Distributionally robust joint chance-constrained programming with Wasserstein metric

Abstract

In this paper, we develop an exact reformulation and a deterministic approximation for distributionally robust joint chance-constrained programmings $(\mathrm{DRCCPs})$ with a general class of convex uncertain constraints under data-driven Wasserstein ambiguity sets. It is known that robust chance constraints can be conservatively approximated by worst-case conditional value-at-risk (CVaR) constraints. It is shown that the proposed worst-case CVaR approximation model can be reformulated as an optimization problem involving biconvex constraints for joint DRCCP. This approximation is essentially exact under certain conditions. We derive a convex relaxation of this approximation model by constructing new decision variables which allows us to eliminate biconvex terms. Specifically, when the constraint function is affine in both the decision variable and the uncertainty, the resulting approximation model is equivalent to a tractable mixed-integer convex reformulation for joint binary DRCCP. Numerical results illustrate the computational effectiveness and superiority of the proposed formulations.

Keywords:

• Distributionally robust optimization problem
• chance-constrained programming
• Wasserstein metric
• conic optimization
• mixed-integer programming

Mathematics Subject Classification (2000):

• 90C15
• 90C11
• 90C25

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Some or all data, models, or code generated or used during the study are available from the first author of the paper by request.

Additional information

Funding

This research was supported by National Natural Science Foundation of China (Grant No. 11271243).

Notes on contributors

Yining Gu

Yining Gu, received the M.S. degree from the School of Mathematics, Tianjin University, Tianjin, China, in 2018. She is currently pursuing the Ph.D. degree with the School of Mathematics, Shanghai University of Finance and Economics, Shanghai, China. Her research interests include stochastic optimization, distributionally robust optimization, and their applications.

Yanjun Wang

Yanjun Wang, received the Ph.D. degree from the School of Science, Xi'an Jiaotong University, Xi'an, China, in 2004. She is currently a Professor with the School of Mathematics, Shanghai University of Finance and Economics, Shanghai, China. Her research interests include stochastic optimization, distributionally robust optimization, non-convex optimization, global optimization, and their applications.