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Dynamic Basis Function Generation for Network Revenue Management

Daniel Adelman, Christiane Barz, Alba V. Olivares-Nadal

TL;DR

This work develops an iterative, basis-function–driven ADP framework for Network Revenue Management, introducing the Nonlinear Incremental Algorithm (NLIAlg) that incrementally adds and optimizes nonlinear basis functions to approximate the value function. To tackle large instances, it proposes a Two-Phase Incremental Algorithm (2PIAlg) that first uses flow-balance principles to estimate fixed basis-function parameters (Phase I) and then solves a linearized master problem to optimize remaining parameters (Phase II), with a heuristic variant (H-2PIAlg) enabling add-on mode on very large problems. A key theoretical insight extends flow-balance concepts to a stochastic NRM setting, guiding new basis-function generation via imbalance maximization and contraction of the nonlinear search space through norm-constrained exponential ridge bases. Computational results show that H-2PIAlg substantially improves policies and upper bounds over benchmarks like AA, SPLA, and NSEP, particularly when capacity is scarce or memory limits hinder alternative methods. The framework also offers a flexible stand-alone or add-on workflow, providing scalable, progressively better value-function approximations for high-dimensional NRM problems.

Abstract

This paper introduces an algorithm that dynamically generates basis functions to approximate the value function in Network Revenue Management. Unlike existing algorithms sampling the parameters of new basis functions, this Nonlinear Incremental Algorithm (NLIAlg) iteratively refines the value function approximation by optimizing these parameters. For larger instances, the Two-Phase Incremental Algorithm (2PIAlg) modifies NLIAlg to leverage the efficiency of LP solvers. It reduces the size of a large-dimensional nonlinear problem and transforms it into an LP by fixing the basis function parameters, which are then optimized in a second phase using the flow imbalance ideas from Adelman and Klabjan (2012). This marks the first application of these techniques in a stochastic setting. The algorithms can operate in two modes: (1) Standalone mode, to construct a value function approximation from scratch, and (2) Add-on mode, to refine an existing approximation. Our numerical experiments indicate that while NLIAlg and 2PIAlg in standalone mode are only feasible for small-scale problems, the heuristic version of 2PIAlg (H-2PIAlg) in add-on mode, using the Affine Approximation and exponential ridge basis functions, can handle extremely large instances that may cause benchmark network revenue management methods to run out of memory. In these scenarios, H-2PIAlg delivers substantially better policies and upper bounds than the Affine Approximation. Furthermore, H-2PIAlg achieves higher average revenues in policy simulations compared to network revenue management benchmarks in instances with limited capacity.

Dynamic Basis Function Generation for Network Revenue Management

TL;DR

This work develops an iterative, basis-function–driven ADP framework for Network Revenue Management, introducing the Nonlinear Incremental Algorithm (NLIAlg) that incrementally adds and optimizes nonlinear basis functions to approximate the value function. To tackle large instances, it proposes a Two-Phase Incremental Algorithm (2PIAlg) that first uses flow-balance principles to estimate fixed basis-function parameters (Phase I) and then solves a linearized master problem to optimize remaining parameters (Phase II), with a heuristic variant (H-2PIAlg) enabling add-on mode on very large problems. A key theoretical insight extends flow-balance concepts to a stochastic NRM setting, guiding new basis-function generation via imbalance maximization and contraction of the nonlinear search space through norm-constrained exponential ridge bases. Computational results show that H-2PIAlg substantially improves policies and upper bounds over benchmarks like AA, SPLA, and NSEP, particularly when capacity is scarce or memory limits hinder alternative methods. The framework also offers a flexible stand-alone or add-on workflow, providing scalable, progressively better value-function approximations for high-dimensional NRM problems.

Abstract

This paper introduces an algorithm that dynamically generates basis functions to approximate the value function in Network Revenue Management. Unlike existing algorithms sampling the parameters of new basis functions, this Nonlinear Incremental Algorithm (NLIAlg) iteratively refines the value function approximation by optimizing these parameters. For larger instances, the Two-Phase Incremental Algorithm (2PIAlg) modifies NLIAlg to leverage the efficiency of LP solvers. It reduces the size of a large-dimensional nonlinear problem and transforms it into an LP by fixing the basis function parameters, which are then optimized in a second phase using the flow imbalance ideas from Adelman and Klabjan (2012). This marks the first application of these techniques in a stochastic setting. The algorithms can operate in two modes: (1) Standalone mode, to construct a value function approximation from scratch, and (2) Add-on mode, to refine an existing approximation. Our numerical experiments indicate that while NLIAlg and 2PIAlg in standalone mode are only feasible for small-scale problems, the heuristic version of 2PIAlg (H-2PIAlg) in add-on mode, using the Affine Approximation and exponential ridge basis functions, can handle extremely large instances that may cause benchmark network revenue management methods to run out of memory. In these scenarios, H-2PIAlg delivers substantially better policies and upper bounds than the Affine Approximation. Furthermore, H-2PIAlg achieves higher average revenues in policy simulations compared to network revenue management benchmarks in instances with limited capacity.

Paper Structure

This paper contains 43 sections, 6 theorems, 38 equations, 3 figures, 7 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Literature Survey
  3. Basis Functions in Approximate Linear Programming
  4. Solution Methods for Approximate Linear Programming
  5. Iterative Approximation Refinement in ADP for NRM
  6. The NRM Problem
  7. The Value Function Approximation
  8. The Approximate LP
  9. Using Flow-Balance Constraints to Refine the Value Function Approximation
  10. The Nonlinear Incremental Algorithm (NLIAlg)
  11. Solving \ref{['AP']} for Given $K$ via Row Generation
  12. Determining Whether to Increase the Number of Basis Functions $K$
  13. A Two-Phase Approach to Reduce the Dimensionality of Nonlinear \ref{['AP']}
  14. The Heuristic Two-Phase Algorithm (H-2PIAlg)
  15. Choosing the Set of Basis Functions
  16. ...and 28 more sections

Key Result

Proposition 1

If there exists a unique optimal solution to D, adding a new basis function $\phi(\vec{x},\vec{\beta}_{K+1})$ with $|\ell_t(\vec{\beta}_{K+1})|>0$ for some $t\in \{1,...,\tau\}$ to the nonlinear approximation NLA reduces the upper bound given by linearizedAP.

Figures (3)

  • Figure 1: Approximated value function through linear and exponential basis functions
  • Figure 2: Approximated value function through linear and exponential basis functions (NLIAlg) in the toy example introduced in Section \ref{['sec:hat_vs_exponential']}
  • Figure 3: Estimates of the upper bound $\hat{Z}_\phi=Z_{\mathcal{B}}+\sum_t \hat{\pi}_t$ and average revenue $\overline{R}$ from simulated policies for different baseline approximations vs. solving CPU times.

Theorems & Definitions (6)

  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Lemma 1