Regularized robust optimization: the optimal portfolio execution case

An uncertainty set is a crucial component in robust optimization. Unfortunately, it is often unclear how to specify it precisely. Thus it is important to study sensitivity of the robust solution to variations in the uncertainty set, and to develop a method which improves stability of the robust solution. In this paper, to address these issues, we focus on uncertainty in the price impact parameters in an optimal portfolio execution problem. We first illustrate that a small variation in the uncertainty set may result in a large change in the robust solution. We then propose a regularized robust optimization formulation which yields a solution with a better stability property than the classical robust solution. In this approach, the uncertainty set is regularized through a regularization constraint, defined by a linear matrix inequality using the Hessian of the objective function and a regularization parameter. The regularized The authors would like to thank anonymous referees whose comments have improved the presentation of this paper. T.F. Coleman acknowledges funding from the Ophelia Lazaridis University Research Chair (which he holds) and the National Sciences and Engineering Research Council of Canada. The views expressed herein are solely from the authors. Y. Li acknowledges funding from Credit Suisse and the National Sciences and Engineering Research Council of Canada. S. Moazeni ( ) Department of Operations Research and Financial Engineering, Princeton University, Sherrerd Hall, Charlton Street, Princeton, NJ 08544, USA e-mail: [email protected] T.F. Coleman Department of Combinatorics and Optimization, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada e-mail: [email protected] Y. Li David R. Cheriton School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada e-mail: [email protected]