The estimation of variance-covariance matrices in situations that involve the optimization of an objective function (e.g. a log-likelihood function) is usually a difficult numerical problem, since the resulting estimates should be positive semi-definite matrices. We can either use constrained optimization, or employ a parameterization that enforces this condition. We describe here five differen...

We show how to compute in a numerically efficient way the maximum risk of a portfolio, given uncertainty in the means and covariances of asset returns. This is a semidefinite programming problem, and is readily solved by interior-point methods for convex optimization developed in recent years. While not as general, this approach is more accurate and much faster than Monte Carlo methods. The com...

Reward-risk ratio optimization is an important mathematical approach in finance [54]. In this paper, we revisit the model by considering a situation where an investor does not have complete information on the distribution of the underlying uncertainty and consequently a robust action is taken against the risk arising from ambiguity of the true distribution. We propose a distributionally robust ...

Traditional portfolio optimization has been often criticized since it does not account for estimation risk. Theoretical considerations indicate that estimation risk is mainly driven by the parameter uncertainty regarding the expected asset returns rather than their variances and covariances. This is also demonstrated by several numerical studies. The global minimum variance portfolio has been a...

Classical modelling approaches in OR/MS under uncertainty assume a full probabilistic characterization. The learning needed to implement the policies derived from these models is accomplished either through (i) classical statistical estimation procedures or (ii) subjective Bayesian priors. When the data available for learning is limited, or the underlying uncertainty is non-stationary, the erro...

A novel data-driven approach for optimization under uncertainty based on multistage adaptive robust optimization (ARO) and nonparametric kernel density M-estimation is proposed. Different from conventional robust optimization methods, the proposed framework incorporates distributional information to avoid over-conservatism. Robust kernel density estimation with Hampel loss function is employed ...

The use of optimization for the propagation of mixed epistemic/aleatory uncertainties is demonstrated within the context of hypersonic flows. Specifically, this work focuses on strategies applicable for models where input parameters can be divided into a set of variables containing only aleatory uncertainties and a set with epistemic uncertainties. With the input parameters divided in this way,...

We propose a robust portfolio optimization approach based on quantile statistics. The proposed method is robust to extreme events in asset returns, and accommodates large portfolios under limited historical data. Specifically, we show that the risk of the estimated portfolio converges to the oracle optimal risk with parametric rate under weakly dependent asset returns. The theory does not rely ...