Robust and Stochastic Optimization with a Hybrid Coherent Risk Measure with an Application to Supervised Learning

This letter considers a hybrid risk measure for decision-making under uncertainties that tradeoffs between the solutions obtained from the robust optimization and the stochastic optimization techniques. In the proposed framework, the risk measure is shown to satisfy the properties of coherent risk measures. We can control the level of guaranteed robustness using a parameter. We formulate the stochastic and robust optimization problem under the proposed risk measure and show its equivalent formulation and sensitivity result. We introduce the sample approximation of our technique by combining scenario program and sample average approximation, making our framework amenable for practical usage. We present a supervised learning problem as a case study to corroborate our results and show the implications of the proposed method in machine learning.