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

TL;DR

This paper introduces a novel gradually reinforced SAA differentiable homotopy method that efficiently solves stochastic equations by balancing accuracy and computational cost through a smooth solution path.

Contribution

It develops a new homotopy-based approach integrating gradual reinforcement in SAA to improve efficiency and convergence in solving stochastic equations.

Findings

Method reduces computational cost compared to traditional SAA.

Numerical experiments demonstrate high effectiveness and efficiency.

The approach ensures a smooth solution path with global convergence.

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.