These are 90 random instances of the Mean-Variance (MV) problem with minimum buy-in constraints and cardinality, a Mixed-Integer Quadratic program used in [FrGe06, FrGe07, FrFG16a, FrFG16b, ZhSL14] to test some approaches based on the "Perspective Reformulation" (PR) to Mixed-Integer Quadratic Problems with specific structure (nonlinear semicontinuous variables). The instances are all contained in a single file (6.6Mb); please refer to the provided file README.txt and the cited references for further details.

To ease life, we now also directly provide .lp files of the instances, besides the raw data to produce them:

200 (2.1Mb) | 300 (4.6Mb) | 400 (7.7Mb) |

Applying PR techniques to MV, which has a convex non-separable quadratic
objective with Positive-Semidefinite (SDP) Hessian *Q*, requires
finding a diagonal non-negative matrix *D* such that *Q - D* is
still SDP. Different ways for finding *D*, with different trade-offs
between the required time and the quality of the corresponding lower bound,
are proposed in [FrGe07] and [ZhSL14]. These require the solution of
nontrivial SDP problems, in particular for [ZhSL14]. To facilitate reproducing
our results, in particular those of [FrFG16b], some pre-computed diagonals for
the above instances are provided here. Please
refer to the included READ_ME.txt file and the cited references for further
details.

A possible approach to solve these problems is to first reformulate them with the Approximated Projected Perspective Reformulation technique [FrFG16a] (after having extracted the diagonal, see above) prior to sending them to any MIQP solver. We now also provide .lp files of the instances after the reformulation, for all the three types of diagonal; check the readme for more details.

"small" c | 200 (2.1Mb) | 300 (4.6Mb) | 400 (7.7Mb) |

"large" c | 200 (2.1Mb) | 300 (4.6Mb) | 400 (7.7Mb) |

"convex" c | 200 (2.1Mb) | 300 (4.6Mb) | 400 (7.7Mb) |

An improvement on the Approximated Projected Perspective Reformulation technique [FrFG16b] uses dual information to further improve the bound. We provide the optimal dual values for each instance and diagonal, as well as the .lp files of the instances after the reformulation, for all the three types of diagonal. Check the readme for more details.

"small" c | 200 (2.1Mb) | 300 (4.6Mb) | 400 (7.7Mb) |

"large" c | 200 (2.1Mb) | 300 (4.6Mb) | 400 (7.7Mb) |

"convex" c | 200 (2.1Mb) | 300 (4.6Mb) | 400 (7.7Mb) |

Last updated: 30/01/2017.

[FrGe06] A. Frangioni, C. Gentile
"Perspective
Cuts for a Class of Convex 0-1 Mixed Integer Programs" *Mathematical
Programming* **106**(2), 225—236, 2006

[FrGe07] A. Frangioni, C. Gentile
"SDP
Diagonalizations and Perspective Cuts for a Class of Nonseparable MIQP",
*Operations Research Letters* **35**(2), 181—185, 2007

[FrFG16a] A. Frangioni, F. Furini, C. Gentile
"Approximated
Perspective Relaxations: a Project&Lift Approach" *Computational
Optimization and Applications* **63**(3), 705—735, 2016

[FrFG16b] A. Frangioni, F. Furini, C. Gentile
"Improving the
Approximated Projected Perspective Reformulation by Dual Information"
*Technical Report*, Dipartimento di Informatica, Università di
Pisa, 2016

[ZhSL14] X. Zheng, X. Sun, D. Li.
"Improving the Performance of MIQP Solvers for Quadratic Programs with
Cardinality and Minimum Threshold Constraints: A Semidefinite Program
Approach" *INFORMS Journal on Computing* **26**(4):690—703, 2014