  This paper considers a distributed adaptive optimization problem, where all
agents only have access to their local cost functions with a common unknown
parameter, whereas they mean to collaboratively estimate the true parameter and
find the optimal solution over a connected network. A general mathematical
framework for such a problem has not been studied yet. We aim to provide
valuable insights for addressing parameter uncertainty in distributed
optimization problems and simultaneously find the optimal solution. Thus, we
propose a novel Prediction while Optimization scheme, which utilizes
distributed fractional Bayesian learning through weighted averaging on the
log-beliefs to update the beliefs of unknown parameters, and distributed
gradient descent for renewing the estimation of the optimal solution. Then
under suitable assumptions, we prove that all agents' beliefs and decision
variables converge almost surely to the true parameter and the optimal solution
under the true parameter, respectively. We further establish a sublinear
convergence rate for the belief sequence. Finally, numerical experiments are
implemented to corroborate the theoretical analysis.
