## Abstract

The standard quadratic optimization problem (StQP), i.e. the problem of minimizing a quadratic form x^{T}Qx on the standard simplex { x≥ 0: x^{T}e= 1 } , is studied. The StQP arises in numerous applications, and it is known to be NP-hard. Chen, Peng and Zhang showed that almost certainly the StQP with a large random matrix Q= Q^{T}, Qi,j,(i≤j) being independent and identically concave-distributed, attains its minimum at a point x with support size | { j: x_{j}> 0 } | bounded in probability. Later Chen and Peng proved that for Q= (M+ M^{T}) / 2 , with M_{i}_{,}_{j} i.i.d. normal, the likely support size is at most 2. In this paper we show that the likely support size is poly-logarithmic in n, the problem size, for a considerably broader class of the distributions. Unlike the cited papers, the mild constraints are put on the asymptotic behavior of the distribution at a single left endpoint of its support, rather than on the distribution’s shape elsewhere. It also covers the distributions with the left endpoint - ∞, provided that the distribution of Qi,j,(i≤j) (of M_{i}_{,}_{j}, if Q= (M+ M^{T}) / 2 resp.) has a (super/sub) exponentially narrow left tail.

Original language | English (US) |
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Pages (from-to) | 309-336 |

Number of pages | 28 |

Journal | Mathematical Programming |

Volume | 186 |

Issue number | 1-2 |

DOIs | |

State | Published - Mar 2021 |

## Keywords

- Probability analysis
- Random matrices
- Sparse solutions
- Standard quadratic programming

## ASJC Scopus subject areas

- Software
- Mathematics(all)