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Compact Optimality Verification for Optimization Proxies

Wenbo Chen, Haoruo Zhao, Mathieu Tanneau, Pascal Van Hentenryck

TL;DR

The paper tackles the problem of verifying the worst-case optimality gap of optimization proxies over a distribution of instances. It introduces a compact, exact optimality-verification formulation that applies to non-convex problems, along with a gradient-based primal heuristic (PGA-VFA) that leverages a conservative value-function approximation. The approach is demonstrated on large-scale DC-OPF and knapsack proxies, with new MILP encodings for the feasibility layers and extensive experiments showing substantial computational gains and improved solution quality. This work enhances the reliability of optimization proxies in critical applications such as power systems and supply chains, by providing rigorous worst-case guarantees and scalable verification tools.

Abstract

Recent years have witnessed increasing interest in optimization proxies, i.e., machine learning models that approximate the input-output mapping of parametric optimization problems and return near-optimal feasible solutions. Following recent work by (Nellikkath & Chatzivasileiadis, 2021), this paper reconsiders the optimality verification problem for optimization proxies, i.e., the determination of the worst-case optimality gap over the instance distribution. The paper proposes a compact formulation for optimality verification and a gradient-based primal heuristic that brings substantial computational benefits to the original formulation. The compact formulation is also more general and applies to non-convex optimization problems. The benefits of the compact formulation are demonstrated on large-scale DC Optimal Power Flow and knapsack problems.

Compact Optimality Verification for Optimization Proxies

TL;DR

The paper tackles the problem of verifying the worst-case optimality gap of optimization proxies over a distribution of instances. It introduces a compact, exact optimality-verification formulation that applies to non-convex problems, along with a gradient-based primal heuristic (PGA-VFA) that leverages a conservative value-function approximation. The approach is demonstrated on large-scale DC-OPF and knapsack proxies, with new MILP encodings for the feasibility layers and extensive experiments showing substantial computational gains and improved solution quality. This work enhances the reliability of optimization proxies in critical applications such as power systems and supply chains, by providing rigorous worst-case guarantees and scalable verification tools.

Abstract

Recent years have witnessed increasing interest in optimization proxies, i.e., machine learning models that approximate the input-output mapping of parametric optimization problems and return near-optimal feasible solutions. Following recent work by (Nellikkath & Chatzivasileiadis, 2021), this paper reconsiders the optimality verification problem for optimization proxies, i.e., the determination of the worst-case optimality gap over the instance distribution. The paper proposes a compact formulation for optimality verification and a gradient-based primal heuristic that brings substantial computational benefits to the original formulation. The compact formulation is also more general and applies to non-convex optimization problems. The benefits of the compact formulation are demonstrated on large-scale DC Optimal Power Flow and knapsack problems.
Paper Structure (33 sections, 2 theorems, 26 equations, 7 figures, 7 tables, 2 algorithms)

This paper contains 33 sections, 2 theorems, 26 equations, 7 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Parametric Optimization
  5. Optimization Proxies
  6. Optimality Verification
  7. Compact Optimality Verification
  8. Projected Gradient Attack
  9. DC Optimal Power Flow
  10. DC-OPF Formulation
  11. Compact Optimality Verification
  12. Empirical Evaluation
  13. Experiment Setup
  14. Numerical Results
  15. Effectiveness of Compact Formulation
  16. ...and 18 more sections

Key Result

Theorem 4.1

Problems eq:opt_verification:bilevel and eq:optimality_verification_compact have same optimum.

Figures (7)

  • Figure 1: Optimization Proxies
  • Figure 2: Value Function and its Piece-wise Linear Approximation
  • Figure 3: Optimization Proxies for DCOPF
  • Figure 4: Evolution of primal objective value for Compact and Bilevel formulations over time on the 300 IEEE system (top row) and the 1354 Pegase system (bottom row) with different sizes of input domains. Higher values indicate better outcomes.
  • Figure 5: An Optimization Proxy for the Knapsack Problem.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Theorem 4.1
  • proof
  • Theorem 5.1