M.Sc Thesis
M.Sc Student Meirav Rimon Robust Methods for Mathematical Programming with Uncertain Constrains: The Case of Portfolio Selection Department of Industrial Engineering and Management Full Professor Golany Boaz Professor Emeritus Ben-Tal Aharon

Abstract

The capital market's behavior is not known in advance; even if the consequences of economic conditions were understood perfectly, non-economic influences could change the level of the market. Nevertheless, the existence of uncertainty does not mean that careful asset analysis is valueless. There are different approaches tried in handling this problem which is known as "the portfolio selection problem" .

However, all these approaches have a major drawback- either a prior knowledge of the data distribution was required or the problem was computationally intractable due to large-scale size. This research is based on "Affinley Adjustable Robust Counterpart" (AARC) methodology developed by A.Ben-Tal and A.Nemirovski. The methodology stems from The Robust Counterpart (RC) Optimization approach which treats uncertainty data as belonging to a given uncertainty set , described by exact functional relations. The RC approach corresponds to the case when all the variables represent decisions that must be made before the actual realization of the uncertain data becomes known. In the Adjustable Robust Counterpart Optimization approach we distinguish between variables that cannot be adjusted to the data ("here and now" decisions) and variables that can adjust themselves to the data, ("wait and see" decisions). The  Affinley Adjustable Robust Counterpart method in particular, allows the adjustable variables to attune themselves to the true data as an affine function of the data. Applying the above approach on the portfolio selection problem we get a Semidefinite program that admits tight computationally tractable approximation. This approximation is exact when the uncertainty set is an ellipsoid. By using this approach we get two meaningful benefits. First, one needs only crude knowledge of the uncertainty . Second, we get a computationally tractable problem even though it is a large problem of Semidefinite programming. The thesis is constructed of three major sections. The first section reviews the main approaches proposed in the literature to analyze portfolio selection problems. The second section examines different ways of handling data uncertainty. The third section presents the implementation of the portfolio selection problem using the AARC approach and discusses its results via simulations.

The simulations, constructed from two assets and "cash" over three time periods with an ellipsoid uncertainty set, were possible thanks to SEDUMI-a matlab toolbox . The results support the theory, which shows that the investors prefer lower risk in the long run and that the AARC method outperforms the stochastic programming methods and the RC optimization .