The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. Nowadays this class of optimization problems is strongly utilized in engineering, biology and finance. In this paper, a stochastic gradient-based method is proposed for the numerical resolution of a nonconvex stochastic optimization problem on a Hilbert space. We show that, under suitable assumptions, strong or weak accumulation points of the iterates produced by the method converge almost surely to stationary points of the original optimization problem. The proof is based on classical results, such as the theorem by Robbins and Siegmund and the theory of stochastic approximation. The novelty of our contribution lies in the convergence analysis extended to some nonconvex infinite dimensional optimization problems. To conclude, the application to an optimal control problem for a class of elliptic semilinear partial differential equations (PDEs) under uncertainty will be addressed in detail.
A stochastic gradient method for a class of nonlinear PDE-constrained optimal control problems under uncertainty
Teresa Scarinci
2021-01-01
Abstract
The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. Nowadays this class of optimization problems is strongly utilized in engineering, biology and finance. In this paper, a stochastic gradient-based method is proposed for the numerical resolution of a nonconvex stochastic optimization problem on a Hilbert space. We show that, under suitable assumptions, strong or weak accumulation points of the iterates produced by the method converge almost surely to stationary points of the original optimization problem. The proof is based on classical results, such as the theorem by Robbins and Siegmund and the theory of stochastic approximation. The novelty of our contribution lies in the convergence analysis extended to some nonconvex infinite dimensional optimization problems. To conclude, the application to an optimal control problem for a class of elliptic semilinear partial differential equations (PDEs) under uncertainty will be addressed in detail.File | Dimensione | Formato | |
---|---|---|---|
2108.11782.pdf
accesso aperto
Tipologia:
Documento in Pre-print
Licenza:
Dominio pubblico
Dimensione
270.67 kB
Formato
Adobe PDF
|
270.67 kB | Adobe PDF | Visualizza/Apri |
Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.