Experiments in Robust Portfolio Optimization

This paper presents numerical experiments using cutting-plane algorithms to solve complex nonconvex robust optimization models arising in portfolio optimization. The models we study are motivated by realistic considerations, and are nominally NP-hard; however we show that using modern optimization methodology one can solve large, real-life models quite efficiently. Robust optimization is an approach for optimization under uncertainty that does not rely on a stochastic model to model the distribution of data. Often one can view a robust optimization model as a two-stage game; in the first stage a one chooses values for decision variables, while in the second stage an adversary (or ’nature’) picks values for data parameters from a given uncertainty set – the process is adversarial in that the worst-case data distribution will be chosen. Robust optimization is sometimes criticized for being overly conservative. It may also not be conservative enough, since we may not effectively guard against data realizations outside of the uncertainty set. The only protection against this particular problem is to (somehow) enlarge the uncertainty set, but this of course may make us even more conservative, possibly in an unexpected way. Furthermore, in an abstract sense, a robust optimization model tends to give the same ’weight’ to all data realizations, which may be unrealistic in practice. This difficulty is reduced in so-called ’ambiguous chance-constrained’ models, but not completely removed. These issues are well-known, and clearly one should, whenever possible, refine an uncertainty set to focus on the deviations of significance. To some degree, a choice of an uncertainty model is a reflection of risk aversion. In this paper we focus on two types of uncertainty sets. The first type allows a modeler to specify frequencies of deviations (from nominal values) as a function of the magnitude of the deviation; essentially outlining a rough histogram of deviations. The second type models correlation among deviations and gives rise to value-at-risk and conditional value-at-risk problems. Both types of uncertainty sets explicitly include non-convexities observed from prior data. We present cutting-plane algorithms for both problems. The algorithms run a sequence of two-step iterations; in the first step we solve an “implementor problem” which picks values for the decision variables, while the second step solves an “adversarial problem” which finds the worstcase data corresponding to the decision variables just selected by the implementor problem. The adversarial problem, in both cases, is a mixed-integer program. We present theoretical evidence and computational experience using large, real-life data, to show that, in fact, the adversarial problem is not at all difficult in spite of its being NP-hard. In the case of the histogram model, our cutting-plane algorithms find near-optimal solutions in a few minutes of computation, and frequently in just a few seconds. In the case of our VaR and CVaR models, our algorithms are always fast and accurate, often only requiring seconds of computation. The focus of this paper is restricted to the methodology, implementation and experimental testing of the performance of our algorithms. Of course, the real-life asset management implications