Research And Application Of Distributionally Robust Stochastic Optimization Problems Under Several Types Of Uncertaintie

Distributtionally robust stochastic optimization is a kind of uncertainty optimization that combines statistics and optimization theory to overcome the conservative of robust optimization problems and the uncertainty in stochastic optimization.In recent years,significant progress has been made in the study of distributionally robust stochastic optimization problems.This paper mainly starts with the construction of uncertainty sets,and studies the distribution robust stochastic optimization problem under Wasserstein uncertainty sets.In this paper,we study the duality problem of distributionally robust stochastic optimization problems for quasi-convex random variables under the uncertainty set constructed by the Wasserstein distance,and for the distributionally robust stochastic problem under the specific Wasserstein uncertainty set,we give the equivalent transformation form,the transformed problem is an tractable finite convex programming problem.Further,applying the Wasserstein uncertainty set to the classification model,we propose a distributionally robust chance-constrained SVM model withl₂-Wasser-stein distance with good robustness and stability.We also construct a robust problem for uncertainty sets with special moment information,and study the bounds of the model.Firstly,we mainly consider the distributionally robust stochastic optimization problem with a quasi-convex cost function.Corresponding to this problem,we give the concrete Lagrange dual form,and provide the necessary and sufficient conditions for strong duality.Under these conditions,we show the strong dual theorem.When the cost function is quasi-convex,the distributionally robust stochastic optimization problem with∞-Wasserstein distance can be approximately transformed into a tractable form using a data-driven method,under the guarantee of the strong duality theorem.Therefore,we give two specific distributionally robust stochastic optimization models for quasi-convex random variables,under the structure of∞-Wasserstein distance,the model is equivalently transformed into finite convex programming by applying the above conclusions,and do some related numerical experiments.Numerical results show that our proposed data-driven model is effective.Second,we propose a distributionally robust chance-constrained SVM model withl₂-Wasserstein uncertainty set.We present equivalent formulations of distributionally robust chance constraints based onl₂-Wasserstein uncertainty.In terms of this method,the distribu-tionally robust chance-constrained SVM model can be transformed into a solvable linear 0-1mixed integer programming problem when thel₂-Wasserstein distance is discrete form.The distributionally robust chance-constrained SVM model could be transformed into a tractable0-1 mixed-integer SOCP programming problem for the continuous case.Finally,numerical experiments show that compared with the classical SVM model and the Soft SVM model,the proposed model has better robustness and stability.Finally,we study the distributionally robust optimization problem under Uncertainty sets constructed with special moment information.Fsirstly,we study the problem of bounding the survival function P（X>t）=1-F（t）,with given mean and geometric mean of the random variable X.We obtain a pair of tight lower bounds by studying the primal-dual moment optimization problem and providing the problem’s corresponding primal-dual optimal solutions.We further analyze the robust pricing problem with the mean and geometric mean of the random varable,which supplements recent results and their analysis by[1].We expect the new bounds of survival function to be helpful in many other applications.Secondly,we study the Nesvendor problem with given mean and geometric mean of the random variable X.