Algorithm For Cardinality Constrained Portfolio Selection Problem

The cardinality constrained optimization problem is an important practical problem that we often face in daily decision-making.It is widely used in financial investment,im-age processing,data analysis,etc.The investment portfolio is stock held by investors or financial institutions.Its purpose is to spread risk.The mean variance model proposed by Markowitz in 1952 laid the foundation for modern securities portfolio theory and has been widely used in the field of financial investment.In practice,due to the limitation of trans-action cost and other factors,the types of stocks held by traders should not be too much.Therefore,it is necessary to introduce the base constraint in the mean variance model.It is practical to solve the problem.This paper studies the following portfolio problem with cardinality constraints min xT Qx+ap(x)s.t.x ?X where p(x)=[max(rp-rTx,0)]2+[max(eTx-1,0)]2,Q is the covariance matrix of the yield of the securities,rp is the expected rate of re-turn,r is the yield of the securities,and e is the vector of 1 for each component,x={x1,x2,…,xn)T ? Rn is the portfolio,k is the non-zero variable in the portfolio x The number of n is the total number of securities,and the investment budget is 1.For this problem,this paper first uses the integral level set method.Since the appli-cation of the integration method requires the robustness of the region,this method cannot be directly applied.By constructing the Q-measure space,the feasible domain of the car-dinality constraint is satisfied with the robustness hypothesis of the integral total extremum method.Further introducing the penalty function,the original problem is transformed into the unconstrained optimization problem in the Q-measure space min xT Qx+ap(x)s.t.x ?X where p(x)=[max(rp-rTx,0)]2+[max(eTx-1,0)]2,The Monte-Carlo simulation is used in the algorithm,that is,the objective function on the level set is estimated by the random point method,and the integral level set method with the basal constraint portfolio model is realized.Aiming at this problem,we use genetic algorithm.Since the new vector generated by directly interleaving the portfolio vector satisfying the cardinality constraint does not necessarily satisfy the cardinality constraint,we introduce the concept of cardinality vector,which is a new representation of portfolio vector.Since the new vector generated by directly interleaving the portfolio vector satisfies the cardinality constraint,the concept of cardinality vector is introduced in this paper,which is a new representation method of portfolio vector.That is,the cardinality vector is a 2k dimension vector,the first k component is the non-zero component of the portfolio vector,and the last k component represents the ordinal corresponding to it.When the problem has a higher dimensionality,the cardinality vector takes up less storage space.In this paper,the crossover operation of genetic algorithm is designed for this representation,which guarantees that the radix vector still falls within the feasible domain corresponding to the cardinality constraint after the genetic operation.Furthermore,the rules of mutation operation are designed for the high and low genes of the cardinality vector,so that the new cardinality vector obtained by the mutation operation is still a feasible solution to the original problem.This paper firstly proves the feasibility of applying genetic algorithm to the problem of basal constraint portfolio,and designs the genetic algorithm and algorithm based on the representation.The first chapter of this paper is the introduction;the second chapter mainly intro-duces the integral method and some basic theories of genetic algorithm;the third chapter constructs a special kind of Q-measure space for the cardinality constrained mean vari-ance problem,so that the cardinality The constraint problem satisfies the theory of robust-ness analysis in this space,and further introduces the penalty function,which transforms the original problem into the unconstrained optimization problem in Q-measure space,and demonstrates the feasibility of the method through numerical experiments.The fourth chap-ter introduces new The cardinality vector represents the original portfolio vector,and the crossover and mutation rules are used to ensure that the new cardinality vector generated by the genetic algorithm still falls within the feasible domain of the cardinality constrained portfolio problem.Numerical experiments show that the genetic algorithm is feasible in the cardinality constraint.Final is the conclusion and outlook.