[Paper Review] Convex Chance-Constrained Programs with Wasserstein Ambiguity

Haoming Shen, Ruiwei Jiang|arXiv (Cornell University)|Nov 3, 2021

Risk and Portfolio Optimization42 references4 citations

TL;DR

This paper establishes sufficient conditions under which chance-constrained programs with Wasserstein ambiguity yield convex feasible regions, enabling efficient solution via conic optimization. It proves convexity for left-hand side uncertainty when the Wasserstein center is Gaussian and for right-hand side uncertainty when the center is log-concave or α-concave (α ≥ -1), and proposes a convergent block coordinate ascent algorithm for such problems.

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.

Motivation & Objective

• To identify sufficient conditions under which chance-constrained programs with Wasserstein ambiguity yield convex feasible regions, overcoming the typical non-convexity of such problems.
• To extend convexity results to both pessimistic and optimistic chance constraints under Wasserstein ambiguity.
• To develop a convergent block coordinate ascent algorithm for solving the resulting convex optimization problems.
• To provide conic representations for the feasible regions when the ambiguity set is centered on specific distributions (Gaussian, log-concave, α-concave).

Proposed method

• Derives a convex and conic representation for pessimistic individual chance constraints when uncertainty arises on the left-hand side and the Wasserstein center is Gaussian.
• Proves convexity of the feasible region for pessimistic joint chance constraints with right-hand side uncertainty when the Wasserstein center is log-concave or α-concave (α ≥ -1).
• Proposes a block coordinate ascent algorithm for solving the optimistic chance-constrained program, proving global convergence.
• Utilizes duality and properties of the Wasserstein distance to reformulate the distributionally robust chance constraint into a tractable form.
• Applies the Moreau-Yosida regularization and directional derivative analysis to establish convexity and continuity of the value function.
• Employs the CVaR reformulation to express the optimistic chance constraint as a single robust constraint involving the distance to the feasible set.

Experimental results

Research questions

• RQ1 Under what conditions does a chance-constrained program with Wasserstein ambiguity yield a convex feasible region?
• RQ2 Can convexity be preserved when uncertainty affects the left-hand side of an individual chance constraint, particularly when the Wasserstein center is Gaussian?
• RQ3 Does convexity hold for joint chance constraints with right-hand side uncertainty when the Wasserstein center is log-concave or α-concave?
• RQ4 Can a convergent algorithm be designed for solving optimistic chance-constrained programs under Wasserstein ambiguity?
• RQ5 What is the conic representation of the feasible region for these convex chance-constrained programs?

Key findings

• For individual chance constraints with left-hand side uncertainty, the feasible region is convex if the Wasserstein ambiguity set is centered on a Gaussian distribution.
• For joint chance constraints with right-hand side uncertainty, the feasible region is convex if the Wasserstein center is a log-concave distribution or more generally an α-concave distribution with α ≥ -1.
• The paper provides a mixed-integer conic representation for the pessimistic case, extending prior work by Xie (2019) and Chen et al. (2018).
• A block coordinate ascent algorithm is proposed for optimistic chance constraints and proven to converge globally to the optimum.
• The optimistic chance constraint is reformulated as a single robust constraint involving the CVaR of the distance to the feasible set, enabling tractable solution.
• The directional derivative of the value function is shown to be continuous and positively homogeneous, supporting the convexity analysis.