Robust Optimization Models With Shared Uncertain Parameters

We mainly study on the optimization problems in which the objective and constraints comprise the same uncertain parameters and rebuild robust optimization models with shared uncertain parameters for several problems in investment portfolio,risk management and other practical problems.Besides,we also compare some classical portfolio optimization models and study relative position of optimal portfolios on efficient frontier.We introduce the classical Markowitz portfolio theory.In addition,we review portfolio models of the equilibrium between reward and risk based on this theory framework and go over some relevant basis knowledge about robust optimization as well.In Chapter 2,we propose a new portfolio optimization model(i.e.,Mean Variance Ratio model),based on Mean-Variance model and Sharpe ratio model.By solving the new model,we can obtain a new portfolio which results in minimal variation per unit reward.It is proved that the optimal solution must be on the efficient frontier according to efficient frontier theoiy as long as the new model is solvable.In addition,we study on some classical models,analyze mean and variance of respective optimal solutions,and discuss the relative location of the optimal portfolios on the efficient frontier.In Chapter 3,a robust optimization model with shared uncertain parameters is presented for problems in which the objective and constraints share the same uncertain parameters.The model is designed to optimize the worst case of objective on the uncertain parameter(partially known)simultaneously satisfying all constraints.Namely,shared parameter in objective and constraints of this new model has a unique value when reaching the optimal which avoid shortcoming of traditional robust optimization(i.e.,each shared parameter in objective and constraints may have different values respectively when reaching the model's optimal).Then,we study on several robust reward risk optimization models with shared uncertain parameters in portfolio.Under some conditions,the models can be reformulated as non-linear optimization problems by the duality theorem.Furthermore.we do numerical experiments in real world to illustrate that the proposed model is more reasonable than the traditional robust model.Specially,robust model with shared uncertain parameter can work finely and return an optimal decision for investors,while the traditional robust model does not work(i.e.infeasible)under the same circumstance.In Chapter 4,we study on distributionally robust optimization model with shared uncertain parameters,we focus on the problem of minimizing risk with the mean absolute standard devi-ation constraint.Under the assumption of prior first order moment and second order moment of uncertain parameters,based on traditional distributionally robust optimization model,we build a distributionally robust optimization model with shared uncertain parameters and further rep-resent it as a non-linear Semidefinite Programming according to the conic duality theorem.The reasonability and applicability of our model is verified by a small scale experiment on the real data.In Chapter 5,we concentrate on multi-stage logistics production and inventory problem in management.Since there exists the same uncertain demand variable in both objective and constraints,we propose an adjustable robust optimization model with shared uncertain param-eters referring to traditional adjustable robust optimization model.It can also be equivalent to a solvable non-linear optimization problem by linear duality theorem under some assumptions.Furthermore,we compare the proposed model with traditional adjustable robust optimization model.Numerical results show that the applicability of our model and its superiority as well.Thus,investors can utilize the result of the new model to guide the production and inventory.