Augmented Progressive Hedging Algorithm

• The Augmented Progressive Hedging Algorithm is a dual decomposition framework for SMIPs that iteratively linearizes and convexifies nonconvex scenario subproblems.
• It employs a Frank–Wolfe based inner loop using the Simplicial Decomposition Method to construct inner approximations of convex hulls, ensuring provable convergence to Lagrangian dual bounds.
• The method reduces computational cost by replacing MIQPs with MILPs and continuous QPs while offering high parallelizability and robustness to large penalty parameters.

The Augmented Progressive Hedging Algorithm (Augmented PH, or FW–PH when using a Frank–Wolfe-type inner loop) is a primal-dual decomposition framework for computing Lagrangian dual bounds in stochastic mixed-integer programming (SMIP) by embedding iterative linearization and convexification steps within the classical Progressive Hedging (PH) structure. The algorithm addresses key limitations of standard PH when applied to nonconvex feasible sets typical in SMIP, in particular providing convergence guarantees to the Lagrangian dual value even when explicit convex hull descriptions are unavailable. The methodology combines augmented Lagrangian subproblems, inner approximation of convexified sets via Frank--Wolfe (specifically, the Simplicial Decomposition Method, SDM), and classical PH-style dual variable updates.

1. Lagrangian Dual Structure in SMIP

Stochastic mixed-integer programs (SMIPs) are often formulated as two-stage problems: $$\min\, c^\top x + \mathbb{E}\Bigl[\min\bigl\{ q_s^\top y_s : (x, y_s) \in K_s \bigr\}\Bigr] \qquad \text{subject to}~ x \in X$$ where for each scenario $s \in S$ the scenario set $K_s$ can be nonconvex due to integrality or combinatorial structure. In the deterministic equivalent, scenario copies $x_s$ and consensus variable $z$ are introduced, with nonanticipativity constraints $x_s = z$.

Relaxing nonanticipativity via multipliers $\omega_s$ leads to the Lagrangian dual function: $$\phi(\omega) = \sum_s p_s\,\phi_s(\omega_s),\quad \phi_s(\omega_s) = \min\Bigl\{ (c+\omega_s)^\top x + q_s^\top y : (x, y) \in K_s \Bigr\}$$ with the dual maximized subject to the scenario probability-weighted sum $\sum_s p_s \omega_s = 0$. The dual value, when $K_s$ are replaced by their convex hulls $\operatorname{conv}(K_s)$, provides the tightest possible decomposable dual bound for use in decomposition algorithms.

In practice, explicit convex hull representations are not available for most nontrivial SMIP scenario sets, limiting the applicability of standard PH, which depends on working over convex feasible sets for convergence guarantees.

2. Algorithmic Synthesis: Embedding the Frank–Wolfe Inner Loop

The FW–PH method replaces the direct use of $\operatorname{conv}(K_s)$ in the PH primal update by constructing an inner approximation via SDM (Simplicial Decomposition Method) for each scenario. The approach comprises:

1. For each scenario $s$, maintain an inner approximation $V_s$ consisting of discovered extreme points of $K_s$ (from prior subproblem solves).
2. At each PH (outer) iteration:

   • SDM Inner Loop (per scenario): a. Linearize the augmented Lagrangian objective at current primal variables (including multiplier and penalty terms):

     $$(x'_s, y'_s) = \arg\min_{(x, y) \in K_s} \Bigl[(c + \omega_s + \rho(x^{(t-1)}_s - z))^\top x + q_s^\top y \Bigr]$$

     This is a MILP because the objective is linear and the feasible set is $K_s$. b. Expand $V_s \leftarrow V_s \cup \{ (x'_s, y'_s) \}$. c. Solve the convex quadratic over $\operatorname{conv}(V_s)$ (a continuous QP):

     $$(x^{(t)}_s, y^{(t)}_s) = \arg\min_{(x, y) \in \operatorname{conv}(V_s)} L^\rho_s(x, y, z, \omega_s)$$

   d. Terminate inner SDM when the decrease measured by

   $$\Gamma^{(t)} = -\nabla L^\rho_s(x^{(t-1)}_s, y^{(t-1)}_s, z, \omega_s)^\top \left( (x'_s, y'_s) - (x^{(t-1)}_s, y^{(t-1)}_s) \right)$$

   is below a tolerance, or after a pre-specified number of SDM iterations.

3. PH Outer Loop: Across all scenarios, the consensus (averaged) variable $z$ is updated:

$z^{(k)} = \sum_s p_s x_s^{(k)}$

and multipliers updated via

$$\omega_s^{(k+1)} = \omega_s^{(k)} + \rho(x_s^{(k)} - z^{(k)})$$

Dual bounds are constructed at every outer iteration by summing scenario Lagrangians computed from the linearized MILP subproblems.

3. Convergence and Theoretical Benefits

The algorithm’s critical innovation—progressively approximating convex hulls without explicit descriptions—enables strong convergence guarantees absent in “vanilla” PH applied to nonconvex sets. If initial SDM approximations share common $x$-components or at least two SDM steps are performed per PH iteration, it can be shown that the Lagrangian bounds $\phi^{(k)}$ converge to the convexified dual value $\hat{LD}$. This is formalized in the paper’s Proposition 3.4.

Core improvements relative to standard PH include:

• Provable Convergence to Dual Bounds: Sequential improvement of $\operatorname{conv}(V_s)$ ensures contraction in dual bound gaps, avoiding cycling behavior common in PH over nonconvex $K_s$.
• Reduced Computation per Iteration: Each primal scenario update requires only a MILP and a continuous QP rather than an MIQP (as in standard PH).
• Robustness to Penalty Parameter: Large penalty $\rho$ expedites consensus formation without risking infeasibility or divergence, provided SDM is optimized at each step.

Additionally, valid Lagrangian bounds can be constructed at negligible extra cost even when the SDM loop is truncated (e.g., $t_{\max} = 1$), formalized in Proposition 3.3.

4. Implementation Details and Practical Usage

In practical terms:

• Scenario subproblem solutions (MILPs) become the bottleneck; thus, using high-performance MILP solvers is essential.
• The convex QP over $\operatorname{conv}(V_s)$ can be efficiently solved via expressing $(x, y)$ as a convex combination of points in $V_s$, leading to a low-dimensional continuous QP.
• Often, only a few inner SDM iterations per PH outer iteration are needed (even $t_{\max}=1$ suffices if initial points are well chosen).
• The algorithm is highly parallelizable due to scenario-wise SDM routines.

A pseudocode outline is:

Initialize {x_s^0, y_s^0}, z^0 = sum_s p_s x_s^0, omega_s^1 = omega_s^0 + rho(x_s^0 - z^0). for k = 1, 2, ... for each scenario s: set linearization point as a convex combination of z^{k-1} and x_s^{k-1} for t = 1 to t_max (SDM step): solve linearized MILP over K_s, add new point to V_s solve convex QP over conv(V_s) check decrease measure; break if below tolerance compute scenario dual bound update z^k and omega_s^{k+1} check convergence

The most computationally intensive operation is the MILP solve per scenario per SDM iteration.

5. Numerical Performance and Empirical Observations

Empirical results reported in the paper show that the FW–PH algorithm consistently and rapidly converges to high-quality Lagrangian dual bounds, outperforming standard PH implementations in the context of SMIP decomposition. Numerical evidence demonstrates:

• Faster convergence of dual bounds, attributable to improved inner approximation and effective use of penalty terms.
• Substantial computational savings for fixed iteration budgets due to the replacement of expensive MIQPs by MILPs plus trivial QPs.
• Valid dual bounds at each iteration, facilitating early stopping if desired.

A key observation is that, even with minimal SDM steps per outer iteration, the progression of dual bounds is monotonic and aligns tightly with the theoretical optimum.

6. Extensions and Integration with Decomposition Strategies

The algorithm's framework, notably the inner-approximation strategy, is extensible beyond two-stage problems, provided scenario-wise decomposition and convexification are meaningful. Its structure is directly compatible with Benders decomposition and other dual-based approaches for SMIP, provided Lagrangian bounding is required. Further, the algorithm offers a template for augmenting PH with modern first-order and projection-based methods in broader classes of large-scale nonconvex stochastic optimization.

The integration of PH and Frank–Wolfe inner steps in “FW–PH” exemplifies a trend towards combining classical decomposition algorithms with tailored convexification and approximation routines, enabling scalable, robust solutions to stochastic combinatorial optimization problems otherwise intractable by direct means.

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The Augmented Progressive Hedging Algorithm, as realized in the FW–PH method, represents a principled union of PH’s dual decomposition with iterative convexification by Frank–Wolfe linearization, equipped with convergence guarantees for Lagrangian dual bounds in SMIPs and allowing for efficient, robust, and parallelizable implementations in real-world high-dimensional stochastic integer optimization (Boland et al., 2017).