## Abstract

We present three different robust frameworks using probabilistic ambiguity descriptions of the data in least squares problems. These probability ambiguity descriptions are given by: (1) confidence region over the first two moments; (2) bounds on the probability measure with moments constraints; (3) the Kantorovich probability distance from a given measure. For the first case, we give an equivalent formulation and show that the optimization problem can be solved using a semidefinite optimization reformulation or polynomial time algorithms. For the second case, we derive the equivalent Lagrangian problem and show that it is a convex stochastic programming problem. We further analyze three special subcases: (i) finite support; (ii) measure bounds by a reference probability measure; (iii) measure bounds by two reference probability measures with known density functions. We show that case (i) has an equivalent semidefinite programming reformulation and the sample average approximations of case (ii) and (iii) have equivalent semidefinite programming reformulations. For ambiguity description (3), we show that the finite support case can be solved by using an equivalent second order cone programming reformulation.

Original language | English (US) |
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Pages (from-to) | 123-141 |

Number of pages | 19 |

Journal | Mathematical Programming |

Volume | 146 |

Issue number | 1-2 |

DOIs | |

State | Published - Aug 2014 |

## Keywords

- 62G35 Robustness
- 62J05 Linear regression
- 62J07 Ridge regression; shrinkage estimators
- 90C15 Stochastic programming
- 90C22 Semidefinite programming
- 93B35 Sensitivity (robustness)
- Mathematics Subject Classification: 93E24 Least squares and related methods

## ASJC Scopus subject areas

- Software
- Mathematics(all)