[Paper Review] Data-Driven Chance Constrained Programs over Wasserstein Balls

Zhi Chen, Daniel Kühn+1 more|arXiv (Cornell University)|Sep 1, 2018

Risk and Portfolio Optimization50 citations

TL;DR

The authors derive exact deterministic reformulations for data-driven chance constrained problems over Wasserstein ambiguity sets, yielding mixed-integer conic programs (MILP when using 1- or ∞-norm), and show competitive performance against existing schemes.

ABSTRACT

We provide an exact deterministic reformulation for data-driven chance constrained programs over Wasserstein balls. For individual chance constraints as well as joint chance constraints with right-hand side uncertainty, our reformulation amounts to a mixed-integer conic program. In the special case of a Wasserstein ball with the $1$-norm or the $\infty$-norm, the cone is the nonnegative orthant, and the chance constrained program can be reformulated as a mixed-integer linear program. Our reformulation compares favourably to several state-of-the-art data-driven optimization schemes in our numerical experiments.

Motivation & Objective

• Motivate distributionally robust optimization when the data-generating distribution is uncertain.
• Develop exact reformulations for data-driven chance constraints under Wasserstein ambiguity sets.
• Handle both individual and joint chance constraints with right-hand side uncertainty.
• Provide computationally tractable mixed-integer conic reformulations.
• Compare proposed approach with state-of-the-art data-driven optimization schemes.

Proposed method

• Model data-driven chance constraints with Wasserstein ambiguity sets centered at the empirical distribution.
• Characterize worst-case probability of unsafe events over the Wasserstein ball and reformulate as a deterministic program (Theorem 2.2).
• Use a linearization trick to express the sum of the εN smallest distances to the unsafe set via a linear program (Lemma 2.4).
• Derive a mixed-integer conic reformulation for the resulting problem; reduce to MILP when the ground norm is 1-norm or ∞-norm (Proposition 2.6 and related discussions).
• Specialize reformulations to (i) affine safety sets leading to ICC problems and (ii) joint chance constraints with right-hand side uncertainty (Sections 2.3 and 2.4).
• Discuss prior and related work, including exact reformulations for Wasserstein DRO and contrasts with φ-divergence approaches.

Experimental results

Research questions

• RQ1 How can data-driven chance constrained programs be reformulated exactly when the uncertainty set is a Wasserstein ball?
• RQ2 What is the computational form of the reformulation for individual and joint chance constraints under Wasserstein ambiguity?
• RQ3 Can these reformulations be expressed as mixed-integer conic programs, and MILP in common norms?
• RQ4 How do these approaches compare to existing data-driven optimization schemes in numerical experiments?

Key findings

• An exact deterministic reformulation is obtained for data-driven chance constraints over Wasserstein balls, yielding a mixed-integer conic program.
• If the ground norm is 1-norm or ∞-norm, the reformulation reduces to a mixed-integer linear program.
• For individual chance constraints with affine safety sets, the reformulation is tractable via a convex-concave transformation leading to ICC-MILP form (Proposition 2.6).
• A convex reformulation is achieved by expressing the sum of the εN smallest distances to the unsafe set through a linear program (Theorem 2.2 and Lemma 2.4).
• The approach applies to both individual and joint chance constraints with right-hand side uncertainty, and is argued to compare favorably with state-of-the-art data-driven schemes in numerical tests.
• The work positions Wasserstein-based DRO as a robust alternative that avoids some limitations of φ-divergence ambiguity sets.