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Optimal network pricing with oblivious users: a new model and algorithm

Yixuan Li, Andersen Ang, Sebastian Stein

TL;DR

This work mathematically derive and prove a new formulation of Onp with decision-dependent modeling that relax certain existing modeling constraints in the literature, and expresses the Onp formulation as a constrained nonconvex stochastic quadratic program with uncertainty.

Abstract

Traffic modeling is important in modern society. In this work we propose a new model on the optimal network pricing (Onp) with the assumption of oblivious users, in which the users remain oblivious to real-time traffic conditions and others' behavior. Inspired by works on transportation research and network pricing for selfish traffic, we mathematically derive and prove a new formulation of Onp with decision-dependent modeling that relax certain existing modeling constraints in the literature. Then, we express the Onp formulation as a constrained nonconvex stochastic quadratic program with uncertainty, and we propose an efficient algorithm to solve the problem, utilizing graph theory, sparse linear algebra and stochastic approximation. Lastly, we showcase the effectiveness of the proposed algorithm and the usefulness of the new Onp formulation. The proposed algorithm achieves a 5x speedup by exploiting the sparsity structure of the model.

Optimal network pricing with oblivious users: a new model and algorithm

TL;DR

This work mathematically derive and prove a new formulation of Onp with decision-dependent modeling that relax certain existing modeling constraints in the literature, and expresses the Onp formulation as a constrained nonconvex stochastic quadratic program with uncertainty.

Abstract

Traffic modeling is important in modern society. In this work we propose a new model on the optimal network pricing (Onp) with the assumption of oblivious users, in which the users remain oblivious to real-time traffic conditions and others' behavior. Inspired by works on transportation research and network pricing for selfish traffic, we mathematically derive and prove a new formulation of Onp with decision-dependent modeling that relax certain existing modeling constraints in the literature. Then, we express the Onp formulation as a constrained nonconvex stochastic quadratic program with uncertainty, and we propose an efficient algorithm to solve the problem, utilizing graph theory, sparse linear algebra and stochastic approximation. Lastly, we showcase the effectiveness of the proposed algorithm and the usefulness of the new Onp formulation. The proposed algorithm achieves a 5x speedup by exploiting the sparsity structure of the model.

Paper Structure

This paper contains 17 sections, 6 theorems, 51 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Derivation and theory of Onp
  3. Monte Carlo, Gradient and Hessian
  4. Deriving the Monte Carlo approximation.
  5. Gradient derivation and computation
  6. Hessian of and
  7. Nonconvexity and Convexity
  8. is nonconvex
  9. Condition for local convexity
  10. is convex almost surely for some
  11. Convexity of the constraint set
  12. TR-SQP algorithm
  13. Experiment
  14. Verification on a toy problem
  15. Speedup via exploiting sparsity
  16. ...and 2 more sections

Key Result

Theorem 1

Using assignment $A$, the vector $c^{\text{coe}}$, Hadamard product $\odot$ and tensor product $\otimes$, then

Figures (5)

  • Figure 1: The plot of \ref{['eqn:fNP_cnvx_eg']} for three different values of $\lambda$. The function is nonconvex globally because of the kink at $p=0$.
  • Figure 2: Left: the surface of a $c_N$ in $\mathbb{R}^2$. Right: the contour plot of $c_N$. The feasible region is a nonconvex polygon with the green boundary.
  • Figure 3: A graph. The top left corner shows the $(x,p)$ value obtained by IPM and TR-SQP of the route highlighted in red.
  • Figure 4: Experimental result on varying $| \mathcal{R}|$: the median curve over 50 random runs with error bar ($\pm$1 std). The result shows that exploiting the sparsity greatly reduces the computational time, with a speedup factor between 5x to 100x.
  • Figure 5: Transportation network of Sioux Falls, South Dakota, US. The graph has $| \mathcal{V}|=24$, $| \mathcal{E}|=76$ and $| \mathcal{R}|=3298$.

Theorems & Definitions (15)

  • Remark
  • Definition 1
  • Theorem 1
  • proof
  • Remark
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • ...and 5 more