| PROBTYPE |
Problem Type |
OVC, where O is
| – L, | if \(Q^0=0\), else |
| – D, | if \(Q^0\succeq 0, \textrm{sense}=\min, Q^0 = {Q^0}^\top\), else |
| – D, | if \(Q^0\preceq 0, \textrm{sense}=\max, Q^0 = {Q^0}^\top\), else |
| – C, | if \(Q^0\succeq 0, \textrm{sense}=\min\), else |
| – C, | if \(Q^0\preceq 0, \textrm{sense}=\max\), else |
| – Q; |
V is
| – C, | if \(\mathcal{Z} = \emptyset\), else |
| – B, | if \(\mathcal{N} = \mathcal{B}\), else |
| – M, | if \(\mathcal{Z} \setminus \mathcal{B} = \emptyset\), else |
| – I, | if \(\mathcal{N} = \mathcal{Z}\), else |
| – G; |
C is
| – N, | if \(\mathcal{M} = \emptyset\), \(l_j=-\infty, u_j=+\infty, j\in\mathcal{N}\setminus\mathcal{B}\), else |
| – B, | if \(\mathcal{M} = \emptyset\), else |
| – L, | if \(\mathcal{Q} = \emptyset\), else |
| – D, | if \(\forall i\in\mathcal{Q}: Q^i\succeq 0 \textrm{ if } c_l^i\neq-\infty\) and \(Q^i\preceq 0 \textrm{ if } c_u^i\neq+\infty\) and \(Q^i={Q^i}^\top\), else |
| – C, | if \(\forall i\in\mathcal{Q}: Q^i\succeq 0 \textrm{ if } c_l^i\neq-\infty\) and \(Q^i\preceq 0 \textrm{ if } c_u^i\neq+\infty\), else |
| – Q. |
|
| SOLOBJVALUE |
Solution point objective value |
\(\frac{1}{2} (x^*)^\top Q^0 x^* + b^0 x^* + q^0\) |
| SOLINFEASIBILITY |
Solution point infeasibility |
The maximum of
- \( \max\{c_l^i - \frac{1}{2}(x^*)^\top Q^i x^* - b^i x^*, \frac{1}{2}(x^*)^\top Q^i x^* + b^i x^* - c_u^i\}, i\in\mathcal{M} \),
- \( \max\{l_j - x_j^*, x_j^* - u_j\}, j\in\mathcal{N} \),
- \( |x_j^* - \text{round} (x_j^*)|, j\in\mathcal{Z} \), and
- 0.
|
| DONOR |
Donor of instance |
Name of person who submitted this instance to QPLIB. |
| NVARS |
Number of Variables |
\(|\mathcal{N}|\) |
| NCONS |
Number of Constraints |
\(|\mathcal{M}|\) |
| NBINVARS |
Number of Binary Variables |
\(|\mathcal{B}|\) |
| NINTVARS |
Number of (General) Integer Variables |
\(|\mathcal{Z} \setminus \mathcal{B}|\) |
| NNLVARS |
Number of Nonlinear Variables |
\(|\mathcal{N}^Q|\) |
| NNLBINVARS |
Number of Nonlinear Binary Variables |
\(|\mathcal{N}^Q\cap\mathcal{B}|\) |
| NNLINTVARS |
Number of Nonlinear Integer Variables |
\(|\mathcal{N}^Q\cap\mathcal{Z}|\) |
| NBOUNDEDVARS |
Number of Bounded Non-Binary Variables |
\(|\{j\in \mathcal{N}\setminus\mathcal{B} : l_j \neq -\infty, u_j \neq \infty\}|\) |
| NSINGLEBOUNDEDVARS |
Number of Variables with only one bound |
\(|\{j\in \mathcal{N} : \textrm{either}\; l_j = -\infty \;\textrm{or}\; u_j = \infty\}|\) |
| OBJSENSE |
Objective sense |
\(\textrm{sense}\) |
| OBJTYPE |
Objective type |
linear if \(Q^0=0\)
quadratic if \(Q^0\neq 0\) |
| NLINCONS |
Number of linear constraints |
\(|\mathcal{L}|\) |
| NQUADCONS |
Number of quadratic constraints |
\(|\mathcal{Q}|\) |
| NDIAGQUADCONS |
Number of diagonal quadratic constraints |
\(|\{i\in\mathcal{Q} : \forall j,k\in\mathcal{N},j\neq k: Q^i_{j,k} = 0\}|\) |
| NOBJNZ |
Number of nonzeros in objective |
\(|\{j\in\mathcal{N} \,:\, b^0_j\neq 0 \;\textrm{or}\; \exists k\in\mathcal{N} : Q^0_{j,k}+Q^0_{k,j} \neq 0\}|\) |
| NOBJNLNZ |
Number of nonlinear nonzeros in objective |
\(|\{j\in\mathcal{N} \,:\, \exists k\in\mathcal{N} : Q^0_{j,k}+Q^0_{k,j} \neq 0\}|\) |
| NOBJQUADNZ |
Number of quadratic terms in objective |
\(|\{(j,k)\in\mathcal{N}\times\mathcal{N} \,:\, Q^0_{j,k}\neq 0\}|\) |
| NOBJQUADDIAGNZ |
Number of square terms in objective |
\(|\{j\in\mathcal{N} \,:\, Q^0_{j,j}\neq 0\}|\) |
| OBJQUADDENSITY |
Density of \(Q^0\) |
\(\frac{2\times\textrm{NOBJQUADNZ}-\textrm{NOBJQUADDIAGNZ}}{\textrm{NOBJNLNZ}^2}\), if \(Q^0\neq 0\), otherwise 0 |
| NJACOBIANNZ |
Number of nonzeros in Jacobian |
\(\sum_{i\in\mathcal{M}} |\{j\in\mathcal{N} \,:\, b^i_j\neq 0 \;\textrm{or}\; \exists k\in\mathcal{N} : Q^i_{j,k}+Q^i_{k,j} \neq 0\}|\) |
| NJACOBIANNLNZ |
Number of nonlinear nonzeros in Jacobian |
\(\sum_{i\in\mathcal{M}} |\{j\in\mathcal{N} \,:\, \exists k\in\mathcal{N} : Q^i_{j,k}+Q^i_{k,j} \neq 0\}|\) |
| NZ |
Number of nonzeros in Objective Gradient and Jacobian |
\(\textrm{NOBJNZ} + \textrm{NJACOBIANNZ}\) |
| NLNZ |
Number of nonlinear nonzeros in Objective Gradient and Jacobian |
\(\textrm{NOBJNLNZ} + \textrm{NJACOBIANNLNZ}\) |
| NLAGHESSIANNZ |
Number of nonzeros in Hessian of Lagrangian |
\(|\{(j,k)\in\mathcal{N}\times\mathcal{N} \,:\, \exists i\in\mathcal{M}\cup\{0\} : Q^i_{j,k}\neq 0\}|\) |
| NLAGHESSIANDIAGNZ |
Number of nonzeros in diagonal of Hessian of Lagrangian |
\(|\{j\in\mathcal{N} \,:\, \exists i\in\mathcal{M}\cup\{0\} : Q^i_{j,j}\neq 0\}|\) |
| OBJCURVATURE |
Convexity/Concavity of objective function |
linear, if \(Q^0=0\), else
convex, if \(Q^0\succeq 0\), else
concave, if \(Q^0\preceq 0\), else
indefinite.
|
| NOBJQUADNEGEV |
Number of negative eigenvalues in \(Q^0\) |
\(|\{\lambda \leq -10^{-12} \,:\, \exists v\neq 0 : Q^0v = \lambda v\}|\)
|
| NOBJQUADPOSEV |
Number of positive eigenvalues in \(Q^0\) |
\(|\{\lambda \geq 10^{-12} \,:\, \exists v\neq 0 : Q^0v = \lambda v\}|\)
|
| OBJQUADPROBLEVFRAC |
Fraction of problematic eigenvalues of \(Q^0\) |
\(\frac{\textrm{NOBJQUADNEGEV}}{|\mathcal{N}|}\), if \(\textrm{sense}=\min\);
\(\frac{\textrm{NOBJQUADPOSEV}}{|\mathcal{N}|}\), if \(\textrm{sense}=\max\)
|
| CONSCURVATURE |
Convexity/Concavity of constraints |
| linear, | if \(\mathcal{L}=\mathcal{M}\), else |
| convex, | if \(\forall i\in\mathcal{Q}, c_u^i\neq+\infty: Q^i\succeq 0\) and \(\forall i\in\mathcal{Q}, c_l^i\neq-\infty: Q^i\preceq 0\), else |
| concave, | if \(\forall i\in\mathcal{Q}, c_u^i\neq+\infty: Q^i\preceq 0\) and \(\forall i\in\mathcal{Q}, c_l^i\neq-\infty: Q^i\succeq 0\), else |
| indefinite. |
|
| CONVEX |
Convexity of continuous relaxation |
| True, | if \(\textrm{CONSCURVATURE}\) = convex and \(\textrm{OBJCURVATURE}\) = convex, and \(\textrm{sense}=\min\), else |
| True, | if \(\textrm{CONSCURVATURE}\) = convex and \(\textrm{OBJCURVATURE}\) = concave, and \(\textrm{sense}=\max\), else |
| False. |
|
| NCONVEXNLCONS |
Number of convex nonlinear constraints |
\(|\{i\in\mathcal{Q} \;:\; Q^i\succeq 0 \textrm{ if } c_u^i\neq+\infty \textrm{ and } Q^i\preceq 0 \textrm{ if } c_l^i\neq-\infty\}|\) |
| NCONCAVENLCONS |
Number of concave nonlinear constraints |
\(|\{i\in\mathcal{Q} \;:\; Q^i\preceq 0 \textrm{ if } c_u^i\neq+\infty \textrm{ and } Q^i\succeq 0 \textrm{ if } c_l^i\neq-\infty\}|\) |
| NINDEFINITENLCONS |
Number of indefinite nonlinear constraints |
\(|\{i\in\mathcal{Q} \;:\; Q^i\not\succeq 0 \textrm{ and } Q^i\not\preceq 0 \}|\) |
| NLAGHESSIANBLOCKS |
Number of blocks in Hessian of Lagrangian |
\(|\mathcal{P}|\) |
| LAGHESSIANMINBLOCKSIZE |
Minimal blocksize in Hessian of Lagrangian |
\(\min\{|P_k| \;:\; k\in\mathcal{P}\}\) |
| LAGHESSIANMAXBLOCKSIZE |
Maximal blocksize in Hessian of Lagrangian |
\(\max\{|P_k| \;:\; k\in\mathcal{P}\}\) |
| LAGHESSIANAVGBLOCKSIZE |
Average blocksize in Hessian of Lagrangian |
\(\frac{1}{|\mathcal{P}|}\sum_{k\in\mathcal{P}} |P_k|\) |