A Homotopy-Penalty Method For Nonconvex Programming And Its Application In Sparse Portfolio

We consider the cardinality-constrained portfolio,and the ratio of the risk and return of the portfolio is used as the objective function to reduce the influence of the investor's subjective preference on risk and return.The cardinality constraint can prevent the investment too scattered to increase the transaction cost and management cost.We approximate the cardinality constraint with a kind of smooth function and attempt to solve the problem using constraint shifting homotopy method.For nonconvex programming with both equality constraints and inequality constraints,the constraint shifting homotopy method is restricted because its positive linear independence condition and normal cone condition are associated with both inequality constraints and equality constraints.Therefore,we propose a new constraint shifting homotopy method to solve this kind of problem.By means of penalty method,we get the penalty problem with only inequality constraints.Starting from the penalty problem,a constraint shifting homotopy-penalty method is constructed and the existence and convergence of the homotopy path are proved theoretically.We calculate some numerical examples,the numerical results verify the feasibility of our method and it is competitive compared with the general constraint shifting homotopy method.We define an appropriate dynamic constraint for the smooth approximation of the cardinality-constrained portfolio,and use the constraint shifting homotopy-penalty method to solve it,of which the convergence of the homotopy path is proved.In numerical experiments,we compare the effectiveness of our method with the SQP algorithm of fmincon for solving this problem.The results show that,in most cases,our method can find the approximate optimal solution with smaller target value than the SQP algorithm,which verifies the feasibility and effectiveness of our method.