**2** Publications

- Page
**1**of**1**

Comput Optim Appl 2021 12;78(3):705-740. Epub 2021 Jan 12.

Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, Via Vetoio - Loc. Coppito, 67010 L'Aquila, Italy.

For finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite-dimensional problems is less understood, particularly for nonconvex objectives. This paper presents convergence results for the stochastic proximal gradient method applied to Hilbert spaces, motivated by optimization problems with partial differential equation (PDE) constraints with random inputs and coefficients. We study stochastic algorithms for nonconvex and nonsmooth problems, where the nonsmooth part is convex and the nonconvex part is the expectation, which is assumed to have a Lipschitz continuous gradient. The optimization variable is an element of a Hilbert space. We show almost sure convergence of strong limit points of the random sequence generated by the algorithm to stationary points. We demonstrate the stochastic proximal gradient algorithm on a tracking-type functional with a -penalty term constrained by a semilinear PDE and box constraints, where input terms and coefficients are subject to uncertainty. We verify conditions for ensuring convergence of the algorithm and show a simulation.

## Download full-text PDF |
Source |
---|---|

http://dx.doi.org/10.1007/s10589-020-00259-y | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7907055 | PMC |

January 2021

Comput Optim Appl 2018 6;69(2):403-422. Epub 2017 Oct 6.

2Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Vienna, Austria.

This paper considers a linear-quadratic optimal control problem where the control function appears linearly and takes values in a hypercube. It is assumed that the optimal controls are of purely bang-bang type and that the switching function, associated with the problem, exhibits a suitable growth around its zeros. The authors introduce a scheme for the discretization of the problem that doubles the rate of convergence of the Euler's scheme. The proof of the accuracy estimate employs some recently obtained results concerning the stability of the optimal solutions with respect to disturbances.

## Download full-text PDF |
Source |
---|---|

http://dx.doi.org/10.1007/s10589-017-9948-z | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6566299 | PMC |

October 2017

-->