Second-order cone programs (SOCPs) are a fundamental class of convex optimization problems widely applied across engineering domains. Classical interior-point methods are robust and accurate, but their iterative nature can be costly in real-time or repeated-solve settings. This has motivated learning-based solvers that approximate optimizer solutions with neural networks. Recent work has increasingly studied graph neural networks (GNNs) for linear, quadratic, and second-order cone optimization, leveraging their ability to model sparse relational structure among variables, constraints, and objective functions. Most existing GNN-based solvers compress the optimization process into a single forward pass, predicting the optimizer directly without using the intermediate trajectory produced by classical solvers. We introduce SOCPNet, a recursive GNN framework that uses solver trajectories to learn an iterative SOCP solution procedure. A shared GNN model is applied repeatedly, refining the current iterate at each step and enabling inference for a variable number of iterations. Rather than predicting the solution in a single pass, SOCPNet learns from intermediate iterates to progressively refine its estimates using a trajectory-regularized training procedure. We evaluate SOCPNet on multiple datasets spanning standard SOCP problem classes and show that it consistently reduces the final solution error relative to a SOCP-GNN baseline. Finally, we demonstrate how trained SOCPNet models can be used in a hybrid optimization pipeline by warm-starting a classical interior-point solver, reducing solver iterations without loss of accuracy.
Figures & tables
Figure 1: A single shared SOCPNet is applied recursively.
Figure 2: Encoding of SOCP instance with parameters u=(f,F,g,l,r,{Ak,bk,ck,dk}k) into its graph.
Figure 4: Unstructured Small benchmark, nine-step ablation (Table 2 ). Median SRE, median objective gap, and mean constraint violation vs. unroll step ( ↓ better; log axes).
Figure 5: Small Structured families: SOCPNet vs. SOCP-GNN vs. IPOPT. Per-step median SRE ( ↓ better; log axis). Bands are 95% percentile intervals.
Figure 6: Hybrid solver: IPOPT iterations to convergence ( ↓ better). Cold start, one-shot SOCP-GNN warm start, and SOCPNet warm start ( d=256 ) at handoff step k , on the Unstructured Small (left) and Medium (right) datasets ( 50 test problems each).
Appendix figures & tables14 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 7: Unstructured Medium benchmark, nine-step ablation. Same conventions and labels as Fig. 4 .
Figure 8: Compute-matched comparison on Unstructured Small ( d=256 ). Median SRE ( ↓ better; log axis) against the number of message passing block applications. SOCP-GNN is plotted as a constant line because it is single-shot, but it uses the same budget of 25 message passing rounds as SOCPNet at the rightmost point. Repeating its block 25 times does not improve accuracy, whereas SOCPNet descends over that budget.
Step
IPOPT
SOCPNet ( d=256 )
SOCPNet ( d=64 )
SOCP-GNN
IPM-LSTM
25
∼0
3.1×10−6
2.1×10−5
3.3×10−3
2.0×10−2
Appendix
Table 3: Median SRE on the Small Unstructured test set ( 500 instances, h=30 ) under identical training and evaluation. IPM-LSTM adapted to SOCP against SOCPNet, the single-shot SOCP-GNN, and the IPOPT reference ( ↓ better).
Figure 9: Median normalized SRE over unroll steps on the Small Unstructured test set ( 500 instances, h=30 ; ↓ better, log axis). IPM-LSTM peaks near step 25 and then drifts, while both SOCPNet variants ( d=64 and d=256 ) converge and hold orders of magnitude below IPM-LSTM.
Step
IPOPT
SOCPNet ( d=256 )
SOCPNet ( d=64 )
SOCP-GNN
IPM-LSTM
21
∼0
9.6×10−4
1.2×10−3
1.3×10−2
3.7×10−2
Appendix
Table 4: Median SRE on the Medium Unstructured test set ( 500 instances) under identical training and evaluation ( ↓ better).
Figure 10: Median normalized SRE over unroll steps on the Medium Unstructured test set ( 500 instances, h=30 ; ↓ better, log axis). Both SOCPNet variants remain about an order of magnitude below IPM-LSTM
Figure 11: On Unstructured Small: mean and median SRE for S9 with and without step-size control.
Figure 12: On Unstructured Medium: mean and median SRE for S9 with and without step-size control.
Figure 13: Medium SOCP families: SOCPNet vs. SOCP-GNN vs. IPOPT. As Fig. 5 , at Medium scale ( n=100 , h=25 ).
Figure 14: Zero-shot scale transfer, no retraining. Per-step median SRE ( ↓ better; log axis) for a model trained at one scale and evaluated at the other. Same conventions as Fig. 5 .
Held-out family
SOCPNet
SOCP-GNN (one-shot)
ratio
Unstructured
3.77×10−3
1.03×10−1
27×
Robust ellipsoid
5.19×10−2
1.70×10−1
3.3×
Block angular
4.27×10−2
5.25×10−1
12×
Portfolio risk
5.65×10−1
1.28×100−
2.3×
Appendix
Table 5: Cross-family leave-one-out at Small scale. Median SRE on the held-out family ( 500 test instances) after training on the other three ( ↓ better). The ratio is the single-shot error divided by SOCPNet’s SRE .
Figure 15: Cross-family leave-one-out at Small scale. Median SRE on the held-out family ( ↓ better; log axis) for SOCPNet and the single-shot SOCP-GNN trained on the other three families.
Figure 16: Hybrid solver: final solution accuracy. Final SRE for cold start vs. SOCPNet warm start at handoff step k , on Unstructured Small (left) and Medium (right) ( 50 test problems each, SOCPNet d=256 ).
k
0
5
10
15
20
25
Small
IPOPT iterations
11
7
6
6
6
6
SOCPNet time (s)
0.00
0.11
0.21
0.32
0.43
0.54
IPOPT time (s)
17.00
8.89
7.85
7.71
7.74
7.89
total time (s)
17.00
9.00
8.16
8.02
8.17
8.47
speed-up
1.0×
1.9×
2.1×
2.1×
2.1×
2.0×
Medium
IPOPT iterations
15
12
10.5
10
10
10
Appendix
Table 6: Hybrid warm start: iteration and wall-clock breakdown. Medians over 50 test problems per dataset. k is the handoff step, k=0 the cold-start baseline. SOCPNet time is the time spent in the k forward passes, IPOPT the subsequent exact solve, and total their sum.
Semidefinite programs (SDPs) are a powerful framework for convex optimization and for constructing strong relaxations of hard combinatorial problems. However, solving large SDPs can be computationally expensive, motivating the use of machine learning models as fast computational surrogates. Graph neural networks (GNNs) are a natural candidate in this setting due to their sparsity-awareness and ability to model variable-constraint interactions. In this work, we study what expressive power is sufficient to recover optimal SDP solutions. We first prove negative results showing that standard GNN architectures fail on recovering linear SDP solutions. We then identify a more expressive architecture that captures the key structure of SDPs and can, in particular, emulate the updates of a standard first-order solver. Empirically, on both synthetic and SdpLib benchmarks of various classes of SDPs, this more expressive architecture achieves consistently lower prediction error and objective gap than theoretically weaker baselines. Finally, using the learned high-quality predictions to warm-start the first-order solver yields practical speedups of up to 80%.
Chendi Qian, Christopher Morris
Faculty of Computer Science, RWTH Aachen University, Germany
Graph-structured optimization with linear constraints is fundamental to critical infrastructure but faces scalability limits due to massive strict hard constraints and high dimensionality. While recent projection-based methods such as Trainable Sampling Kaczmarz-Motzkin Net (T-SKM-Net) guarantee feasibility, they face high computational costs in dynamic environments by processing the entire constraint set and requiring expensive matrix factorizations. To bridge this gap, we propose the Accelerated Trainable-SKM (AT-SKM) Net framework. To concentrate computation on the active constraints and eliminate redundant calculations, we introduce a hybrid sampling strategy guided by a topology-aware heterogeneous GNN model. To efficiently handle topological shifts in graph-based constraints, we employ a Cholesky Update mechanism that theoretically reduces the equality projection complexity from O(N^3) to O(N^2) under low-rank perturbations. Experiments on random geometric graphs, N-1 Security-Constrained DC-OPF, and minimum-cost gas transport problem demonstrate that AT-SKM reduces iteration counts by up to 85% and achieves 2.95x-7.29x SKM layer speedups, while maintaining zero constraint violations.
Nonlinear model predictive control (NMPC) can be applied to large nonlinear systems with complex objectives while enforcing constraints, but the cost of solving the underlying optimization problems limits real-time application for systems with fast dynamics or many states. Neural networks can reduce this online cost by approximating the control law, but may violate constraints and typically require many precomputed optimizer solutions for training. The learning-based iterative solver for constrained optimization (LISCO) instead learns to predict primal-dual steps toward satisfaction of the Karush-Kuhn-Tucker (KKT) conditions through self-supervised training without precomputed solutions, but its dense, monolithic architecture grows with the prediction horizon. We remove this horizon dependence by exploiting the temporal structure of the problem in the solver architecture, which makes the network size independent of the horizon while leaving the training loss and the solver iteration unchanged. A closed-loop resampling strategy further aligns the training distribution with closed-loop operation at no additional sampling cost. On an economic NMPC benchmark of a CSTR cascade, the proposed solver scales to larger problems than monolithic LISCO and achieves median speedups of up to a factor of four over IPOPT. In closed loop, it achieves comparable control performance to IPOPT with no observed constraint violations.
Lukas Lüken, Sergio Lucia
Chair of Process Automation Systems, TU Dortmund University, Emil-Figge Straße 70, Dortmund, 44227, North Rhine-Westphalia, Germany