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A Gradually Reinforced Sample-Average-Approximation Differentiable Homotopy Method for a System of Stochastic Equations

Peixuan Li, Chuangyin Dang, Yang Zhan

TL;DR

The GRSAA differentiable homotopy method serves as a bridge to link the gradually reinforced SAA scheme and a differentiable homotopy method and retains the nice property of global convergence the homotopy method possesses while greatly reducing the computational cost for attaining a desired solution to the original SSE.

Abstract

This paper intends to apply the sample-average-approximation (SAA) scheme to solve a system of stochastic equations (SSE), which has many applications in a variety of fields. The SAA is an effective paradigm to address risks and uncertainty in stochastic models from the perspective of Monte Carlo principle. Nonetheless, a numerical conflict arises from the sample size of SAA when one has to make a tradeoff between the accuracy of solutions and the computational cost. To alleviate this issue, we incorporate a gradually reinforced SAA scheme into a differentiable homotopy method and develop a gradually reinforced sample-average-approximation (GRSAA) differentiable homotopy method in this paper. By introducing a series of continuously differentiable functions of the homotopy parameter $t$ ranging between zero and one, we establish a differentiable homotopy system, which is able to gradually increase the sample size of SAA as $t$ descends from one to zero. The set of solutions to the homotopy system contains an everywhere smooth path, which starts from an arbitrary point and ends at a solution to the SAA with any desired accuracy. The GRSAA differentiable homotopy method serves as a bridge to link the gradually reinforced SAA scheme and a differentiable homotopy method and retains the nice property of global convergence the homotopy method possesses while greatly reducing the computational cost for attaining a desired solution to the original SSE. Several numerical experiments further confirm the effectiveness and efficiency of the proposed method.

A Gradually Reinforced Sample-Average-Approximation Differentiable Homotopy Method for a System of Stochastic Equations

TL;DR

The GRSAA differentiable homotopy method serves as a bridge to link the gradually reinforced SAA scheme and a differentiable homotopy method and retains the nice property of global convergence the homotopy method possesses while greatly reducing the computational cost for attaining a desired solution to the original SSE.

Abstract

This paper intends to apply the sample-average-approximation (SAA) scheme to solve a system of stochastic equations (SSE), which has many applications in a variety of fields. The SAA is an effective paradigm to address risks and uncertainty in stochastic models from the perspective of Monte Carlo principle. Nonetheless, a numerical conflict arises from the sample size of SAA when one has to make a tradeoff between the accuracy of solutions and the computational cost. To alleviate this issue, we incorporate a gradually reinforced SAA scheme into a differentiable homotopy method and develop a gradually reinforced sample-average-approximation (GRSAA) differentiable homotopy method in this paper. By introducing a series of continuously differentiable functions of the homotopy parameter ranging between zero and one, we establish a differentiable homotopy system, which is able to gradually increase the sample size of SAA as descends from one to zero. The set of solutions to the homotopy system contains an everywhere smooth path, which starts from an arbitrary point and ends at a solution to the SAA with any desired accuracy. The GRSAA differentiable homotopy method serves as a bridge to link the gradually reinforced SAA scheme and a differentiable homotopy method and retains the nice property of global convergence the homotopy method possesses while greatly reducing the computational cost for attaining a desired solution to the original SSE. Several numerical experiments further confirm the effectiveness and efficiency of the proposed method.
Paper Structure (13 sections, 10 theorems, 31 equations, 9 figures, 2 tables)

This paper contains 13 sections, 10 theorems, 31 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A Gradually Reinforced SAA Differentiable Homotopy Method
  3. Background of the SAA and Homotopy Methods
  4. A Gradually Reinforced SAA Differentiable Homotopy System
  5. Development of a Smooth Path
  6. Convergence Properties
  7. A GRSAA Differentiable Homotopy Path
  8. A Gradually Reinforced SAA Simplicial Homotopy Method
  9. Numerical Performance
  10. A Stochastic Market Equilibrium Problem
  11. A Stochastic System of Equations
  12. A Stochastic Variational Inequality
  13. Conclusions

Key Result

Lemma 1

$d(x,t)$ is continuously differentiable on $\mathbb{R}^n\times [0,1]$.

Figures (9)

  • Figure 1: A GRSAA Differentiable Homotopy Method
  • Figure 2: Numerical Results for the Market Equilibrium Problem
  • Figure 3: Changes of Variables in Iterations
  • Figure 4: Computational Time for GRSAA-DH and GRSAA-SH
  • Figure 5: Number of Iterations for GRSAA-DH and GRSAA-SH
  • ...and 4 more figures

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Corollary 1
  • Theorem 1
  • proof
  • Theorem 2: Browder's Fixed Point Theorem
  • Corollary 2
  • Lemma 2
  • proof
  • Theorem 3
  • ...and 8 more