Technical Summary: WARP: A Benchmark for Primal-Dual Warm-Starting of Interior-Point Solvers

Problem Statement

Solving the Alternating Current Optimal Power Flow (AC-OPF) problem is central to electricity market operations, typically performed every 5–15 minutes by Independent System Operators. The standard solver for this non-convex nonlinear program (NLP) is the primal-dual Interior-Point Method (IPM), specifically IPOPT. While a growing body of literature attempts to accelerate these solvers using machine learning to predict "warm-start" iterates, existing approaches suffer from fundamental methodological flaws.

Prior works train neural networks to predict only the primal variables ($\hat{x}$) and benchmark their performance against a "flat start" initialization ($V_m = 1, V_a = 0$). The authors argue this baseline is inappropriate because the actual default initialization for IPOPT and MIPS is the variable-bound midpoint, $(l+u)/2$. Furthermore, theoretical results (Yildirim and Wright, 2002) suggest that providing the ground-truth optimal primal solution without corresponding dual variables can cause IPMs to diverge, as the optimal solution lies on the constraint boundary where the log-barrier function is singular.

Methodology and Diagnosis

The paper presents a systematic diagnosis of why primal-only warm-starting fails and proposes a corrected evaluation framework.

1. Correcting the Baseline and Diagnosing Failure

The authors demonstrate that prior reported gains (30–46% iteration reductions) are artifacts of comparing against the weak flat start. When benchmarked against the solver's true midpoint default:

• No primal-only method improves convergence: Various strategies, including normalization, feasibility projection, and constraint screening, fail to reduce iterations below the midpoint baseline.
• Primal accuracy is anticorrelated with convergence: Models with lower prediction error often yield more solver iterations. Providing the ground-truth optimal solution $x^*$ without dual variables causes the solver to diverge.
• Geometric Bottleneck: The failure is attributed to the "centrality" of the starting point. The midpoint maximizes the minimum slack, providing a well-centred starting point for the log-barrier. Primal predictions, even if accurate, often displace the voltage magnitude ($V_m$) from the midpoint toward the constraint boundaries, degrading the barrier centrality.

2. Oracle Experiments

To establish the theoretical ceiling for warm-starting, the authors conducted oracle experiments using the complete primal-dual-barrier state $(x^*, \lambda^*, z^*, \mu^*)$.

• Primal-only ($x^*$): Increases iterations by ~5% compared to cold start.
• Primal + Equality Duals ($x^*, \lambda^*$): Reduces iterations by 45%.
• Primal + All Duals ($x^*, \lambda^*, z^*$): Reduces iterations by 79%.
• Full State ($x^*, \lambda^*, z^*, \mu^*$): Reduces iterations by 85% (from 23 to 3).
  This establishes that the entire iteration gap resides in the dual variables and the barrier parameter, making dual prediction necessary and sufficient for effective warm-starting.

3. The WARP Benchmark Suite

To enable rigorous evaluation, the authors release:

• Dual-Labeled Datasets: AC-OPF instances (based on pglib_opf_case118_ieee) augmented with IPOPT-extracted solutions for the full state $(x^*, \lambda^*, z^*, \mu^*)$.
• Corrected Protocol: A standardized evaluation benchmarking against the midpoint default and measuring IPOPT iteration counts.

4. WARP Architecture

The authors introduce WARP, a topology-aware encode-process-decode (EPD) interaction network designed to predict the full IPM state.

• Architecture: It operates on a heterogeneous graph representing the power grid (buses, generators, loads, lines). It uses unshared weights across 15 interaction network blocks.
• Key Design Choices:
  • No Node Residuals: Unlike standard MeshGraphNets, WARP omits node residual connections. Ablation studies show this improves dual prediction by 5$\times$ and reduces validation loss from 0.45 to 0.09.
  • Edge Updates: Per-step edge feature updates are identified as the dominant architectural feature, contributing a 9$\times$ impact on performance.
  • Binding-Mask Loss: The training loss decomposes bound multipliers ($z$) into binding and non-binding components to handle sparsity.
• Topology Conditioning: WARP processes the graph structure directly, allowing it to handle N-1 contingency topologies (single line removals) without retraining.

Key Results

• Iteration Reduction: WARP achieves a 76% reduction in IPOPT iterations (from 22.6 to 5.4 mean iterations), approaching the 85% reduction of the oracle.
• Primal-Only vs. Primal-Dual: When the same WARP architecture is trained only on primal targets, it yields 21.1 iterations (no improvement over the midpoint), confirming that the gains are driven by dual prediction, not architectural capacity.
• Topology Robustness: WARP successfully processes 20/20 N-1 contingency scenarios with physically reasonable predictions. In contrast, a fixed-dimension LSTM baseline (IPM-LSTM) fails on all contingencies due to input dimension mismatches.
• Ablation Insights:
  • Removing node residuals is the single most impactful change.
  • Constraint screening (removing non-binding constraints) is counterproductive for IPMs, increasing iterations by 2.3$\times$.
  • Diffusion-based variants did not outperform deterministic regression.

Significance and Claims

The paper claims three primary contributions:

1. Methodological Correction: It exposes the inappropriate evaluation baseline (flat start) used in prior work and demonstrates that primal-only warm-starting is structurally misaligned with IPM convergence properties.
2. Theoretical and Empirical Diagnosis: It establishes that the complete primal-dual-barrier state is necessary and sufficient for warm-starting, with dual variables accounting for the entire iteration gap.
3. Benchmark and Baseline: It releases the WARP benchmark suite (datasets, protocol, and code) and a strong baseline (WARP) that achieves state-of-the-art iteration reduction while natively supporting topology variations.

The authors conclude that while WARP achieves a 76% reduction, the remaining gap to the oracle (3.2 iterations) is likely due to the extreme sensitivity of IPMs to small absolute errors in high-magnitude dual variables. The work emphasizes that future progress in learning-based IPM acceleration requires predicting the full dual-barrier state rather than just primal solutions.