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Risk-Adaptive Local Decision Rules

Johannes O. Royset, Miguel A. Lejeune

TL;DR

The paper develops Risk-Adaptive Local Decision Rules for parameterized mixed-binary optimization by solving risk-measure–based training problems over flexible mappings $(F,G)$. It proves local consistency and nonasymptotic error bounds, and introduces a decomposition algorithm to solve large-scale training problems efficiently, demonstrated on a nonlinear search-theory model. The resulting decision rules (notably MDR and AMDR) provide near-optimal, feasible prescriptions across parameter sets while accommodating inexact evaluations and various risk preferences, with AMDR showing strong robustness to distributional shifts in out-of-sample tests. The framework enables sensitivity and stability analyses for broad parameterized problems and yields solver-friendly, scalable approaches for practical decision support in complex combinatorial settings.

Abstract

For parameterized mixed-binary optimization problems, we construct local decision rules that prescribe near-optimal courses of action across a set of parameter values. The decision rules stem from solving risk-adaptive training problems over classes of continuous, possibly nonlinear mappings. In asymptotic and nonasymptotic analysis, we establish that the decision rules prescribe near-optimal decisions locally for the actual problems, without relying on linearity, convexity, or smoothness. The development also accounts for practically important aspects such as inexact function evaluations, solution tolerances in training problems, regularization, and reformulations to solver-friendly models. The decision rules also furnish a means to carry out sensitivity and stability analysis for broad classes of parameterized optimization problems. We develop a decomposition algorithm for solving the resulting training problems and demonstrate its ability to generate quality decision rules on a nonlinear binary optimization model from search theory.

Risk-Adaptive Local Decision Rules

TL;DR

The paper develops Risk-Adaptive Local Decision Rules for parameterized mixed-binary optimization by solving risk-measure–based training problems over flexible mappings . It proves local consistency and nonasymptotic error bounds, and introduces a decomposition algorithm to solve large-scale training problems efficiently, demonstrated on a nonlinear search-theory model. The resulting decision rules (notably MDR and AMDR) provide near-optimal, feasible prescriptions across parameter sets while accommodating inexact evaluations and various risk preferences, with AMDR showing strong robustness to distributional shifts in out-of-sample tests. The framework enables sensitivity and stability analyses for broad parameterized problems and yields solver-friendly, scalable approaches for practical decision support in complex combinatorial settings.

Abstract

For parameterized mixed-binary optimization problems, we construct local decision rules that prescribe near-optimal courses of action across a set of parameter values. The decision rules stem from solving risk-adaptive training problems over classes of continuous, possibly nonlinear mappings. In asymptotic and nonasymptotic analysis, we establish that the decision rules prescribe near-optimal decisions locally for the actual problems, without relying on linearity, convexity, or smoothness. The development also accounts for practically important aspects such as inexact function evaluations, solution tolerances in training problems, regularization, and reformulations to solver-friendly models. The decision rules also furnish a means to carry out sensitivity and stability analysis for broad classes of parameterized optimization problems. We develop a decomposition algorithm for solving the resulting training problems and demonstrate its ability to generate quality decision rules on a nonlinear binary optimization model from search theory.
Paper Structure (15 sections, 6 theorems, 89 equations, 2 figures, 13 tables)

This paper contains 15 sections, 6 theorems, 89 equations, 2 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Framework
  3. Training Problem
  4. Implementation and Solution of Training Problems
  5. Local Consistency
  6. Nonasymptotic Error Bounds
  7. Decomposition Algorithm and Computational Results
  8. Decomposition Algorithm
  9. Examples of Convergence and Infeasibility
  10. Examples of Feasible Decision Rules
  11. Conclusions
  12. Supplementary Material
  13. Proofs
  14. Implementation Details
  15. Additional Numerical Results

Key Result

Proposition 2.1

(solution recovery). Suppose that each ${\cal R}_1, \dots, {\cal R}_q$ is the worst-case risk measure and there exists $(\bar{F}, \bar{G}) \in {\cal F} \times {\cal G}$ such that for every $\omega\in \Omega$ one has Then, the following hold:

Figures (2)

  • Figure 1: Summary of suboptimality for AMDR across training problems and instances: uu = uniform training data and uniform test data, ub = uniform training data and beta test data, etc.; "-d" indicates the dispersed target case. Left portion gives average suboptimality and right portion gives 95% quantile of suboptimality levels.
  • Figure 2: Cell numbers in discretized environment with 81 cells.

Theorems & Definitions (6)

  • Proposition 2.1
  • Theorem 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Theorem 4.1
  • Theorem 4.2