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Regularized robust optimization: the optimal portfolio execution case

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Abstract

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 robust solution is then more stable with respect to variation in the uncertainty set specification, in addition to being more robust to estimation errors in the price impact parameters. The regularized robust optimal execution strategy can be computed by an efficient method based on convex optimization. Improvement in the stability of the robust solution is analyzed. We also study implications of the regularization on the optimal execution strategy and its corresponding execution cost. Through the regularization parameter, one can adjust the level of conservatism of the robust solution.

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Notes

  1. The second summation of the objective function in problem (9), at k=1, yields the term \(\frac{1}{\tau}\bar{S}^{T}\tilde{H}\bar{S}\).

  2. The units for H and G are $ per share2 and $ per day per share2, respectively.

  3. Note that the problem of minimizing a non-convex quadratic function is known to be NP-hard [45].

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Authors and Affiliations

  1. Department of Operations Research and Financial Engineering, Princeton University, Sherrerd Hall, Charlton Street, Princeton, NJ, 08544, USA

    Somayeh Moazeni

  2. Department of Combinatorics and Optimization, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada

    Thomas F. Coleman

  3. David R. Cheriton School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada

    Yuying Li

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  1. Somayeh Moazeni
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  2. Thomas F. Coleman
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  3. Yuying Li
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Correspondence to Somayeh Moazeni.

Additional information

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.

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Moazeni, S., Coleman, T.F. & Li, Y. Regularized robust optimization: the optimal portfolio execution case. Comput Optim Appl 55, 341–377 (2013). https://doi.org/10.1007/s10589-012-9526-3

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