Online Quadratically Constrained Convex Optimization with Applications to Risk Adjusted Portfolio Selection

While online convex optimization has emerged as a powerful large scale optimization approach, much of existing literature assumes a simple way to project onto a given feasible set. The assumption is often not true, and the projection step usually becomes the key computational bottleneck. Motivated by applications in risk-adjusted portfolio selection, in this paper we consider online quadratically constrained convex optimization problems, where the feasible set involves intersections of ellipsoids. We show that regret guarantees for the online problem can be achieved by solving a suitable quadratically constrained quadratic program (QCQP) at each step, and present an efficient algorithm for solving QCQPs based on the alternating directions method. We then specialize the general framework to risk adjusted portfolio selection. Through extensive experiments on two real world stock datasets, our proposed algorithm RAMP is shown to significantly outperform existing approaches at any given risk level and match the performance of the best heuristics which do not accommodate risk constraints.