A Gradually Reinforced Sample-Average-Approximation Differentiable Homotopy Method for a System of Stochastic Equations
arxiv(2024)
摘要
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.
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