### Abstract

The second-order tangent set is an important concept in describing the curvature of the set involved. Due to the existence of the complementarity condition, the second-order cone (SOC) complementarity set is a nonconvex set. Moreover, unlike the vector complementarity set, the SOC complementarity set is not even the union of finitely many polyhedral convex sets. Despite these difficulties, we succeed in showing that like the vector complementarity set, the SOC complementarity set is second-order directionally differentiable and an exact formula for the second-order tangent set of the SOC complementarity set can be given. We derive these results by establishing the relationship between the second-order tangent set of the SOC complementarity set and the second-order directional derivative of the projection operator over the SOC, and calculating the second-order directional derivative of the projection operator over the SOC. As an application, we derive second-order necessary optimality conditions for the mathematical program with SOC complementarity constraints.

Original language | English |
---|---|

Pages (from-to) | 2986-3011 |

Number of pages | 26 |

Journal | SIAM Journal on Optimization |

Volume | 29 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2019 Jan 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Mathematical program with second-order cone complementarity constraints
- Projection operator
- Second-order cone complementarity sets
- Second-order directional derivatives
- Second-order necessary optimality conditions
- Second-order tangent sets

### ASJC Scopus subject areas

- Software
- Theoretical Computer Science

### Cite this

^{∗}

*SIAM Journal on Optimization*,

*29*(4), 2986-3011. https://doi.org/10.1137/17M1140479