Abstract
Portfolio Optimization is based on the efficient allocation of several assets, which can get heavily affected by the uncertainty in input parameters. So we must look for such solutions which can give us steady results in uncertain conditions too. Recently, the uncertainty based optimization problems are being dealt with robust optimization approach. With this development, the interest of researchers has been shifted toward the robust portfolio optimization. In this paper, we study the robust counterparts of the uncertain mean-variance problems under box and ellipsoidal uncertainties. We convert those uncertain problems into bi-level optimization models and then derive their robust counterparts. We also solve a problem using this methodology and compared the optimal results of box and ellipsoidal uncertainty models with the nominal model.
Highlights
Portfolio Optimization deals with the decision making problems of efficient distribution of financial assets
The proposed methodology. 1: Define the uncertain set with its centre as the nominal value u0 of uncertain parameters u and radius as the perturbation amount δ. 2: Convert the uncertain problem into a bilevel optimization form, where the lower level problem represents the worst realization of the uncertain parameters (u) in the domain of perturbation and the upper level problem represents our main problem. 3: Replace the lower level problem with its KKT conditions. 4: Solve the KKT conditions to get the worst case realization of the uncertain parameter u in terms of u0and δ. 5: Use this value of u in the upper level problem to get the robust counterpart of our uncertain optimization problem. 6: Solve the robust counterpart problem to get the robust solution
We have proposed a methodology for finding the robust solutions of uncertain portfolio optimization problems
Summary
Portfolio Optimization deals with the decision making problems of efficient distribution of financial assets. In last two decades this problem has been addressed properly with the help of Robust optimization It is first introduced by Ben-Tal for solving uncertain linear problems [2] and after that it is being used in many disciplines of science and engineering [5]. The study showed that the robust maximum risk-adjusted return problem with this uncertainty set can be solved as a conic programming problem. In this paper our main focus is to propose a methodology for solving robust optimization problems and to compare the box and ellipsoidal uncertainty models. Our proposed methodology includes the transformation of uncertain mean-variance problems into bilevel optimization form and the use of single level reduction approach for solving this.
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