# MINLPLib

### 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
• $$\mathcal{N} = \{ 1, \ldots, n \}$$ is the set of (indices) of variables,
• $$\mathcal{M} = \{ 1, \ldots, m \}$$ is the set of (indices) of constraints expressed as algebraic equations,
• $$x = [x_j]_{j = 1}^n \in \mathbb{R}^n$$ is a finite vector of variables,
• $$\textrm{sense}\in\{\min,\max\}$$ is the objective sense,
• $$f_0:\dom(f_0)\to\mathbb{R}$$ is the objective function given as an algebraic expression, $$\dom(f_0)\subseteq\mathbb{R}^n$$,
• $$f_i:\dom(f_i)\to\mathbb{R}$$ is a constraint function given as an given as an algebraic expression, $$\dom(f_i)\subseteq\mathbb{R}^n$$, $$i\in\mathcal{M}$$,
• $$\simeq_i\in\{\leq,\geq,=\}$$ is a relation symbol, $$i\in\mathcal{M}$$,
• $$r_i\in\mathbb{R}$$ is the right-hand-side value of a constraint, $$i\in\mathcal{M}$$
• $$l_j\in\mathbb{R}\cup\{-\infty\}, u_j\in\mathbb{R}\cup\{+\infty\}$$, $$l_j\leq u_j$$ are the lower and upper bounds, respectively, on each variable $$x_j$$, $$j \in \mathcal{N}$$,
• $$\mathcal{S} \subseteq \mathcal{N}$$ are the semi-continuous/semi-integer variables,
• $$\mathcal{Z} \subseteq \mathcal{N}$$ are the variables restricted to only attain integer values,
• $$\mathcal{T}_k$$ is the set of special-ordered-set (SOS) conditions of type $$k$$, $$k\in\{1,2\}$$,
• $$t\in\mathcal{T}_k$$ is a vector of length $$\dim(t)$$ with pairwise distinct entries from $$\mathcal{N}$$,
• $$\mathrm{SOS1}(v)$$ is true, iff at most one entry of the vector v is not zero ($$|\{k \in \{1,\ldots,\dim(v)\} : v_k > 0 \}| \leq 1$$),
• $$\mathrm{SOS2}(v)$$ is true, iff either at most one entry of the vector v is not zero or two consecutive entries of the vector v are not zero.
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
• $$\mathcal{M}_0 := \mathcal{M}\cup\{0\}$$,
• $$\mathcal{B} := \{j\in\mathcal{Z} \,:\, \{l_j,u_j\}=\{0,1\}\}$$ be the set of binary variables,
• $$\mathcal{N}^\mathrm{NZ}(h) := \{j\in\mathcal{N} \,:\, \exists \hat x \in\dom(h) : \frac{\partial h}{\partial x_j}(\hat x) \neq 0 \}$$ be the set of variables that appear in a function $$h:\mathbb{R}^n\to\mathbb{R}$$,
• $$\mathcal{N}^\mathrm{NL}(h) := \{j\in\mathcal{N} \,:\, \exists \hat x, \hat x'\in\dom(h) : \hat x\neq \hat x', \frac{\partial h}{\partial x_j}(\hat x) \neq \frac{\partial h}{\partial x_j}(\hat x')\}$$ be the set of variables that appear nonlinearly in a function $$h:\mathbb{R}^n\to\mathbb{R}$$,
• $$\mathrm{constant}(h) := (\exists c_0\in\mathbb{R} : \forall x\in\mathbb{R}^n : h(x) = c_0$$ indicate that a function $$h:\mathbb{R}^n\to\mathbb{R}$$ is constant,
• $$\mathrm{linear}(h) := (\exists c\in\mathbb{R}^n, c_0\in\mathbb{R} : \forall x\in\mathbb{R}^n : h(x) = c_0 + c^\top x)$$ indicate that a function $$h:\mathbb{R}^n\to\mathbb{R}$$ is affine-linear,
• $$\mathrm{quadratic}(h) := (\exists Q \in \mathbb{R}^{n\times n}, q\in\mathbb{R}^n, q_0\in\mathbb{R} : \forall x\in\mathbb{R}^n : h(x) = q_0 + q^\top x + x^\top Qx)$$ indicate that a function $$h:\mathbb{R}^n\to\mathbb{R}$$ is quadratic,
• $$\mathrm{polynomial}(h) := (\exists K, |K|<\infty: \exists a_{k}\in\mathbb{N}_{\geq 0}^n, b_k\in\mathbb{R} : \forall x\in\mathbb{R}^n : h(x) = \sum_{k\in K} b_k \prod_{i\in \mathcal{N}} x_i^{a_{k,i}})$$ indicate that a function $$h:\mathbb{R}^n\to\mathbb{R}$$ is polynomial,
• $$\mathrm{signomial}(h) := (\exists K, |K|<\infty: \exists a_{k}\in\mathbb{R}^n, b_k\in\mathbb{R} : \forall x\in\mathbb{R}^n : h(x) = \sum_{k\in K} b_k \prod_{i\in \mathcal{N}} x_i^{a_{k,i}})$$ indicate that a function $$h:\mathbb{R}^n\to\mathbb{R}$$ is signomial,
• curvature$$(h)\in$$ {linear, convex, concave, indefinite, nonconvex, nonconcave, unknown} is the best known "curvature" property of a function $$h:\mathbb{R}^n\to\mathbb{R}$$ on $$\dom(h)\cap[l,u]$$. Since the curvature cannot always be decided easily, the latter three values are used for (partially) inconclusive knownledge. For example, a function is found to be nonconvex if a point $$\hat x\in \dom(h)\cap[l,u]$$ is known such that $$\nabla^2 h(\hat x)\not\succeq 0$$, but neither concavity ($$\forall \hat x \in \dom(h)\cap[l,u] : \nabla^2 h(\hat x)\preceq 0$$) nor indefiniteness ($$\exists \hat x,\hat x' \in \dom(h)\cap[l,u] : \nabla^2 h(\hat x)\preceq 0, \nabla^2 h(\hat x')\succeq 0)$$ could be proven. However, since proving indefiniteness of an indefinite function is likely (thus, not surely) to succeed, a nonconvex or nonconcave curvature is likely to indicate a concave or convex, respectively, function.
• $$P := \{P_k\}_{k\in\mathcal{P}}$$, $$\mathcal{P} := \{1,\ldots,p\}$$, be the finest partitioning of $$\bigcup_{i\in\mathcal{M}_0}\mathcal{N}^\mathrm{NL}(f_i)$$ such that $$\forall i\in\mathcal{M}\cup\{0\} \;\forall \hat x \in \dom(f_i) \;\forall j,k\in\mathcal{N}^\mathrm{NL}(f_i), \frac{\partial^2 f_i}{\partial x_j\partial x_k}(\hat x)\neq 0 : \exists p\in\mathcal{P} : j,k\in P_p$$. That is, $$P$$ describes the block structure of the Hessian of the Lagrangian.

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 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.
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].

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.
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.