Discrete approximation scheme in distributionally robust optimization

Liu，Yongchao1; Yuan，Xiaoming2; Zhang，Jin3

2021-05-01

10.4208/NMTMA.OA-2020-0125

Numerical Mathematics-Theory Methods and Applications   影响因子和分区

1004-8979

2079-7338

Discrete approximation, which has been the prevailing scheme in stochastic programming in the past decade, has been extended to distributionally robust optimization (DRO) recently. In this paper, we conduct rigorous quantitative stability analysis of discrete approximation schemes for DRO, which measures the approximation error in terms of discretization sample size. For the ambiguity set defined through equality and inequality moment conditions, we quantify the discrepancy between the discretized ambiguity sets and the original set with respect to the Wasserstein metric. To establish the quantitative convergence, we develop a Hoffman error bound theory with Hoffman constant calculation criteria in a infinite dimensional space, which can be regarded as a byproduct of independent interest. For the ambiguity set defined by Wasserstein ball and moment conditions combined with Wasserstein ball, we present similar quantitative stability analysis by taking full advantage of the convex property inherently admitted by Wasserstein metric. Efficient numerical methods for specifically solving discrete approximation DRO problems with thousands of samples are also designed. In particular, we reformulate different types of discrete approximation problems into a class of saddle point problems with completely separable structures. The stochastic primal-dual hybrid gradient (PDHG) algorithm where in each iteration we update a random subset of the sampled variables is then amenable as a solution method for the reformulated saddle point problems. Some preliminary numerical tests are reported.

Discrete approximation method Distributionally robust optimization Hoffman lemma Quantitative stability analysis Stochastic primal-dual hybrid gradient

SCI

英语

通讯

NSFC[11971090,11971220] ; General Research Fund from Hong Kong Research Grants Council[12313516] ; Fundamental Research Funds for the Central Universities[DUT19LK24] ; Guangdong Basic and Applied Basic Research Foundation[2019A1515011152]

Mathematics

Mathematics, Applied ; Mathematics

WOS:000613945700001

GLOBAL SCIENCE PRESS

2-s2.0-85101166385

Scopus

1.School of Mathematical Sciences,Dalian University of Technology,Dalian,116024,China
2.Department of Mathematics,University of Hong Kong,Hong Kong
3.SUSTech International Center for Mathematics,Department of Mathematics,Southern University of Science and Technology,Shenzhen,China

Liu，Yongchao,Yuan，Xiaoming,Zhang，Jin. Discrete approximation scheme in distributionally robust optimization[J]. Numerical Mathematics-Theory Methods and Applications,2021,14(2):285-320.

Liu，Yongchao,Yuan，Xiaoming,&Zhang，Jin.(2021).Discrete approximation scheme in distributionally robust optimization.Numerical Mathematics-Theory Methods and Applications,14(2),285-320.

Liu，Yongchao,et al."Discrete approximation scheme in distributionally robust optimization".Numerical Mathematics-Theory Methods and Applications 14.2(2021):285-320.