liang et al., 2021, Liang et al., 2 Apr 2024).

Open challenges include the extension to general nonconvex regimes, scalable bundle management in very high-dimensional problems, and further integration of acceleration and adaptive proximal parameter schemes. The recent body of work suggests that the surrogate-based bundle principle is a unifying theme in first-order methods for structured large-scale optimization.