MINLPLib

A Library of Mixed-Integer and Continuous Nonlinear Programming Instances

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Documentation

MINLPLib provides instances of optimization problems of the form \[ \newcommand{\dom}{\mathrm{dom}} \begin{align*} \textrm{sense} \;\; & f_0(x) \\ \textrm{such that}\;\; & f_i(x) \simeq_i r_i & i \in \mathcal{M} \\ & l_j \leq x_j \leq u_j & j \in \mathcal{N}\setminus\mathcal{S} \\ & x_j = 0 \vee l_j \leq x_j \leq u_j & j \in \mathcal{S} \\ & x_j \in \mathbb{Z} & j \in \mathcal{Z} \\ & \mathrm{SOS1}((x_{t_1}, \ldots, x_{t_{\dim(t)}})) & t \in \mathcal{T}_1 \\ & \mathrm{SOS2}((x_{t_1}, \ldots, x_{t_{\dim(t)}})) & t \in \mathcal{T}_2 \end{align*} \] where Note, that for the sake of simplicity, some of the definitions on this page assume differentiability (once or twice) of a function on its domain. These definitions may also be extended to functions that are not differentiable everywhere, though all functions that appear in instances on MINLPLib are differentiable almost everywhere on their domain. Further, let

For each instance, we collect the following information. Most of it is shown on the detailed page for each instance, while the instances listing page shows partially aggregated information.

Identifier Meaning Definition
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 feasible solution points are available. These are listed here, together with their maximal absolute constraint violation. The best primal bound is set bold. 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 \(|\mathcal{T}_2|\)
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].

For each solution point, we also collect some information:

Identifier Meaning Definition
FORMATS Available file formats Each solution point is available in GAMS Data Exchange (GDX) binary format and an easy to parse ASCII text format that lists all nonzero solution values.
ADDDATE Date of addition The date when the point was added to the library.
OBJVALUE Objective value Value of \(f_0(x)\).
INFEASIBILITY Constraint violation The maximal absolute violation of all problem constraints, including variable bounds, SOS, etc. Usually, a solution polishing process is applied to each point before addition to the library. This ensures that for each point, constraints on integrality, variable bounds, semicontinuity, and special-ordered-sets are exactly satisfied and only violations of the algebraic constraints \(f_i(x)\simeq_i r_i\), \(i\in\mathcal{M}\), may remain.

Last updated: 2018-09-14 Git hash: ac5a5314
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