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Convex Chance-Constrained Programs with Wasserstein Ambiguity

Haoming Shen, Ruiwei Jiang

TL;DR

This paper studies convexity properties of chance constraints under Wasserstein ambiguity, identifying conditions under which P-CC and O-CC yield convex feasible sets. It shows that joint P-CC with RHS uncertainty is convex when the reference distribution is α-concave with α ≥ -1, and that two-sided P-CC with LHS uncertainty is convex when the reference is elliptical and star-unimodal; it also develops practical solution methods, including a block coordinate ascent algorithm and SOC inner approximations with provable convergence and asymptotic exactness. The optimistic counterparts are shown to inherit these convexity properties under analogous assumptions, broadening applicability. Numerical experiments in production and hydro planning illustrate improved out-of-sample reliability and computational efficiency relative to standard CC and moment-ambiguous approaches, highlighting the practical impact of the proposed convexity results and inner-approximation schemes.

Abstract

Chance constraints yield non-convex feasible regions in general. In particular, when the uncertain parameters are modeled by a Wasserstein ball, arXiv:1806.07418 and arXiv:1809.00210 showed that the distributionally robust (pessimistic) chance constraint admits a mixed-integer conic representation. This paper identifies sufficient conditions that lead to convex feasible regions of chance constraints with Wasserstein ambiguity. First, when uncertainty arises from the right-hand side of a pessimistic joint chance constraint, we show that the ensuing feasible region is convex if the Wasserstein ball is centered around a log-concave distribution (or, more generally, an $α$-concave distribution with $α\geq -1$). In addition, we propose a block coordinate ascent algorithm and prove its convergence to global optimum, as well as the rate of convergence. Second, when uncertainty arises from the left-hand side of a pessimistic two-sided chance constraint, we show the convexity if the Wasserstein ball is centered around an elliptical and star-unimodal distribution. In addition, we propose a family of second-order conic inner approximations, and we bound their approximation error and prove their asymptotic exactness. Furthermore, we extend the convexity results to optimistic chance constraints.

Convex Chance-Constrained Programs with Wasserstein Ambiguity

TL;DR

This paper studies convexity properties of chance constraints under Wasserstein ambiguity, identifying conditions under which P-CC and O-CC yield convex feasible sets. It shows that joint P-CC with RHS uncertainty is convex when the reference distribution is α-concave with α ≥ -1, and that two-sided P-CC with LHS uncertainty is convex when the reference is elliptical and star-unimodal; it also develops practical solution methods, including a block coordinate ascent algorithm and SOC inner approximations with provable convergence and asymptotic exactness. The optimistic counterparts are shown to inherit these convexity properties under analogous assumptions, broadening applicability. Numerical experiments in production and hydro planning illustrate improved out-of-sample reliability and computational efficiency relative to standard CC and moment-ambiguous approaches, highlighting the practical impact of the proposed convexity results and inner-approximation schemes.

Abstract

Chance constraints yield non-convex feasible regions in general. In particular, when the uncertain parameters are modeled by a Wasserstein ball, arXiv:1806.07418 and arXiv:1809.00210 showed that the distributionally robust (pessimistic) chance constraint admits a mixed-integer conic representation. This paper identifies sufficient conditions that lead to convex feasible regions of chance constraints with Wasserstein ambiguity. First, when uncertainty arises from the right-hand side of a pessimistic joint chance constraint, we show that the ensuing feasible region is convex if the Wasserstein ball is centered around a log-concave distribution (or, more generally, an -concave distribution with ). In addition, we propose a block coordinate ascent algorithm and prove its convergence to global optimum, as well as the rate of convergence. Second, when uncertainty arises from the left-hand side of a pessimistic two-sided chance constraint, we show the convexity if the Wasserstein ball is centered around an elliptical and star-unimodal distribution. In addition, we propose a family of second-order conic inner approximations, and we bound their approximation error and prove their asymptotic exactness. Furthermore, we extend the convexity results to optimistic chance constraints.

Paper Structure

This paper contains 47 sections, 51 theorems, 266 equations, 7 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Key Definitions and Examples
  3. Pessimistic Chance Constraint
  4. Joint (P-CC) with RHS Uncertainty
  5. Two-Sided (P-CC) with LHS Uncertainty
  6. Solution Approaches for Pessimistic Chance Constraint
  7. Block Coordinate Ascent Algorithm
  8. SOC Inner Approximation
  9. Inner Approximations for $\mathcal{C}_\delta$ and $\mathcal{X}^p_\text{T}$
  10. Approximation Error and Asymptotic Exactness of $\widehat{\mathcal{X}}^p_\text{T}$
  11. Optimistic Chance Constraint
  12. Joint (O-CC) with RHS Uncertainty
  13. Two-Sided (O-CC) with LHS Uncertainty
  14. Numerical Experiments
  15. Production Planning
  16. ...and 32 more sections

Key Result

Proposition 1

For $\delta > 0$, it holds that Here, random variable ${\zeta}$ follows the reference distribution $\mathbb{P{}}$ and $\mathbf{d}\left( \zeta, \mathcal{S}^c(x) \right)$ represents the distance from $\zeta$ to the "unsafe" set $\mathcal{S}^c(x)$chen-2018-data-driven, and $\Xi$ is the support of $\xi$.

Figures (7)

  • Figure 1: Contours of $g_{\epsilon}(\ell, u)$ with varying $\delta$ and a polyhedral inner approximation $\widehat{\mathcal{C}}_N$ of $\mathcal{C}_\delta$ with $N = 5$ and $\delta = 0.050$
  • Figure 2: Visualization of random variables $\zeta_1$ and $\zeta_2$ in Examples \ref{['exa:counter-1']}--\ref{['exa:counter-2']}
  • Figure 3: Visualization of $\gamma_{\tau}$ and $(\bar{g}_{\epsilon})^{-1}(s)$
  • Figure 4: Convergence of Algorithm \ref{['algo:bcm-for-rho-u']} on Production Planning Instances; solid line = average of the difference $\phi(x_k, y_k) - {\phi}^{\ast}$ across five runs, error bar = standard deviation of the difference
  • Figure 5: Risk envelopes under different risk thresholds
  • ...and 2 more figures

Theorems & Definitions (112)

  • Example 1
  • Example 2
  • Definition 1
  • Definition 2
  • Example 3
  • Example 4
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 1: Adapted from Theorem $1$ in xie-2019-distr-robus
  • ...and 102 more