Robust Conic Satisficing

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 of an evaluation function that depends on both our 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 it yields solutions with superior out-of-sample performance. We generalize the robust satisficing framework to conic optimization problems with recourse, which has broader applications in predictive and prescriptive analytics including risk management, statistical supervised learning, among others. We derive an exact semidefinite optimization formulation for biconvex quadratic evaluation function, with quadratic penalty and ellipsoidal support set. More importantly, under complete and bounded recourse, and reasonably chosen polyhedral support set and penalty function, we propose safe approximations that do not lead to infeasible problems for any reasonably chosen target. We demonstrate that the assumption of complete and bounded recourse is however not unimpeachable, and then introduce a novel perspective casting technique to derive an equivalent conic optimization problem that satisfies the stated assumptions. Finally, we showcase the computational study on data-driven portfolio optimization that maximizes the expected exponential utility of the portfolio returns. We demonstrate that the solutions obtained via robust satisficing can significantly improve over the solutions obtained by stochastic optimization models, including the celebrated Markowitz model, which is prone to overfitting.



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