| FORMATS |
Available file formats |
Each instance is available in a number of file formats. As not every format may allow to express an instance, some formats are not be available for some instances. |
| POINTS |
Solution points |
For most instances, one or several solution points are available. These are listed here, together with their maximal absolute constraint violation.
Points with a constraint violation below the feasibility tolerance are used determine the best primal bound, which is set bold.
Points with a higher constraint violation are also listed, as their objective value can give some indication on how sensitive the objective function is w.r.t. feasibility tolerances.
See also the FAQ. |
| PRIMALBOUND |
Primal Bound |
The objective value of the best known feasible solution point. See also POINTS. |
| DUALBOUNDS |
Dual Bounds |
Dual bounds on the optimal value as reported by some solvers. The 1st, 2nd, and 3rd best bound are in bold. See also the FAQ. |
| S |
"Solved" status of instance |
Whether for the best known feasible solution, at least 3 solvers claim global optimality (up to a relative optimal tolerance (gap) of \(10^{-6}\)), or at least 3 solvers claim infeasibility of the instance. |
| REFERENCES |
Literature references |
Literature references regarding the source of the instance. |
| SOURCE |
Instance source |
Information on where the instance was obtained from. |
| APPLICATION |
Application area |
Information on an application area that this instance belongs to, if any. |
| ADDDATE |
Date of addition |
The date when the instance was added to the library. |
| REMOVEDATE |
Date of removal |
The date when the instance was removed from the library. |
| REMOVEREASON |
Reason of removal |
The reason why the instance was removed from the library. |
| PROBTYPE |
Problem type |
A classification of the instance type that is more specific than "MINLP". It is given by the concatentation of
| B, | if \(\mathcal{N}=\mathcal{B}\), else |
| I, | if \(\mathcal{N}=\mathcal{Z}\), else |
| MI, | if \(\mathcal{Z}\setminus\mathcal{B}\neq 0\), else |
| MB, | if \(\mathcal{B}\neq 0\), else |
| S, | if \(\mathcal{S}\cup\mathcal{T}_1\cup\mathcal{T}_2\neq 0\), else |
| [empty] |
and
| NLP, | if \(\exists i\in\mathcal{M}_0: \neg\mathrm{quadratic}(f_i) \), else |
| QCQP, | if \(\neg\mathrm{linear}(f_0) \wedge \exists i\in\mathcal{M}: \neg\mathrm{linear}(f_i) \), else |
| QP, | if \(\neg\mathrm{linear}(f_0) \), else |
| QCP, | if \(\exists i\in\mathcal{M}: \neg\mathrm{linear}(f_i) \), else |
| P. |
|
| NVARS |
Number of Variables |
\(|\mathcal{N}|\) |
| NBINVARS |
Number of Binary Variables |
\(|\mathcal{B}|\) |
| NINTVARS |
Number of (General) Integer Variables |
\(|\mathcal{Z} \setminus \mathcal{B}|\) |
| NNLVARS |
Number of Nonlinear Variables |
\(|\bigcup_{i\in\mathcal{M}_0} \mathcal{N}^\mathrm{NL}(f_i)|\) |
| NNLBINVARS |
Number of Nonlinear Binary Variables |
\(|\bigcup_{i\in\mathcal{M}_0} \mathcal{N}^\mathrm{NL}(f_i)\cap\mathcal{B}|\) |
| NNLINTVARS |
Number of Nonlinear Integer Variables |
\(|\bigcup_{i\in\mathcal{M}_0} \mathcal{N}^\mathrm{NL}(f_i)\cap\mathcal{Z}|\) |
| OBJSENSE |
Objective sense |
\(\textrm{sense}\) |
| OBJTYPE |
Objective type |
| constant, | if \(\mathrm{constant}(f_0)\), else |
| linear, | if \(\mathrm{linear}(f_0)\), else |
| quadratic, | if \(\mathrm{quadratic}(f_0)\), else |
| polynomial, | if \(\mathrm{polynomial}(f_0)\), else |
| signomial, | if \(\mathrm{signomial}(f_0)\), else |
| nonlinear. | |
|
| OBJCURVATURE |
Objective curvature |
\(\mathrm{curvature}(f_0)\) |
| NOBJNZ |
Number of nonzeros in objective Gradient |
\(|\mathcal{N}^{NZ}(f_0)|\) |
| NOBJNLNZ |
Number of nonlinear nonzeros in objective |
\(|\mathcal{N}^{NL}(f_0)|\) |
| NCONS |
Number of Constraints |
\(|\mathcal{M}|\) |
| NLINCONS |
Number of linear constraints |
\(|\{i \in\mathcal{M} : \mathrm{linear}(f_i) \wedge \neg \mathrm{constant}(f_i)\} \) |
| NQUADCONS |
Number of quadratic constraints |
\(|\{i \in\mathcal{M} : \mathrm{quadratic}(f_i) \wedge \neg \mathrm{linear}(f_i)\} \) |
| NPOLYNOMCONS |
Number of polynomial constraints |
\(|\{i \in\mathcal{M} : \mathrm{polynomial}(f_i) \wedge \neg \mathrm{quadratic}(f_i)\} \) |
| NSIGNOMCONS |
Number of signomial constraints |
\(|\{i \in\mathcal{M} : \mathrm{signomial}(f_i) \wedge \neg \mathrm{polynomial}(f_i)\} \) |
| NGENNLCONS |
Number of general nonlinear constraints |
\(|\{i \in\mathcal{M} : \neg \mathrm{signomial}(f_i)\}|\) |
| NLOPERANDS |
Nonlinear operands |
The operands that appear in the GAMS specification of general nonlinear functions (\(f_0\) and \(f_i, i\in\mathcal{M}\)). |
| CONSCURVATURE |
Constraints curvature |
| linear, | if \(\forall i \in \mathcal{M}: \mathrm{linear}(f_i)\), else |
| convex, | if \(\forall i \in \mathcal{M}, \simeq_i = "=" : \mathrm{linear}(f_i)\) and \(\forall i \in \mathcal{M}, \simeq_i = "\leq" : \mathrm{curvature}(f_i) = \mathrm{convex}\) and \(\forall i\in\mathcal{M}, \simeq_i = "\geq" : \mathrm{curvature}(f_i) = \mathrm{concave}\), else |
| concave, | if \(\forall i \in \mathcal{M}, \simeq_i = "=" : \mathrm{linear}(f_i)\) and \(\forall i \in \mathcal{M}, \simeq_i = "\leq" : \mathrm{curvature}(f_i) = \mathrm{concave}\) and \(\forall i\in\mathcal{M}, \simeq_i = "\geq" : \mathrm{curvature}(f_i) = \mathrm{convex}\), else |
| indefinite, | if \(\exists i \in \mathcal{M}, \simeq_i = "=" : \neg\mathrm{linear}(f_i)\) or \(\exists i \in \mathcal{M}, \simeq_i = "\leq" : \mathrm{curvature}(f_i) \in\{\mathrm{concave},\mathrm{indefinite}\}\) or \(\exists i\in\mathcal{M}, \simeq_i = "\geq" : \mathrm{curvature}(f_i) \in \{\mathrm{convex},\mathrm{indefinite}\}\), else |
| unknown. |
|
| NJACOBIANNZ |
Number of nonzeros in Jacobian |
\(\sum_{i\in\mathcal{M}} |\mathcal{N}^\mathrm{NZ}(f_i)|\) |
| NJACOBIANNLNZ |
Number of nonlinear nonzeros in Jacobian |
\(\sum_{i\in\mathcal{M}} |\mathcal{N}^\mathrm{NL}(f_i)|\) |
| NNZ |
Number of nonzeros in Jacobian and objective Gradient |
\(\sum_{i\in\mathcal{M}_0} |\mathcal{N}^\mathrm{NZ}(f_i)|\) |
| NLAGHESSIANNZ |
Number of nonzeros in (Upper-Left) Hessian of Lagrangian |
\(|\{(j,k)\in\mathcal{N}\times\mathcal{N} : j\geq k, \exists \hat x \in \bigcap_{i\in\mathcal{M}_0} \dom(f_i) : \exists i\in\mathcal{M}_0 : \frac{\partial^2 f_i}{\partial x_j\partial x_k} (\hat x)\neq 0 \}|\) |
| NLAGHESSIANDIAGNZ |
Number of nonzeros in diagonal of Hessian of Lagrangian |
\(|\{j\in\mathcal{N} : \exists \hat x \in \bigcap_{i\in\mathcal{M}_0} \dom(f_i) : \exists i\in\mathcal{M}_0 : \frac{\partial^2 f_i}{\partial x_j^2} (\hat x)\neq 0 \}|\) |
| 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 |
\(\frac{1}{|\mathcal{P}|}\sum_{k\in\mathcal{P}} |P_k|\) |
| NSEMI |
Number of semicontinuous/semiinteger variables |
\(|\mathcal{S}|\) |
| NNLSEMI |
Number of nonlinear semicontinuous/semiinteger variables |
\(|\bigcup_{i\in\mathcal{M}_0} \mathcal{N}^\mathrm{NL}(f_i)\cap\mathcal{S}|\) |
| NSOS1 |
Number of SOS constraints of type 1 |
\(|\mathcal{T}_1|\) |
| NSOS2 |
Number of SOS constraints of type 2 |
\(|\mathcal{T}_2|\) |
| MINCOEF |
Minimal coefficient |
The smallest absolute value of nonzero coefficients in objective and constraints functions (\(f_i(x)\)).
For nonlinear functions, the numeric constants in the algebraic expression are checked.
|
| MAXCOEF |
Maximal coefficient |
The largest absolute value of coefficients in objective and constraints functions (\(f_i(x)\)).
For nonlinear functions, the numeric constants in the algebraic expression are checked.
|
| INITINFEASIBILITY |
Infeasibility of initial point |
Instances may come with initial values for \(x\). This value is the maximal absolute violation of all constraints in this initial point. |
| SPARSITYJACOBIAN |
Sparsity pattern of objective Gradient and Jacobian |
A picture of size \(n \times (m+1)\) that shows the sparsity pattern of the Gradient of the objective function and the Jacobian. Red color indicates nonlinearity.
That is, the color of the pixel in row \(i\), \(i\in\mathcal{M}_0\), and column \(j\), \(j\in\mathcal{N}\), is
| red, | if \(j\in\mathcal{N}^{NL}(f_i)\), else |
| black, | if \(j\in\mathcal{N}^{NZ}(f_i)\), else |
| white. |
The sparsity pattern may not be available on very large instances.
|
| SPARSITYHESSIAN |
Sparsity pattern of Hessian of Lagrangian |
A picture of size \(n \times n\) that shows the sparsity pattern of the upper-right triangle of the Hessian of the Lagrangian. That is, the color of the pixel in row \(j\) and column \(k\), \(j,k\in\mathcal{N}\), \(j\geq k\), is
| black, | if \(\exists \hat x \in \bigcap_{i\in\mathcal{M}_0} \dom(f_i) : \exists i\in\mathcal{M}_0 : \frac{\partial^2 f_i}{\partial x_j\partial x_k} (\hat x)\neq 0 \), else |
| white. |
The sparsity pattern may not be available on very large instances.
|
| C |
Convexity of continuous relaxation |
| True (✔), | if CONSCURVATURE = convex and OBJCURVATURE = convex, and sense = min, else |
| True (✔), | if CONSCURVATURE = convex and OBJCURVATURE = concave, and sense = max, else |
| False (-), | if CONSCURVATURE \(\in\) {concave,indefinite,nonconvex}, else |
| False (-), | if OBJCURVATURE \(\in\) {concave,indefinite,nonconvex} and sense = min, else |
| False (-), | if OBJCURVATURE \(\in\) {convex,indefinite,nonconcave} and sense = max, else |
| [empty]. |
|