Robust Conic Satisficing

Abstract: In practical optimization problems, we typically model uncertainty as a random variable though its true probability distribution is unobservable to the decision maker. Historical data provides some information of this distribution that we can use to approximately quantify the risk that depends on both the decision and the uncertainty. This empirical optimization approach is vulnerable to the issues of overfitting, which could be overcome by several data-driven robust optimization techniques. To tackle overfitting, Long et.al.(2022) propose a robust satisficing model, which is specified by a performance target and a penalty function that measures the deviation of the uncertainty from its nominal value, and yields solutions with superior out-of-sample performance. We generalize the robust satisficing framework to conic optimization problems with recourse, which has broad applications in predictive and prescriptive analytics. We derive an exact semidefinite optimization formulation for a biconvex quadratic evaluation function, with quadratic penalty and ellipsoidal support set. More importantly, under complete and bounded recourse, and a reasonably chosen polyhedral support set and penalty function, we propose safe approximations that are feasible for any reasonably chosen target. We then demonstrate that the assumption of complete and bounded recourse is not unimpeachable, and then introduce a novel perspective casting technique to derive an equivalent conic optimization problem satisfying the stated assumptions. Computationally, we showcase a study on data-driven portfolio optimization and demonstrate that the robust satisficing solutions can provide significant improvements over the solutions obtained by stochastic optimization models, including the celebrated Markowitz model, which is prone to overfitting.