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Pessimistic bilevel optimization approach for decision-focused learning

Diego Jiménez, Bernardo K. Pagnoncelli, Hande Yaman

TL;DR

This work proposes a pessimistic bilevel approach for solving general decision-focused formulations of combinatorial optimization problems and benchmarks its performance on the 0-1 knapsack problem against estimate-then-optimize and decision-focused methods, including the popular SPO+ approach.

Abstract

The recent interest in contextual optimization problems, where randomness is associated with side information, has led to two primary strategies for formulation and solution. The first, estimate-then-optimize, separates the estimation of the problem's parameters from the optimization process. The second, decision-focused optimization, integrates the optimization problem's structure directly into the prediction procedure. In this work, we propose a pessimistic bilevel approach for solving general decision-focused formulations of combinatorial optimization problems. Our method solves an $\varepsilon$-approximation of the pessimistic bilevel problem using a specialized cut generation algorithm. We benchmark its performance on the 0-1 knapsack problem against estimate-then-optimize and decision-focused methods, including the popular SPO+ approach. Computational experiments highlight the proposed method's advantages, particularly in reducing out-of-sample regret.

Pessimistic bilevel optimization approach for decision-focused learning

TL;DR

This work proposes a pessimistic bilevel approach for solving general decision-focused formulations of combinatorial optimization problems and benchmarks its performance on the 0-1 knapsack problem against estimate-then-optimize and decision-focused methods, including the popular SPO+ approach.

Abstract

The recent interest in contextual optimization problems, where randomness is associated with side information, has led to two primary strategies for formulation and solution. The first, estimate-then-optimize, separates the estimation of the problem's parameters from the optimization process. The second, decision-focused optimization, integrates the optimization problem's structure directly into the prediction procedure. In this work, we propose a pessimistic bilevel approach for solving general decision-focused formulations of combinatorial optimization problems. Our method solves an -approximation of the pessimistic bilevel problem using a specialized cut generation algorithm. We benchmark its performance on the 0-1 knapsack problem against estimate-then-optimize and decision-focused methods, including the popular SPO+ approach. Computational experiments highlight the proposed method's advantages, particularly in reducing out-of-sample regret.

Paper Structure

This paper contains 20 sections, 19 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical framework and formulations
  3. Preliminaries and notation
  4. Bilevel integrated estimation-optimization
  5. Pessimistic bilevel IEO formulation
  6. Solution methodology
  7. Valid relaxations for pessimistic IEO
  8. Column-and-constraint generation algorithm for pessimistic IEO
  9. Branch--and--cut algorithm for pessimistic IEO
  10. Additional settings
  11. Linearization of $f_{\mathbf{w}}(\mathbf{x}_i)^{\intercal} \bar{\mathbf{z}}_i$
  12. Warmstart initialization
  13. Maximum number of points allowed
  14. Generation of multiple valid cuts
  15. Computational experiments
  16. ...and 5 more sections

Figures (2)

  • Figure 1: Out--of--sample results, $n_y = 10$, $n_{test} = 1000$.
  • Figure 2: Out--of--sample results, $n_y = 20$, $n_{test} = 1000$.

Theorems & Definitions (2)

  • proof
  • proof