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Data-driven Policies For Two-stage Stochastic Linear Programs

Chhavi Sharma, Harsha Gangammanavar

Abstract

A stochastic program typically involves several parameters, including deterministic first-stage parameters and stochastic second-stage elements that serve as input data. These programs are re-solved whenever any input parameter changes. However, in practical applications, quick decision-making is necessary, and solving a stochastic program from scratch for every change in input data can be computationally costly. This work addresses this challenge for two-stage stochastic linear programs (2-SLPs) with varying right-hand sides for the first-stage constraints. We construct a Piecewise Linear Difference-of-Convex (PLDC) policy by leveraging optimal bases from previous solves. This PLDC policy retains optimal solutions for previously encountered parameters and provides high-quality solutions for new right-hand-side vectors. Our proposed policy directly applies to the extensive form of the 2-SLP. When stage decomposition algorithms, such as the L-Shaped and Stochastic Decomposition, are applied to solve the 2-SLPs, we develop L-Shaped- and Stochastic-Decomposition-guided static procedures to train the policy. We also develop a sequential procedure that iteratively tracks the quality of the learned policy and incorporates new basis information to improve it. We assess the performance of our policy through analytical and numerical techniques. Our compelling experimental results show that the policy prescribes solutions that are feasible and optimal for a significant percentage of new instances.

Data-driven Policies For Two-stage Stochastic Linear Programs

Abstract

A stochastic program typically involves several parameters, including deterministic first-stage parameters and stochastic second-stage elements that serve as input data. These programs are re-solved whenever any input parameter changes. However, in practical applications, quick decision-making is necessary, and solving a stochastic program from scratch for every change in input data can be computationally costly. This work addresses this challenge for two-stage stochastic linear programs (2-SLPs) with varying right-hand sides for the first-stage constraints. We construct a Piecewise Linear Difference-of-Convex (PLDC) policy by leveraging optimal bases from previous solves. This PLDC policy retains optimal solutions for previously encountered parameters and provides high-quality solutions for new right-hand-side vectors. Our proposed policy directly applies to the extensive form of the 2-SLP. When stage decomposition algorithms, such as the L-Shaped and Stochastic Decomposition, are applied to solve the 2-SLPs, we develop L-Shaped- and Stochastic-Decomposition-guided static procedures to train the policy. We also develop a sequential procedure that iteratively tracks the quality of the learned policy and incorporates new basis information to improve it. We assess the performance of our policy through analytical and numerical techniques. Our compelling experimental results show that the policy prescribes solutions that are feasible and optimal for a significant percentage of new instances.
Paper Structure (29 sections, 3 theorems, 39 equations, 5 figures, 8 tables, 2 algorithms)

This paper contains 29 sections, 3 theorems, 39 equations, 5 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Relations to Parametric Optimization and Decision Rules
  3. Contributions
  4. A Motivating Example
  5. Notations and Background
  6. Policy Design
  7. Data-driven Policy Approximation
  8. PLDC Policy for Two-Stage Stochastic Linear Programming
  9. Description of Policy Design using L-Shaped
  10. Creating Cells
  11. PLDC Policy
  12. Description of Policy Design using Stochastic Decomposition
  13. Creating Cells
  14. Sequential Procedure for Policy Design
  15. Evaluating L-Shaped-Guided Policy
  16. ...and 14 more sections

Key Result

Lemma 2.1

Let $\phi(b) = \phi_1(b) - \phi_2(b)$ be a continuous piecewise-linear function, and let the right-hand side set $\widehat{\mathcal{B}}$ be partitioned into $L$ cells. For every cell ${\mathcal{C}}^\ell$, there exists $u^{\ell} = [u^\ell_1;\ldots;u_{d_x}^\ell] \in \mathbb{R}^{d_x \times m_1}$, $v^{\

Figures (5)

  • Figure 1: (Color online) Approximate cells for an LP problem with two constraints and five variables. $b_1$ and $b_2$ denote the right-hand-side values of the first and second constraints, respectively.
  • Figure 2: (Color online) Stage decomposition method: L-Shaped, Instance name: PGP2, Feasibility tolerance: $10^{-6}$, Optimality tolerance: $0.001$, CI tolerance: $10^{-4}$.
  • Figure 3: (Color online) Stage Decomposition Method: SD, Instance name: PGP2, Feasibility tolerance: $10^{-6}$, Relative mean value difference tolerance: $10^{-8}$, CI tolerance: $10^{-4}$.
  • Figure 4: Stage decomposition method: L-Shaped, Instance name: CEP, Feasibility tolerance: $10^{-6}$, Optimality tolerance: $0.001$, CI tolerance: $10^{-4}$.
  • Figure 5: Stage Decomposition Method: SD, Instance name: CEP, Feasibility tolerance: $10^{-6}$, Relative mean value difference tolerance: $10^{-8}$, CI tolerance: $10^{-4}$.

Theorems & Definitions (5)

  • Lemma 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2