# General mixed-integer quadratically constrained problem instances

Problem description

This library contains instances of quadratically constrained Mixed Integer Quadratic Programs (MIQCP_m) that can be formulated as follows:

Min f(x) = xt Q0 x + ct x
(MIQCPm) s.t.
 xt Qr x + c r t x = bqr r=1, ..., mq : mq quadratic equalities equalities xt Qr x + c r t x ≤ bqr r=mq+1, ..., (mq+pq) : pq quadratic inequalities 0 ≤ ℓ ≤ x ≤ u n positive and lower and upper bounded variables xi ∈ N i=1,..,nb_int integer variables xi ∈ R i=nb_int,...,n real variables

Where Qr ∈ Sn, cr ∈ Rn, bq ∈ Rmq+pq, ℓ ∈ Rn, and u ∈ Rn. In these instances, for all r= 0, ..., mq+pq, the submatrix of pure real quadratic terms of each matrix Qr defined by (q(rij) ∈ {nb_int..n}x{nb_int..n} is positive semidefinite.

Files format

Each .dat file contains one instance, in the format of solver SMIQCP:

n nb_int 0 0 mq pq
u
u1, u2... un

1, ℓ2... &elln
Q
nnzQ0
i j q0ij
C
nnzc0
i c0i
Aq
nnzQr + nnzcr r=1..mq
r 0 i+1 cri
r i+1 j+1 qrij
bq
nnzbr r=1..mq
r i bri
Dq
nnzQr + nnzcr r=1,..,pq
r 0 i+1 cri
r i+1 j+1 qrij
eq
nnzer r=1..pq
r i eri
where nnzM is the number of non zero elements of matrix/vector M.

An example is available here

The optimal solution values are reported here :

• References

 S. Elloumi and A. Lambert. Global solution of non-convex quadratic=ally constrained quadratic programs Optimization Methods and Software 34 (1): 98-114, (2019) DOI: 10.1080/10556788.2017.1350675

 A. Billionnet, S. Elloumi and A. Lambert. Exact quadratic convex reformulations of mixed-integer quadratically constrained problems Mathematical Programming serie A. 158 (1), 235-266 (2016) DOI: 10.1007/s10107-015-0921-2

 A. Billionnet, S. Elloumi and A. Lambert. Convex reformulations of Integer Quadratically Constrained Problems, ISMP (21th International Symposium of Mathematical programming), Berlin, Germany, august 19-24, 2012.

 A. Billionnet, S. Elloumi and A. Lambert. A solution method for quadratically constrained integer problems. Optimization 2011, Lisbon, july, 2011.

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