Convex Chance-Constrained Programs with Wasserstein Ambiguity
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866913733139234816 |
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| author | Shen, Haoming Jiang, Ruiwei |
| author_facet | Shen, Haoming Jiang, Ruiwei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_02486 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Convex Chance-Constrained Programs with Wasserstein Ambiguity Shen, Haoming Jiang, Ruiwei Optimization and Control 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. |
| title | Convex Chance-Constrained Programs with Wasserstein Ambiguity |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2111.02486 |