Polynomial Optimization Algorithms For Two Kinds Of Portfolio Problems

With the rapid economic development of various countries,the discussion of investment portfolio has become an important topic in the financial theory circle.Since Markowitz put forward the mean-variance(M-V)model in 1952,more and more scholars have been studying risk measures that can better substitute variance and portfolio models that are more suitable for market conditions.To fit the actual market better,this paper constructed two types of portfolio models,namely,high-order moment portfolio model and branch bar portfolio model based on Shortfall Risk(SR).And polynomial optimization algorithm was proposed to solve the two types of portfolio models based on moment-SOS slack method.The first chapter is the introduction,mainly introduces the research background,content and status of the high order moment portfolio problem and the distributionally robust SR portfolio problem based on moment fuzzy set.The chapter 2 is the preparatory knowledge,mainly introduces the symbol description of polynomial optimization,the basic concept and some important property theorems.In chapter 3,a high-moment portfolio model is established.Considering the high-moment in the portfolio selection model can overcome the shortcoming of M-V model(assuming that the return rate of the portfolio follows normal distribution).We can consider arbitrary moments in the model according to actual needs.Since the high-order moment portfolio model is a random polynomial optimization model with non-convex objective function,we construct a sample average approximation(PSAA)model with regular terms.The optimal solution set of PSAA model can approach the optimal solution set of the original problem well by appropriate perturbation.At the same time,we propose a polynomial optimization algorithm for PSAA model based on Moment-SOS algorithm.The convergence analysis of the algorithm is given,and the numerical results verify the validity of the new model and the proposed method.In chapter 4,the SR portfolio model is established by using SR to measure the risk of portfolio.Since the real value or probability distribution of random parameters of real investment problems can not be determined in advance,the idea of distributionally robust optimization is applied to SR portfolio model and fuzzy set is constructed based on generalized moments.In addition,We prove that under certain conditions,the distributionally robust SR portfolio problem can be equivalent to a polynomial optimization problem with conical constraints,and then consider using the theory of polynomial optimization and Lasserre moment-SOS relaxation algorithm to solve the problem.Numerical results verify the validity of the model and the proposed method.