### Abstract

Optimization problems face random constraint violations when uncertainty arises in constraint parameters. Effective ways of controlling such violations include risk constraints, e.g., chance constraints and conditional Value-at-Risk constraints. This paper studies these two types of risk constraints when the probability distribution of the uncertain parameters is ambiguous. In particular, we assume that the distributional information consists of the first two moments of the uncertainty and a generalized notion of unimodality. We find that the ambiguous risk constraints in this setting can be recast as a set of second-order cone (SOC) constraints. In order to facilitate the algorithmic implementation, we also derive efficient ways of finding violated SOC constraints. Finally, we demonstrate the theoretical results via computational case studies on power system operations.

Original language | English (US) |
---|---|

Pages (from-to) | 1-42 |

Number of pages | 42 |

Journal | Mathematical Programming |

DOIs | |

State | Accepted/In press - Nov 24 2017 |

Externally published | Yes |

### Fingerprint

### Keywords

- Ambiguity
- Chance constraints
- Conditional Value-at-Risk
- Golden section search
- Second-order cone representation
- Separation

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Mathematical Programming*, 1-42. https://doi.org/10.1007/s10107-017-1212-x

**Ambiguous risk constraints with moment and unimodality information.** / Li, Bowen; Jiang, Ruiwei; Mathieu, Johanna L.

Research output: Contribution to journal › Article

*Mathematical Programming*, pp. 1-42. https://doi.org/10.1007/s10107-017-1212-x

}

TY - JOUR

T1 - Ambiguous risk constraints with moment and unimodality information

AU - Li, Bowen

AU - Jiang, Ruiwei

AU - Mathieu, Johanna L.

PY - 2017/11/24

Y1 - 2017/11/24

N2 - Optimization problems face random constraint violations when uncertainty arises in constraint parameters. Effective ways of controlling such violations include risk constraints, e.g., chance constraints and conditional Value-at-Risk constraints. This paper studies these two types of risk constraints when the probability distribution of the uncertain parameters is ambiguous. In particular, we assume that the distributional information consists of the first two moments of the uncertainty and a generalized notion of unimodality. We find that the ambiguous risk constraints in this setting can be recast as a set of second-order cone (SOC) constraints. In order to facilitate the algorithmic implementation, we also derive efficient ways of finding violated SOC constraints. Finally, we demonstrate the theoretical results via computational case studies on power system operations.

AB - Optimization problems face random constraint violations when uncertainty arises in constraint parameters. Effective ways of controlling such violations include risk constraints, e.g., chance constraints and conditional Value-at-Risk constraints. This paper studies these two types of risk constraints when the probability distribution of the uncertain parameters is ambiguous. In particular, we assume that the distributional information consists of the first two moments of the uncertainty and a generalized notion of unimodality. We find that the ambiguous risk constraints in this setting can be recast as a set of second-order cone (SOC) constraints. In order to facilitate the algorithmic implementation, we also derive efficient ways of finding violated SOC constraints. Finally, we demonstrate the theoretical results via computational case studies on power system operations.

KW - Ambiguity

KW - Chance constraints

KW - Conditional Value-at-Risk

KW - Golden section search

KW - Second-order cone representation

KW - Separation

UR - http://www.scopus.com/inward/record.url?scp=85035135920&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85035135920&partnerID=8YFLogxK

U2 - 10.1007/s10107-017-1212-x

DO - 10.1007/s10107-017-1212-x

M3 - Article

SP - 1

EP - 42

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

ER -