### Abstract

In this paper, we survey known results and present new ones. In particular we give an easy proof of the fact that Slater’s condition holds generically in linear conic programming. We further discuss the problem of stability of uniqueness, nondegeneracy, and strict complementarity. We also comment on the fact that in general, a conic program cannot be treated as a smooth problem and that techniques from nonsmooth geometric measure theory are needed.

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

Pages (from-to) | 77 |

Number of pages | 94 |

Journal | Mathematics of operations research |

Volume | 42 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Feb 2017 |

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### Keywords

- conic optimization
- generic properties
- Slater’s condition
- uniqueness and nondegeneracy of optimal solutions
- strict complementarity
- stability

### Cite this

*Mathematics of operations research*,

*42*(1), 77. https://doi.org/10.1287/moor.2016.0793

}

*Mathematics of operations research*, vol. 42, no. 1, pp. 77. https://doi.org/10.1287/moor.2016.0793

**Genericity Results in Linear Conic Programming : A Tour d’Horizon.** / Dür, Mirjam; Jargalsaikhan, Bolor; Still, Georg J.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Genericity Results in Linear Conic Programming

T2 - A Tour d’Horizon

AU - Dür, Mirjam

AU - Jargalsaikhan, Bolor

AU - Still, Georg J.

N1 - Finally, they gratefully acknowledge support of the Netherlands Organisation for Scientific Research [Vici Grant 639.033.907].

PY - 2017/2/1

Y1 - 2017/2/1

N2 - This paper is concerned with so-called generic properties of general linear conic programs. Many results have been obtained on this subject during the last two decades. For example, it is known that uniqueness, strict complementarity, and nondegeneracy of optimal solutions hold for almost all problem instances. Strong duality holds generically in a stronger sense, i.e., it holds for a generic subset of problem instances.In this paper, we survey known results and present new ones. In particular we give an easy proof of the fact that Slater’s condition holds generically in linear conic programming. We further discuss the problem of stability of uniqueness, nondegeneracy, and strict complementarity. We also comment on the fact that in general, a conic program cannot be treated as a smooth problem and that techniques from nonsmooth geometric measure theory are needed.

AB - This paper is concerned with so-called generic properties of general linear conic programs. Many results have been obtained on this subject during the last two decades. For example, it is known that uniqueness, strict complementarity, and nondegeneracy of optimal solutions hold for almost all problem instances. Strong duality holds generically in a stronger sense, i.e., it holds for a generic subset of problem instances.In this paper, we survey known results and present new ones. In particular we give an easy proof of the fact that Slater’s condition holds generically in linear conic programming. We further discuss the problem of stability of uniqueness, nondegeneracy, and strict complementarity. We also comment on the fact that in general, a conic program cannot be treated as a smooth problem and that techniques from nonsmooth geometric measure theory are needed.

KW - conic optimization

KW - generic properties

KW - Slater’s condition

KW - uniqueness and nondegeneracy of optimal solutions

KW - strict complementarity

KW - stability

U2 - 10.1287/moor.2016.0793

DO - 10.1287/moor.2016.0793

M3 - Article

VL - 42

SP - 77

JO - Mathematics of operations research

JF - Mathematics of operations research

SN - 0364-765X

IS - 1

ER -