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IPM-LSTM: A Learning-Based Interior Point Method for Solving Nonlinear Programs
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Proposes IPM-LSTM, which uses a coordinate-wise LSTM trained self-supervised to approximately solve the Newton linear systems inside an interior point method, then warm-starts IPOPT with the resulting primal-dual iterate.
Problem
Solving constrained nonlinear programs with interior point methods is bottlenecked by the O(n^3) linear system solves at each iteration, and existing learning-based warm-start methods yield solutions that are not well-centered for IPMs.
Approach
Replace the exact Newton linear-system solve with an LSTM that minimizes the least-squares residual ||J y + F||^2 unrolled over IPM iterations and LSTM steps, with diagonal Ruiz preconditioning, and use its well-centered primal-dual output to warm-start IPOPT.
Key results
- On non-convex QPs, warm-starting IPOPT with IPM-LSTM output reduces iterations by up to 63.9% and solve time by up to 70.5%.
- On convex QPs (RHS), IPM-LSTM reduces IPOPT iterations from 12.5 to 7.2 (42.4%/15.1% total gain including LSTM time), the best warm-start iteration reduction among compared L2O methods.
- On convex QPs (ALL), warm-starting IPOPT reduces iterations by 35.7% and total solve time by 7.5%.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Warm-start IPOPT with IPM-LSTM solutions | IPOPT on Convex QPs (RHS), 200 variables · Convex QPs (RHS), 200 variables | total time (s): 0.642 -> 0.545; 15.1% | Helped |
| Warm-start IPOPT with IPM-LSTM solutions | IPOPT on Convex QPs (ALL), 200 variables · Convex QPs (ALL), 200 variables | solution time reduction: 7.5% | Helped |
| Warm-start IPOPT with IPM-LSTM solutions | IPOPT on Convex QCQPs (RHS), 200 variables · Convex QCQPs (RHS), 200 variables | total time (s): 1.098 -> 1.030; 6.2% | Helped |
| Warm-start IPOPT with IPM-LSTM solutions | IPOPT on Convex QCQPs (ALL), 200 variables · Convex QCQPs (ALL), 200 variables | solution time reduction: 11.4% | Helped |
| Warm-start IPOPT with IPM-LSTM solutions | IPOPT on non-convex QPs · Non-convex QPs | solution time reduction: up to 70.5% | Helped |
| Warm-start IPOPT with IPM-LSTM solutions | IPOPT on Convex QPs (RHS), 100 variables · Convex QPs (RHS), 100 variables | total time (s): 0.269 -> 0.160; 40.5% | Helped |
| Warm-start IPOPT with IPM-LSTM solutions | IPOPT on Convex QPs (ALL), 100 variables · Convex QPs (ALL), 100 variables | solution time reduction: 28.0% | Helped |
| Warm-start IPOPT with IPM-LSTM solutions | IPOPT on Convex QCQPs (RHS), 100 variables · Convex QCQPs (RHS), 100 variables | solution time reduction: 25.8% | Helped |
| Warm-start IPOPT with IPM-LSTM solutions | IPOPT on Convex QCQPs (ALL), 100 variables · Convex QCQPs (ALL), 100 variables | solution time reduction: 31.6% | Helped |
| Increase LSTM time steps from 50 to 60 (ablation) | IPM-LSTM on Convex QPs (RHS) · Convex QPs (RHS), 100 variables | total time (s): 0.160 -> 0.154 | Helped |
| Increase LSTM time steps from 50 to 70 (ablation) | IPM-LSTM on Convex QPs (RHS) · Convex QPs (RHS), 100 variables | total time (s): 0.160 -> 0.164 | Hurt |
Surprising findings
- Solutions from DeepLDE warm-start worsened IPOPT iteration count in several settings, and some configurations (e.g., non-convex programs ALL) showed negative total time gains for IPM-LSTM.
Limitations
- IPM-LSTM is relatively computationally expensive because it uses multi-cell LSTMs, so end-to-end runtime lags simple MLP L2O baselines.
- Assumption 1 (residual and boundedness conditions) is only empirically checked, not guaranteed by the LSTM.
- Condition numbers grow in later IPM iterations, so the method is terminated after finitely many iterations.
Experimental setup
Datasets: Chen and Burer 2012 RandQP Donti et al. 2021 generator Liang et al. 2023 generator GlobalLib MINLPLib
Benchmarks: convex QPs (RHS) convex QPs (ALL) convex QCQPs (RHS) convex QCQPs (ALL) non-convex QPs (GlobalLib, RandQP) simple non-convex programs
Metrics: objective value max and mean inequality violation max and mean equality violation solve time IPOPT iterations warm-start gain (iterations/time reduction)
Baselines: OSQP IPOPT NN DC3 DeepLDE PDL LOOP-LC H-Proj
Builds on: Andrychowicz et al. 2016 learned optimizers inexact interior point method (Bellavia 1998) IPOPT DC3
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