[Paper Review] Distributionally Robust Chance-Constrained Programmings for Non-Linear Uncertainties with Wasserstein Distance

Yining Gu, Yanjun Wang|arXiv (Cornell University)|Mar 8, 2021

Risk and Portfolio Optimization26 references4 citations

TL;DR

This paper proposes an exact reformulation and deterministic approximation for distributionally robust chance-constrained programs (DRCCPs) with convex non-linear uncertainties using Wasserstein ambiguity sets. By leveraging biconvex relaxation and variable transformation, it achieves a tractable convex reformulation that becomes exact under specific conditions, such as binary decisions and linear constraints, enabling efficient solution via convex optimization.

ABSTRACT

In this paper, we develop an exact reformulation and a deterministic approximation for distributionally robust chance-constrained programmings (DRCCPs) with convex non-linear 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 this approximation can be reformulated as an optimization problem involving biconvex constraints for joint DRCCP. We also demonstrate that this approximation amounts to a convex programming by constructing new decision variables, which allows us to eliminate biconvex terms. Meanwhile, we present how to transform many kinds of non-linear uncertain constraints into cone inequalities. Besides, it turns out that this approximation becomes essentially exact under certain conditions for joint DRCCP. Specifically, when the decision variables are binary and the uncertain constraints are linear, then this approximation is equivalent to a tractable mixed-integer convex reformulation. Numerical results illustrate that the proposed formulations can be solved efficiently.

Motivation & Objective

• To address the challenge of solving distributionally robust chance-constrained programs (DRCCPs) with convex non-linear uncertain constraints under data-driven Wasserstein ambiguity sets.
• To develop a deterministic approximation that transforms the inherently non-convex biconvex reformulation into a tractable convex program.
• To enable efficient solution of joint DRCCPs by eliminating biconvex terms through novel decision variable construction.
• To provide a systematic method for transforming various non-linear uncertain constraints into cone inequalities.
• To identify conditions under which the proposed approximation becomes exactly equivalent to the original problem.

Proposed method

• Proposes a biconvex reformulation of joint DRCCPs by leveraging worst-case conditional value-at-risk (CVaR) constraints as a conservative approximation.
• Introduces new decision variables to eliminate biconvex terms, transforming the problem into a convex programming formulation.
• Employs the Wasserstein distance to define ambiguity sets, enabling data-driven distributional robustness for non-linear constraints.
• Transforms non-linear uncertain constraints into second-order cone or second-order cone representable inequalities for tractability.
• Applies a mixed-integer convex reformulation when decision variables are binary and constraints are linear, ensuring exactness.
• Validates the approach through numerical experiments demonstrating efficient solvability using standard convex optimization solvers.

Experimental results

Research questions

• RQ1 Can a conservative approximation of distributionally robust chance-constrained programs with non-linear uncertainties be transformed into a tractable convex optimization problem?
• RQ2 Under what conditions does the proposed approximation become exactly equivalent to the original DRCCP formulation?
• RQ3 How can non-linear uncertain constraints be systematically converted into cone inequalities for use in robust optimization?
• RQ4 What is the role of the Wasserstein distance in enabling data-driven ambiguity sets for non-linear constraints?
• RQ5 Can the biconvex structure arising from joint chance constraints be eliminated via variable transformation to yield a convex reformulation?

Key findings

• The proposed method transforms the biconvex reformulation of joint DRCCPs into a convex program through the introduction of new decision variables, enabling efficient solution.
• The approximation becomes exactly equivalent to the original problem when decision variables are binary and uncertain constraints are linear, yielding a tractable mixed-integer convex program.
• The method successfully converts various types of non-linear uncertain constraints into cone inequalities, enhancing computational tractability.
• Numerical results confirm that the proposed formulations are efficiently solvable using standard convex optimization solvers.
• The use of Wasserstein ambiguity sets ensures data-driven robustness while maintaining structural tractability for non-linear constraints.