The Study On The Global Algorithm For Optimal Trade-off Portfolio Selection With Parameter Sensitivity

In 1952,Markowitz proposed the Mean-Variance(MV)model for portfolio selection.The basic parameters in the model,expected returns and covariance matrices,are usually derived from historical data.The lack of historical data may lead to inaccurate parameter estimates in the MV model,and the accuracy of parameter estimates directly affects the measurement of returns and risks.To overcome the impact of parameter estimation errors,scholars have proposed a MV model with parameter sensitivity constraints,but it is difficult to choose the upper and lower bounds of parameter sensitivity in this model.In order to overcome the difficulty in selecting the upper and lower bounds of parameter sensitivity,an optimal trade-off portfolio problem with parameter sensitivity has been proposed in the literature.The optimization model is a non-convex non-differentiable optimization problem,in which the objective function contains maximum and minimum functions,and it is very difficult to find its global solution.This thesis aims to study the global algorithm and its numerical implementation for the optimal trade-off portfolio problem with parameter sensitivity.The following are the main results of this thesis:(1)We study the semi-definite programming(SDP)method for the optimal trade-off portfolio problem with parameter sensitivity.We first transform the problem into an equivalent non-convex quadratic constrained programming(QCQP)problem by introducing auxiliary variables.By combining the eigenvalue decomposition and the secant cut technique,we propose an SDP relaxation for this QCQP and estimate the gap between it and the original problem.The preliminary numerical results then show that the SDP relaxation can provide a tighter lower bound of the original problem and effectively find the global optimal solution for most test instances,and the computational time is less than that of the solver GUROBI.(2)We study the successive convex optimization(SCO)algorithm for the optimal trade-off portfolio problem with parameter sensitivity and its global convergence.We first cast the transformed QCQP problem as an equivalent DC(Difference of convex functions,DC)programming by using the eigenvalue decomposition.A second-order programming(SOCP)approximation of the equivalent DC problem is derived by linearizing the concave quadratic term in the constraints of the DC problem.Based on the SOCP approximation problem,an SCO algorithm for solving the original problem is proposed.It is proved that the SCO algorithm converges to a KKT point of the QCQP transformed problem.(3)We investigate the global algorithm for the optimal trade-off portfolio problem with parameter sensitivity and its global convergence and complexity.We first propose a global algorithm for the problem by combining the SCO algorithm,the SDP relaxation method and the branch-and-bound framework.We show that the proposed algorithm converges to an (?)-global optimal solution of the original problem and estimate the complexity of the algorithm.The numerical results then show that the algorithm can effectively find the global optimal solution of the original problem and the calculation time is much better than the solver GUROBI.In addition,the KKT point obtained by the SCO method is usually the global optimal solution of the original problem.