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Necessary Optimality Conditions for Integrated Learning and Optimization Problem in Contextual Optimization

Yuan Tao, Huifu Xu

TL;DR

This work derives first-order necessary optimality conditions for integrated learning and optimization (ILO), a two-stage stochastic bilevel framework that jointly learns a predictive distribution and optimizes downstream decisions. It develops convex-case conditions using Mordukhovich coderivatives for a VI-based reformulation and nonconvex-case conditions via a value-function approach under stochastic partial calmness, with measurability and selection results to support sample-based analysis. The theory is instantiated in practical SPO settings, including portfolio optimization and newsvendor problems, and demonstrates how explicit coderivative expressions can be obtained for polyhedral and orthant feasible sets. The results enable gradient-based algorithms for ILO and provide a principled way to incorporate predictive models (linear, Gaussian mixtures, kernel regression) into decision problems with uncertainty.

Abstract

Integrated learning and optimization (ILO) is a framework in contextual optimization which aims to train a predictive model for the probability distribution of the underlying problem data uncertainty, with the goal of enhancing the quality of downstream decisions. This framework represents a new class of stochastic bilevel programs, which are extensively utilized in the literature of operations research and management science, yet remain underexplored from the perspective of optimization theory. In this paper, we fill the gap. Specifically, we derive the first-order necessary optimality conditions in terms of Mordukhovich limiting subdifferentials. To this end, we formulate the bilevel program as a two-stage stochastic program with variational inequality constraints when the lower-level decision-making problem is convex, and establish an optimality condition via sensitivity analysis of the second-stage value function. In the case where the lower level optimization problem is nonconvex, we adopt the value function approach in the literature of bilevel programs and derive the first-order necessary conditions under stochastic partial calmness conditions. The derived optimality conditions are applied to several existing ILO problems in the literature. These conditions may be used for the design of gradient-based algorithms for solving ILO problems.

Necessary Optimality Conditions for Integrated Learning and Optimization Problem in Contextual Optimization

TL;DR

This work derives first-order necessary optimality conditions for integrated learning and optimization (ILO), a two-stage stochastic bilevel framework that jointly learns a predictive distribution and optimizes downstream decisions. It develops convex-case conditions using Mordukhovich coderivatives for a VI-based reformulation and nonconvex-case conditions via a value-function approach under stochastic partial calmness, with measurability and selection results to support sample-based analysis. The theory is instantiated in practical SPO settings, including portfolio optimization and newsvendor problems, and demonstrates how explicit coderivative expressions can be obtained for polyhedral and orthant feasible sets. The results enable gradient-based algorithms for ILO and provide a principled way to incorporate predictive models (linear, Gaussian mixtures, kernel regression) into decision problems with uncertainty.

Abstract

Integrated learning and optimization (ILO) is a framework in contextual optimization which aims to train a predictive model for the probability distribution of the underlying problem data uncertainty, with the goal of enhancing the quality of downstream decisions. This framework represents a new class of stochastic bilevel programs, which are extensively utilized in the literature of operations research and management science, yet remain underexplored from the perspective of optimization theory. In this paper, we fill the gap. Specifically, we derive the first-order necessary optimality conditions in terms of Mordukhovich limiting subdifferentials. To this end, we formulate the bilevel program as a two-stage stochastic program with variational inequality constraints when the lower-level decision-making problem is convex, and establish an optimality condition via sensitivity analysis of the second-stage value function. In the case where the lower level optimization problem is nonconvex, we adopt the value function approach in the literature of bilevel programs and derive the first-order necessary conditions under stochastic partial calmness conditions. The derived optimality conditions are applied to several existing ILO problems in the literature. These conditions may be used for the design of gradient-based algorithms for solving ILO problems.
Paper Structure (18 sections, 15 theorems, 105 equations)

This paper contains 18 sections, 15 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Mordukhovich Subdifferential
  4. Variational analysis
  5. Set-valued mappings and measurability
  6. First-order Necessary Optimality Condition: Convex Case
  7. Characterization of the Mordukhovich coderivative for polyhedral feasible set
  8. First-order Necessary Optimality Condition: Nonconvex Case
  9. Uniform/stochastic partial calmness.
  10. Necessary optimality conditions under uniform/stochastic partial calmness
  11. Applications
  12. Finite-dimensional parameterized predictive models
  13. Optimality conditions
  14. The portfolio selection problem with the linear prediction model.
  15. The newsvendor problem with a kernel regression.
  16. ...and 3 more sections

Key Result

Proposition 2.3

Let $\Phi:{\cal X}\to 2^{\mathbb{R}^m}$ be uniformly compact near $\bar{x}$. Then $\Phi$ is upper semi-continuous at $\bar{x}$ if and only if $\Phi$ is closed.

Theorems & Definitions (21)

  • Definition 2.1: Coderivatives
  • Definition 2.2: Subdifferentials
  • Proposition 2.3
  • Definition 3.1: M-multipliers
  • Definition 3.2: NNAMCQ
  • Proposition 3.3
  • Proposition 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Lemma 3.7
  • ...and 11 more