Theory, Algorithms And Applications Of Generalized Moment Problems: A New Primal-Dual Perspectiv

The moment problem is an important part in distributionally robust optimization with applica-tions in various areas,such as probability inequality,distributionally robust optimization,revenue management,and inventory management.The moment problem is to achieve the worst-case dis-tribution that satisfies some given set of moment constraints.For the problem with the polynomial moment constraints,semi-definite programming is an effective method to solve the dual of the mo-ment problem.However,for the non-polynomial moment problem,the previous works focus on the specific types of moments without an effective approach to find the analytic solution or provide an accurate numerical solution.To address the gap,this thesis studies the generalized moment problem’s theory,algorithm,and application.First of all,we propose a new algorithm framework for solving the generalized moment problem.The algorithm framework can effectively obtain the closed-form solution,which is also effective for some non-polynomial moment problems.Secondly,we focus on the optimal solution’s structural characteristics,especially,we provide the sparse property of the optimal solu-tion’s support.Finally,we extend the novel framework to the probability inequality and newsvendor problem to show the approach’s potential.Our framework can also work and provide some exciting results.We expect the new approach to be functional in many other applications.·We propose a novel algorithm framework for the one-dimensional generalized moment prob- lem,which can solve the non-polynomial generalized moment problem and obtain the closed- form solution.In addition,we extend the novel algorithm framework to generalized moment problems with structure information,such as unimodal,symmetric and convex.Furthermore,our algorithm framework combines the primal feasible condition,complementary slackness condition,and first-order condition and transforms the generalized moment problem into a deterministic equation system.The transformation can reduce the difficulty of solving the generalized moment problem,as far as we know,the transformation is a new exploration for solving generalized moment problem.·We study the sparsity of the optimal support in the generalized moment problem.we find that the optimal support has at most₂ⁿ⁺¹points.In addition,the sparsity results of the optimal support play a vital role in reducing the complexity of solving generalized moment problem.·We apply the novel algorithm framework to distributionally robust newsvendor problems.For the distributionally robust newsvendor problem with mean and t moment constraints,we use the novel algorithm framework to obtain the closed-form solutions;Further,we propose a new newsvendor model with the mean and exponential moment constraints,it is a non-polynomial moment problem.We also use the novel algorithm framework to obtain the optimal solution.In addition,the new model can better use the light tail information and provide the much more stable decisions than other models,especially when the critical ratio is high.·We apply the new algorithm framework in the studying of probability inequalities.We come up with a new probability inequality containing the mean,variance,and semivariance.We prove that the upper and lower bounds are tight and provide the corresponding optimal distribution;For the robust pricing problem with mean and geometric mean constraints,we construct an effective algorithm to solve this problem.