Low-precision first-order method-based fix-and-propagate heuristics for large-scale mixed-integer linear optimization

Abstract: We investigate the use of low-precision first-order methods (FOMs) within a fix-and-propagate (FP) framework for solving mixed-integer programming problems (MIPs). FOMs, using only matrix-vector products instead of matrix factorizations, are well suited for GPU acceleration and have recently gained more attention for their application to large-scale linear programming problems (LPs). We employ PDLP, a variant of the Primal-Dual Hybrid Gradient (PDHG) method specialized to LP problems, to solve the LP-relaxation of our MIPs to low accuracy. This solution is used to motivate fixings within our fix-and-propagate framework. We implemented four different FP variants using primal and dual LP solution information. We evaluate the performance of our heuristics on MIPLIB 2017, showcasing that the low-accuracy LP solution produced by the FOM does not lead to a loss in quality of the FP heuristic solutions when compared to a high-accuracy interior-point method LP solution. Further, we use our FP framework to produce high-accuracy solutions for large-scale (up to 243 million non-zeros and 8 million decision variables) unit-commitment energy-system optimization models created with the modeling framework REMix. For the largest problems, we can generate solutions with under 2% primal-dual gap in less than 4 hours, whereas commercial solvers cannot generate feasible solutions within two days of runtime. This study represents the first successful application of FOMs in large-scale mixed-integer optimization, demonstrating their efficacy and establishing a foundation for future research in this domain.

• The paper introduces a novel heuristic integrating low-precision first-order methods into fix-and-propagate frameworks for large-scale MIP solving.
• It demonstrates that using the PDLP approach achieves over 20% optimality gap reduction and 2–3x speedup compared to traditional IPM-based relaxations.
• Empirical results on MIPLIB benchmarks and energy system models validate the method’s scalability and effectiveness in GPU-accelerated environments.

Low-Precision First-Order Method-Based Fix-and-Propagate Heuristics for Large-Scale MIP

Problem Statement and Motivation

The study centers on advancing heuristic techniques for mixed-integer linear programs (MIPs) by leveraging first-order methods (FOMs) to obtain low-precision LP relaxations and integrating these within fix-and-propagate (FP) frameworks. The motivation is grounded in the limitations of classical LP solvers (Simplex and Interior-Point Methods, IPMs), both in speed and scalability, particularly for large-scale instances arising in energy system optimization models (ESOMs) with millions of variables and constraints. Since real-world problem data often contain uncertainties and modest optimality gaps are acceptable, the paper questions whether high-precision LP solutions are necessary for effective MIP heuristics and explores FOMs as a computationally advantageous substitute, especially in the context of GPU acceleration.

First-Order Methods and LP Relaxations

FOMs, specifically the Primal-Dual Hybrid Gradient (PDHG) method as implemented in PDLP [ApplegatePDLP], are characterized by their exclusive reliance on matrix-vector products and their suitability for GPU hardware. Unlike IPMs, they do not require matrix factorizations, making them highly scalable. However, they are typically only capable of low-to-moderate accuracy. The paper demonstrates the feasibility of employing PDLP for solving the LP relaxation of MIPs at low precision, resulting in significant computational savings when compared to classical methods.

Fix-and-Propagate Heuristic Framework

The FP heuristic operates by iteratively fixing integer variables and applying domain propagation to reduce search space—potentially detecting infeasibility early or tightening bounds for other unfixed variables. The heuristic is initialized by solving the LP relaxation of the MIP to obtain primal and dual solutions, then these are used to inform variable selection and fixing strategies via four LP-based FP variants:

• Type: Orders variables by domain/type, requiring minimal LP information.
• Frac: Leverages fractionality in the LP solution as a selection criterion.
• RedCost: Exploits reduced cost information to prioritize fixings.
• Dual: Incorporates dual variable and constraint activity into variable selection.

The fixing strategies, selection mechanics, backtracking, and propagation are agnostic to the underlying LP solver but benefit from information extracted via (approximate) LP relaxations.

Numerical Evaluation and Results

The study conducts extensive computational experiments on MIPLIB 2017 and large-scale ESOM instances from the REMix framework. Key empirical findings include:

• MIPLIB Benchmarks: FP heuristics guided by low-precision PDLP and high-precision IPM relaxations exhibit comparable solution quality (optimality gaps), with an average gap reduction of over 20% versus uninformed heuristics. Time savings are substantial with PDLP; reduced accuracy yields 2–3x speedup.
• Unit Commitment Energy System Models: For models scaling up to 243 million non-zeros and 8 million variables, PDLP-based FP heuristics achieve gaps below 2% within 4 hours, outperforming commercial MIP solvers (Gurobi, CPLEX, COPT) that fail to produce feasible solutions within two days. The gap remained robust with lower LP solution accuracy, unchanged from high-accuracy settings.
• LP Solver Comparison: PDLP is increasingly competitive as model size grows, often outperforming IPMs, especially when IPMs reach memory or compute bottlenecks.

These results rigorously support the claim that low-accuracy FOM-based LP relaxations are valuable for large-scale MIP heuristics, both in solution quality and computational efficiency, with particular benefits for GPU-enabled environments.

Practical and Theoretical Implications

The study establishes that FOMs, especially GPU-accelerated PDLP, can serve as an LP solver within primal heuristic frameworks for MIPs, without any observable degradation in feasible solution quality compared to classical implementations. This overturns traditional assumptions requiring high-precision LP relaxations for effective MIP heuristics. The fast, scalable computation uncovers new tractable domains, notably for real-world, high-dimensional energy system models.

Theoretically, this revises the guidance for embedding LP relaxations in MIP heuristics and motivates further exploration of FOMs for guiding search, diving, and neighborhood search. The compositional design of fix-and-propagate heuristics enables additional integration with dual information, reduced costs, and propagation mechanisms, reinforcing a versatile paradigm for heuristic MIP solving.

Future Prospects

The findings prompt further development in several directions:

• Fully GPU-native FP implementations, including domain propagation.
• Expansion of FOM-based heuristic schemes to other classes of combinatorial optimization beyond classical MIPs.
• Revisiting traditional MIP solver components (branching, diving, neighborhood search) with FOM guidance.
• Integration with machine learning-based approaches for variable selection and fixing [bengio_machine_2020, Gasse2019].
• Broad adoption in ESOMs, logistics, and industrial applications requiring rapid generation of high-quality feasible solutions.

Conclusion

The paper demonstrates that low-precision FOMs can successfully substitute classical LP solvers in FP heuristics for MIPs, allowing scalable treatment of ultra-large optimization instances while preserving solution quality. This marks a significant advance in both practical optimization for large-scale applications and theoretical understanding of LP solver requirements within heuristic frameworks. The work establishes a foundation for future efforts in leveraging FOMs and GPU acceleration in mixed-integer optimization (2503.10344).