Data-driven distributionally robust optimization

Abstract

Embodiments of the disclosure include a method for providing data-driven distributionally robust optimization. The method includes receiving a plurality of samples of one or more uncertain parameters for a complex system and calculating a distribution uncertainty set for the one or more uncertain parameters. The method also includes receiving a deterministic problem model associated with the complex system that includes an objective and one or more constraints and creating a distributionally robust counterpart (DRC) model based on the distribution uncertainty set and the deterministic problem model. The method further includes formulating the DRC as a generalized problem of moments (GPM), applying a semi-definite programing (SDP) relaxation to the GPM and generating an approximation for a globally optimal distributionally robust solution to the complex system.

What is claimed is: 
     
         1 . A method for providing data-driven distributionally robust optimization, the method comprising:
 receiving a plurality of samples of one or more uncertain parameters for a complex system;   calculating a distribution uncertainty set for the one or more uncertain parameters;   receiving a deterministic problem model associated with the complex system that includes an objective and one or more constraints, wherein the plurality of samples is described by an unknown distribution;   creating a distributionally robust counterpart (DRC) model based on the distribution uncertainty set and the deterministic problem model;   formulating the DRC as a generalized problem of moments (GPM);   applying a semi-definite programing (SDP) relaxation to the GPM; and   generating an approximation for a globally optimal distributionally robust solution to the complex system.   
     
     
         2 . The method of  claim 1 , wherein formulating the DRC as the GPM comprises:
 calculating a dual minimization problem of an inner maximization problem;   transforming a feasible set of an inner minimization problem to match a structure of the feasible set of an outer minimization problem; and   reducing a minimization-minimization problem to a minimization problem, which constitutes the GPM.   
     
     
         3 . The method of  claim 1 , wherein formulating the DRC as the GPM comprises:
 calculating a dual-maximization problem of an inner minimization problem, transforming a feasible set of a inner maximization problem to match the structure of the feasible set of an outer maximization problem; and   reducing a maximization-maximization problem to a maximization problem, which constitutes the GPM.   
     
     
         4 . The method of  claim 1 , wherein calculating the distribution uncertainty set for the one or more uncertain parameters is based on a polynomial estimate of a probability density function. 
     
     
         5 . The method of  claim 1 , wherein calculating the distribution uncertainty set for the one or more uncertain parameters is based on statistical estimates for a plurality of moments of the unknown distribution of the uncertain system parameters up to an arbitrary order. 
     
     
         6 . The method of  claim 1 , wherein calculating the distribution uncertainty set for the one or more uncertain parameters is based on histogram estimates for the unknown distribution of the uncertain system parameters. 
     
     
         7 . The method of  claim 1 , wherein the distribution uncertainty set includes a support that is described by one or more multivariate polynomial inequality constraints. 
     
     
         8 . The method of  claim 1 , wherein the objective is described as multivariate polynomial. 
     
     
         9 . The method of  claim 1 , wherein the equality and/or inequality constraints are described as multivariate polynomials. 
     
     
         10 . The method of  claim 1 , wherein the approximation for a distributionally robust solution includes precision level.