Distributionally Robust Optimization Based on Kernel Density Estimation and Mean-Entropic Value-at-Risk

In this paper, a distributionally robust optimization model based on kernel density estimation (KDE) and mean entropic value-at-risk (EVaR) is proposed, where the ambiguity set is defined as a KDE-$\varphi$-divergence “ball” centered at the empirical distribution in the weighted KDE distribution function family, which is a finite-dimensional set. Instead of the joint probability distribution of the random vector, the one-dimensional probability distribution of the random loss function is approximated by the univariate weighted KDE for dimensionality reduction. Under the mild conditions of the kernel and $\varphi$-divergence function, the computationally tractable reformulation of the corresponding distributionally robust mean-EVaR optimization model is derived by Fenchel’s duality theory. Convergence of the optimal value and the solution set of the distributionally robust optimization problem based on KDE and mean-EVaR to those of the corresponding stochastic programming problem with the true distribution is proved. For some special cases, including portfolio selection, newsvendor problem, and linear two-stage stochastic programming problem, concrete tractable reformulations are given. Primary empirical test results for portfolio selection and project management problems show that the proposed model is promising.

Funding: This work was funded by the National Natural Science Foundation of China [Grants 11971092 and 11571061] and the Fundamental Research Funds for the Central Universities [Grants DUT15RC(3)037 and DUT18RC(4)067].