A stochastic regularized second-order iterative scheme for optimal control and inverse problems in stochastic partial
Marc Dambrine1, Akhtar A Khan2, Miguel Sama3
1Laboratoire de Mathématiques et de leurs Applications, Université de Pau et des Pays de l'Adour, E2S UPPA, CNRS, LMAP, 64013 Pau Cedex, France.
Summary
This study introduces a novel iterative method for solving stochastic optimization problems, crucial for fields like machine learning and optimal control. The new approach demonstrates reliable convergence in Hilbert spaces, enhancing problem-solving capabilities.
Area of Science:
- Optimization Theory
- Applied Mathematics
- Numerical Analysis
Background:
- Many applied models in optimal control, inverse problems, and machine learning result in stochastic optimization problems within Hilbert spaces.
- A key condition for these problems, under convexity, is a stochastic variational inequality.
Purpose of the Study:
- To present a new stochastic regularized second-order iterative scheme for solving variational inequalities in stochastic environments.
- To analyze the convergence properties of this scheme in a Hilbert space setting.
Main Methods:
- Development of a stochastic regularized second-order iterative scheme.
- Utilizing sampling techniques to access the primary operator.
- Applying stochastic approximation framework for convergence analysis.
Main Results:
- The proposed iterative scheme is shown to have almost-sure convergence in a Hilbert space.
- Demonstrated feasibility and efficacy through applications to stochastic optimal control and inverse problems governed by SPDEs.
Conclusions:
- The novel iterative scheme provides an effective tool for addressing complex stochastic optimization problems.
- The findings are relevant to various scientific and engineering disciplines relying on stochastic modeling.
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