Research map / Learning theory and algorithms / Combinatorial and search optimization
Learning to Solve Bilevel Programs with Binary Tender
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Learns the lower-level optimal value function of bilevel programs with binary tender using GNN and input supermodular neural networks (ISNN), embeds the ReLU network as a mixed-integer program, and refines with enhanced sampling and iteration.
Problem
Bilevel programs whose linking (tender) variables are binary are hard to solve because the lower-level optimal value function has no tractable closed form when the lower level contains discrete variables, and gradient methods do not apply to discrete formulations.
Approach
Train a neural network to approximate the lower-level value function over binary tender, embed its ReLU form as a mixed-integer single-level reformulation, and introduce ISNN whose non-negative weight constraints make the output supermodular so the optimality constraint becomes linear inequalities; combine with an enhanced sampling method guided by an upper bound and an iterative refinement loop.
Key results
- With LP lower level, ISNN reduces objective difference by more than 10% versus GNN when n>=40, and ISNN always outperforms GNN in average and minimum objective difference at equal iteration counts.
- For n=10, both GNN and ISNN recover the true optimal solution in all 10 replications because all feasible x are enumerated.
- Computational time is significantly shorter than MiBS when n>=40 (LP lower level); with MILP lower level, the method achieves under 5% objective difference in most instances and outperforms MiBS for n=20, 40, 50.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ISNN instead of GNN with the same iteration count (ablation) | Neural Bilevel Algorithm · Randomly generated instances, LP lower level, n ≥ 40 | objective difference: more than 10% | Helped |
| Increase the number of iterations (ablation) | Neural Bilevel Algorithm · Randomly generated instances, LP lower level | average objective difference | Helped |
| Use enhanced sampling instead of naive sampling (ablation) | Neural Bilevel Algorithm · Randomly generated instances, LP lower level | average objective difference | Helped |
| Use GNN or ISNN instead of MiBS | Neural Bilevel Algorithm · Randomly generated instances, LP lower level, n ≥ 40 | computational time | Helped |
| Use ISNN instead of GNN with the same iteration count (ablation) | Neural Bilevel Algorithm · Randomly generated instances, LP lower level | objective difference | Helped |
| Use the proposed method instead of MiBS | Neural Bilevel Algorithm · Randomly generated instances, MILP lower level | objective difference: less than 5% | Helped |
| Use the proposed method instead of MiBS | Neural Bilevel Algorithm · Randomly generated instances, MILP lower level, n=20, 40, 50 | objective difference | Helped |
| Use the proposed method instead of MiBS | Neural Bilevel Algorithm · Randomly generated instances, MILP lower level | computational time | Helped |
Surprising findings
- With MILP lower levels, MiBS reports incorrect optimal solutions in some instances, which the proposed method beats.
Limitations
- Objective difference exceeds 15% for n=30, 50, 60 with LP lower level
- Exponential number of supermodular cuts unless separated on-the-fly; scalability depends on separation
- Approximation quality with limited samples; marginal gains from enhanced sampling shrink as n grows
Experimental setup
Datasets: randomly generated bilevel instances (n=10 to 60, m=20)
Benchmarks: randomly generated BP instances with LP lower level randomly generated BP instances with MILP lower level illustrative example
Metrics: objective difference relative to MiBS computational time
Baselines: MiBS GNN
Builds on: input convex neural network MiBS Latin hypercube sampling
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