A Distributed Robust Portfolio Optimization Problem Based On WCVaR Constraint

Robust optimization,as a special type of optimization problem,has always been a hot issue studied by people,and it has very important applications in the modeling process of practical problems.In many practical problems,the decision-making is often guided by the optimization model.In these models,there are some parameters that need to be specified or estimated.These parameters are limited to a set of distributions as the random variables studied,conservative decision-making comprehensively considers the optimal solution under the worst case distribution in the set.Therefore,the key to this kind of problem is to construct uncertain sets.This paper mainly studies the equivalent forms of distributed robust portfolio optimization problems based on the worst-case conditional value-at-risk(WCVa R)constraints under several distributed sets.We study the following models:where,r(χ,ξ)is the return function,x∈X is the decision vector that represents investors’ choice of assets,the random variable ξ is the return vector,X∈R~N is the convex compact set,a,β are the given confidence levels,P represents the probability distribution of ξ,P is a fuzzy set consisting of a probability distribution.Due to the uncertainty of returns and risks,investors need to effectively allocate assets in an uncertain environment in order to maximize returns.We call this type of problem of portfolio optimization with unknown distribution as a distributed robust portfolio Optimization.The purpose of distributed robust optimization is to transform the original problem into a solvable equivalent problem.The main idea of this paper is to represent the WCVa R distribution robust portfolio optimization problem with an unknown distribution as a solvable optimization problem without robustness by constructing different distribution sets P.First,for the distribution set defined by JS-divergence distance,through divergence theory,measure conversion theory and Lagrange duality theory,the above distributed robust portfolio optimization model is transformed into a constrained optimization problem underempirical distribution;Second,for a set of distributions with known expectations and covariance,and uncertainty set based on moment estimation,on the basis of known expectation and covariance,add the distribution set of support set information,the model can be transformed into a semi-definite programming problem that is not robust and easy to solve through moment theory,Lagrange duality theory,etc.This article mainly describes from two aspects:The first chapter introduces the research background of portfolio and risk management and the prerequisite knowledge required for the equivalence transformation of the robust distribution optimization problem.the second chapter gives the model studied in this article,and gives the model equivalent from four different perspective form.