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Conformal Prediction for Early Stopping in Mixed Integer Optimization

Stefan Clarke, Bartolomeo Stellato

TL;DR

The paper tackles the inefficiency of proving optimality in mixed-integer programming by learning when to terminate solvers early. It trains a neural predictor to estimate the true optimality gap from solver state and uses conformal prediction to calibrate a stopping threshold, guaranteeing that near-optimal solutions are returned with probability at least $1-oldsymbol{lpha}$ for a specified tolerance $\epsilon$. Theoretical results bound expected suboptimality and runtime and ensure a high success probability on carried-out instances, conditioned on calibration data. Empirical evaluation on distributional MIPLIB families shows speedups exceeding $60\%$ while maintaining $0.1\%$-optimality with $95\%$ probability, highlighting practical impact for time-critical optimization tasks.

Abstract

Mixed-integer optimization solvers often find optimal solutions early in the search, yet spend the majority of computation time proving optimality. We exploit this by learning when to terminate solvers early on distributions of similar problem instances. Our method trains a neural network to estimate the true optimality gap from the solver state, then uses conformal prediction to calibrate a stopping threshold with rigorous probabilistic guarantees on solution quality. On five problem families from the distributional MIPLIB library, our method reduces solve time by over 60% while guaranteeing 0.1%- optimal solutions with 95% probability

Conformal Prediction for Early Stopping in Mixed Integer Optimization

TL;DR

The paper tackles the inefficiency of proving optimality in mixed-integer programming by learning when to terminate solvers early. It trains a neural predictor to estimate the true optimality gap from solver state and uses conformal prediction to calibrate a stopping threshold, guaranteeing that near-optimal solutions are returned with probability at least for a specified tolerance . Theoretical results bound expected suboptimality and runtime and ensure a high success probability on carried-out instances, conditioned on calibration data. Empirical evaluation on distributional MIPLIB families shows speedups exceeding while maintaining -optimality with probability, highlighting practical impact for time-critical optimization tasks.

Abstract

Mixed-integer optimization solvers often find optimal solutions early in the search, yet spend the majority of computation time proving optimality. We exploit this by learning when to terminate solvers early on distributions of similar problem instances. Our method trains a neural network to estimate the true optimality gap from the solver state, then uses conformal prediction to calibrate a stopping threshold with rigorous probabilistic guarantees on solution quality. On five problem families from the distributional MIPLIB library, our method reduces solve time by over 60% while guaranteeing 0.1%- optimal solutions with 95% probability
Paper Structure (31 sections, 4 theorems, 38 equations, 4 figures, 2 tables)

This paper contains 31 sections, 4 theorems, 38 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related works
  3. Learning to accelerate mixed-integer optimization
  4. Conformal prediction
  5. Parametric mixed integer optimization
  6. Parametric mixed-integer programs
  7. Solver behavior over time
  8. Standard termination criteria
  9. Probabilistic termination criteria
  10. Learning termination criteria
  11. Conformal prediction for termination
  12. Training the predictor
  13. Method overview
  14. Theoretical guarantees
  15. Expected suboptimality and runtime
  16. ...and 16 more sections

Key Result

Theorem 4.1

Let $c \in \mathbf{Z}_+$ and $n \in \{1, \dots, c\}$. Let $\theta_1, \dots, \theta_c, \theta_{c+1} \sim \vartheta$ be independent. Let Then $\mathbf{P}[g_{\theta_{c+1}}(\hat{\tau}_{\kappa}(\theta_{c+1})) \le \epsilon] \ge n/(c + 1)$.

Figures (4)

  • Figure 1: Bounds for one solve of one instance of the distributional MIPLIB optimal transmission switching problem family Huang2024DistributionalMA. The optimal solution is found after about 57 seconds, but the solve does not terminate until nearly 120 seconds have passed because the gap is not within the tolerance. Our method with a learned lower-bound terminates the solve much earlier, around 57 seconds. The upper-bound remains large until around 20 seconds, after which it quickly decreases as the solver finds feasible solutions.
  • Figure 2: Diagram of the proposed conformal prediction method for accelarating MIP solvers by learning to terminate early. The LSTM predicts the suboptimality-gap and the solver terminates and returns the current best solution when the predicted gap is below a threshold. The threshold is chosen by conformal prediction.
  • Figure 3: Suboptimality, solve-time, and number of nodes explored at termination across all problem families. The dashed line in the top panel marks the target suboptimality $\epsilon = 0.1\%$; values are clipped to $10^{-6}$ when an optimal solution is found. Our methods (shaded grey) achieve substantially faster solve-times than the global solvers Gurobi (GRB) and COPT while maintaining low suboptimality.
  • Figure 4: The number of instances solved to within $2\epsilon = 0.2\%$-optimality over time for each method on each dataset. Since $\epsilon$ was the conformal target tolerance, we expect almost all problems from each method to be solved to this level of optimality. The conformal prediction method solves most problems very quickly, but it is not guaranteed to solve all problems to within $0.2\%$-optimality due to the adjustment made by Theorem \ref{['thm:conformal']}, which occurs with probability around $\alpha$.

Theorems & Definitions (8)

  • Theorem 4.1
  • Lemma 4.2
  • proof
  • proof : Proof of Theorem \ref{['thm:conformal']}
  • Theorem 5.1
  • proof : Proof of Theorem \ref{['thm:expect']}
  • Theorem 5.2
  • proof