The Study On The Global Optimization Algorithm For Portfolio Selection Problem With Factor Risk Control

In modern financial markets,portfolio selection has always been one of the main concerns of individual investors or investment institutions: how to allocate a reasonable amount of capital and allocate it to various risky assets to maximize profits and minimize risks.In 1952,Markowitz's mean-variance portfolio selection model was the foundation of modern investment theory.Since then,based on the mean-variance model,the various risk measurement methods have been proposed so that the portfolio theory are greatly improved and developed.Markowitz's mean-variance model only measured the systemic risk of a portfolio,while it did not consider the contribution of a single asset or risk factor to the overall risk of the portfolio.Based on the factor model,the concept of factor risk is proposed to measure the contribution of individual factors to the overall risk of portfolios,and the mean-variance portfolio selection model with factor risk constraints is constructed in the literature.The resulting model is a quadratic programming problem with non-convex quadratic constraints,thus finding its global optimal solution is NP-hard.In this paper,we will study the new global optimization algorithm for the mean-variance portfolio selection problem with factor risk control and its numerical implementation.By exploiting the special structure of the model,we develop a new convex relaxation based on non-redundant matrix separation and a new branch-and-bound global algorithm based on this convex relaxation.We analyze the global convergence of the proposed algorithm.The effectiveness of the algorithm is verified by numerical experiments.The main results of this paper are as follows:First,we investigate the convex relaxation approach based on a non-redundant matrix splitting for the portfolio selection model with factor risk control.Especially,we investigate how to choose a suitable matrix splitting approach so that the resulting relaxation model can provide a stronger lower bound for it.First,we show that for any given redundant matrix splitting,there exists a corresponding non-redundant matrix splitting,so that the convex relaxation based on it can provide a stronger lower bound.To find such a non-redundant matrix splitting,we propose to solve some auxiliary SDP problem.We show that the matrix splitting derived via solving an auxiliary SDP problem is non-redundant.We present a new convex relaxation based on non-redundant matrix splitting and the properties of its optimal solutions and optimal values.Numerical experiments demonstrate that the new convex relaxation provides a tighter lower bound than the exiting convex relaxation in the literature.Secondly,by combining the new convex relaxation scheme and the inner convex approximation technique based on non-redundant matrix splitting,we propose a new global algorithm in the branch-and-bound framework for the mean-variance portfolio selection problem with factor risk constraint.We show that the proposed algorithm converges to the global optimal solution of the original problem.The effectiveness of the new algorithm is verified by real data and random data of S&P 500.Preliminary numerical results demonstrate that our proposed algorithm can find effectively the global optimal solution of the original problem.