  Optimization theory has been widely studied in academia and finds a large
variety of applications in industry. The different optimization models in their
discrete and/or continuous settings has catered to a rich source of research
problems. Robust convex optimization is a branch of optimization theory in
which the variables or parameters involved have a certain level of uncertainty.
In this work, we consider the online robust optimization meta-algorithm by
Ben-Tal et al. and show that for a large range of stochastic subgradients, this
algorithm has the same guarantee as the original non-stochastic version. We
develop a quantum version of this algorithm and show that an at most quadratic
improvement in terms of the dimension can be achieved. The speedup is due to
the use of quantum state preparation, quantum norm estimation, and quantum
multi-sampling. We apply our quantum meta-algorithm to examples such as robust
linear programs and robust semidefinite programs and give applications of these
robust optimization problems in finance and engineering.
