基于CVaR约束的分布鲁棒投资组合优化问题的等价形式

Equivalent form of distributed robust portfolio optimization problem based on CVaR constraint

在许多实际问题中经常通过优化模型来指导决策.在这些模型中,存在着需要指定或估计的参数.而这些参数作为随机变量要限制在一个分布集合内,保守决策综合考虑了集合中分布最坏的情况下进行的优化求解.所以,此类问题的关键就是不确定集的构造.在本文中,研究了概率分布集合由JS-散度定义的CVaR分布鲁棒优化问题.对目标函数中的期望值函数,经过适当的度量测度的选取、Lagrange对偶理论将问题转化为经验分布下的约束优化问题,从而得到期望值函数的等价形式.对于约束中的CVaR函数,类似的方法也可以得到其等价形式.因此,最终可得到基于JS-散度的CVaR分布鲁棒投资组合优化问题的等价形式.

In many practical problems,the optimization model is often used to guide the decision.In these models,there are parameters that need to be specified or estimated.These parameters are restricted as a random variable in a distribution set.Conservative decision considers the optimal solution for the worst distribution case in the set.Therefore,the key to such problems is the construction of the uncertainty sets.In this paper,we study the robust optimization problem of CVaR distribution which the probability distribution set is defined by JS-divergence.For the expected value function in the objective function,by applying the change-of-measure technique and Lagrange duality theory,the problem is transformed into the constrained optimization problem under the nominal distribution,so the equivalent form of the expected value function is obtained.For the CVaR function in the constraint,a similar method can also get its equivalent form.Hence,we finally get the equivalent form of the CVaR distribution robust portfolio optimization problem that is based on the JS-divergence function.