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[docs] minor formatting changes (#2939)
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‎.vale.ini‎

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# TODO(odow): investigate
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[docs/src/manual/standard_form.md]
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Vale.Spelling = OFF
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[docs/src/submodules/Bridges/implementation.md]
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Vale.Spelling = OFF
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‎docs/src/manual/standard_form.md‎

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The function types implemented in MathOptInterface.jl are:
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| Function | Description |
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| :--------------- | :---------- |
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| [`VariableIndex`](@ref) | ``x_j``, the projection onto a single coordinate defined by a variable index ``j``. |
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| [`VectorOfVariables`](@ref) | The projection onto multiple coordinates (that is, extracting a sub-vector). |
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| [`ScalarAffineFunction`](@ref) | ``a^T x + b``, where ``a`` is a vector and ``b`` scalar. |
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| [`ScalarNonlinearFunction`](@ref) | ``f(x)``, where ``f`` is a nonlinear function. |
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| [`VectorAffineFunction`](@ref) | ``A x + b``, where ``A`` is a matrix and ``b`` is a vector. |
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| [`ScalarQuadraticFunction`](@ref) | ``\frac{1}{2} x^T Q x + a^T x + b``, where ``Q`` is a symmetric matrix, ``a`` is a vector, and ``b`` is a constant. |
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| [`VectorQuadraticFunction`](@ref) | A vector of scalar-valued quadratic functions. |
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| [`VectorNonlinearFunction`](@ref) | ``f(x)``, where ``f`` is a vector-valued nonlinear function. |
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| Function | Description |
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| :-------------------------------- | :----------------------------------------------------------------------------------------------------------------- |
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| [`VariableIndex`](@ref) | ``x_j``, the projection onto a single coordinate defined by a variable index ``j`` |
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| [`ScalarAffineFunction`](@ref) | ``a^T x + b``, where ``a`` is a vector and ``b`` scalar |
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| [`ScalarQuadraticFunction`](@ref) | ``\frac{1}{2} x^T Q x + a^T x + b``, where ``Q`` is a symmetric matrix, ``a`` is a vector, and ``b`` is a constant |
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| [`ScalarNonlinearFunction`](@ref) | ``f(x)``, where ``f`` is a nonlinear function |
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| [`VectorOfVariables`](@ref) | The projection onto multiple coordinates (that is, extracting a sub-vector) |
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| [`VectorAffineFunction`](@ref) | ``A x + b``, where ``A`` is a matrix and ``b`` is a vector |
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| [`VectorQuadraticFunction`](@ref) | A vector of scalar-valued quadratic functions |
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| [`VectorNonlinearFunction`](@ref) | ``f(x)``, where ``f`` is a vector-valued nonlinear function |
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## One-dimensional sets
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The one-dimensional set types implemented in MathOptInterface.jl are:
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| Set | Description |
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| :------------------------------------------------------------- | :------------------------------------- |
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| [`LessThan(u)`](@ref MathOptInterface.LessThan) | ``(-\infty, u]`` |
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| [`GreaterThan(l)`](@ref MathOptInterface.GreaterThan) | ``[l, \infty)`` |
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| [`EqualTo(v)`](@ref MathOptInterface.GreaterThan) | ``\{v\}`` |
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| [`Interval(l, u)`](@ref MathOptInterface.Interval) | ``[l, u]`` |
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| [`Integer()`](@ref MathOptInterface.Integer) | ``\mathbb{Z}`` |
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| [`ZeroOne()`](@ref MathOptInterface.ZeroOne) | ``\{ 0, 1 \}`` |
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| [`Semicontinuous(l, u)`](@ref MathOptInterface.Semicontinuous) | ``\{ 0\} \cup [l, u]`` |
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| [`Semiinteger(l, u)`](@ref MathOptInterface.Semiinteger) | ``\{ 0\} \cup \{l,l+1,\ldots,u-1,u\}`` |
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| Set | Description |
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| :----------------------------- | :------------------------------------- |
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| [`LessThan(u)`](@ref) | ``(-\infty, u]`` |
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| [`GreaterThan(l)`](@ref) | ``[l, \infty)`` |
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| [`EqualTo(v)`](@ref) | ``\{v\}`` |
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| [`Interval(l, u)`](@ref) | ``[l, u]`` |
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| [`Integer()`](@ref) | ``\mathbb{Z}`` |
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| [`ZeroOne()`](@ref) | ``\{ 0, 1 \}`` |
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| [`Semicontinuous(l, u)`](@ref) | ``\{ 0\} \cup [l, u]`` |
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| [`Semiinteger(l, u)`](@ref) | ``\{ 0\} \cup \{l,l+1,\ldots,u-1,u\}`` |
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## Vector cones
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The vector-valued set types implemented in MathOptInterface.jl are:
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| Set | Description |
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| :---------------------------------------------------------------------| :---------- |
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| [`Reals(d)`](@ref MathOptInterface.Reals) | ``\mathbb{R}^{d}`` |
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| [`Zeros(d)`](@ref MathOptInterface.Zeros) | ``0^{d}`` |
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| [`Nonnegatives(d)`](@ref MathOptInterface.Nonnegatives) | ``\{ x \in \mathbb{R}^{d} : x \ge 0 \}`` |
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| [`Nonpositives(d)`](@ref MathOptInterface.Nonpositives) | ``\{ x \in \mathbb{R}^{d} : x \le 0 \}`` |
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| [`SecondOrderCone(d)`](@ref MathOptInterface.SecondOrderCone) | ``\{ (t,x) \in \mathbb{R}^{d} : t \ge \lVert x \rVert_2 \}`` |
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| [`RotatedSecondOrderCone(d)`](@ref MathOptInterface.RotatedSecondOrderCone) | ``\{ (t,u,x) \in \mathbb{R}^{d} : 2tu \ge \lVert x \rVert_2^2, t \ge 0,u \ge 0 \}`` |
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| [`ExponentialCone()`](@ref MathOptInterface.ExponentialCone) | ``\{ (x,y,z) \in \mathbb{R}^3 : y \exp (x/y) \le z, y > 0 \}`` |
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| [`DualExponentialCone()`](@ref MathOptInterface.DualExponentialCone) | ``\{ (u,v,w) \in \mathbb{R}^3 : -u \exp (v/u) \le \exp(1) w, u < 0 \}`` |
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| [`GeometricMeanCone(d)`](@ref MathOptInterface.GeometricMeanCone) | ``\{ (t,x) \in \mathbb{R}^{1+n} : x \ge 0, t \le \sqrt[n]{x_1 x_2 \cdots x_n} \}`` where ``n`` is ``d - 1`` |
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| [`PowerCone(α)`](@ref MathOptInterface.PowerCone) | ``\{ (x,y,z) \in \mathbb{R}^3 : x^{\alpha} y^{1-\alpha} \ge \|z\|, x \ge 0,y \ge 0 \}`` |
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| [`DualPowerCone(α)`](@ref MathOptInterface.DualPowerCone) | ``\{ (u,v,w) \in \mathbb{R}^3 : \left(\frac{u}{\alpha}\right)^{\alpha}\left(\frac{v}{1-\alpha}\right)^{1-\alpha} \ge \|w\|, u,v \ge 0 \}`` |
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| [`NormOneCone(d)`](@ref MathOptInterface.NormOneCone) | ``\{ (t,x) \in \mathbb{R}^{d} : t \ge \sum_i \lvert x_i \rvert \}`` |
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| [`NormInfinityCone(d)`](@ref MathOptInterface.NormInfinityCone) | ``\{ (t,x) \in \mathbb{R}^{d} : t \ge \max_i \lvert x_i \rvert \}`` |
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| [`RelativeEntropyCone(d)`](@ref MathOptInterface.RelativeEntropyCone) | ``\{ (u, v, w) \in \mathbb{R}^{d} : u \ge \sum_i w_i \log (\frac{w_i}{v_i}), v_i \ge 0, w_i \ge 0 \}`` |
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| [`HyperRectangle(l, u)`](@ref MathOptInterface.HyperRectangle) | ``\{x \in \bar{\mathbb{R}}^d: x_i \in [l_i, u_i] \forall i=1,\ldots,d\}`` |
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| [`NormCone(p, d)`](@ref MathOptInterface.NormCone) | ``\{ (t,x) \in \mathbb{R}^{d} : t \ge \left(\sum\limits_i \lvert x_i \rvert^p\right)^{\frac{1}{p}} \}`` |
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| [`VectorNonlinearOracle`](@ref MathOptInterface.VectorNonlinearOracle)| ``\{x \in \mathbb{R}^{dimension}: l \le f(x) \le u \}`` |
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| Set | Description |
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| :---------------------------------- | :----------------------------------------------------------------------------------------------------------------------------------------- |
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| [`Reals(d)`](@ref) | ``\mathbb{R}^{d}`` |
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| [`Zeros(d)`](@ref) | ``0^{d}`` |
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| [`Nonnegatives(d)`](@ref) | ``\{ x \in \mathbb{R}^{d} : x \ge 0 \}`` |
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| [`Nonpositives(d)`](@ref) | ``\{ x \in \mathbb{R}^{d} : x \le 0 \}`` |
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| [`SecondOrderCone(d)`](@ref) | ``\{ (t,x) \in \mathbb{R}^{d} : t \ge \lVert x \rVert_2 \}`` |
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| [`RotatedSecondOrderCone(d)`](@ref) | ``\{ (t,u,x) \in \mathbb{R}^{d} : 2tu \ge \lVert x \rVert_2^2, t \ge 0,u \ge 0 \}`` |
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| [`ExponentialCone()`](@ref) | ``\{ (x,y,z) \in \mathbb{R}^3 : y \exp (x/y) \le z, y > 0 \}`` |
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| [`DualExponentialCone()`](@ref) | ``\{ (u,v,w) \in \mathbb{R}^3 : -u \exp (v/u) \le \exp(1) w, u < 0 \}`` |
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| [`GeometricMeanCone(d)`](@ref) | ``\{ (t,x) \in \mathbb{R}^{1+n} : x \ge 0, t \le \sqrt[n]{x_1 x_2 \cdots x_n} \}`` where ``n`` is ``d - 1`` |
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| [`PowerCone(α)`](@ref) | ``\{ (x,y,z) \in \mathbb{R}^3 : x^{\alpha} y^{1-\alpha} \ge \|z\|, x \ge 0,y \ge 0 \}`` |
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| [`DualPowerCone(α)`](@ref) | ``\{ (u,v,w) \in \mathbb{R}^3 : \left(\frac{u}{\alpha}\right)^{\alpha}\left(\frac{v}{1-\alpha}\right)^{1-\alpha} \ge \|w\|, u,v \ge 0 \}`` |
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| [`NormOneCone(d)`](@ref) | ``\{ (t,x) \in \mathbb{R}^{d} : t \ge \sum_i \lvert x_i \rvert \}`` |
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| [`NormInfinityCone(d)`](@ref) | ``\{ (t,x) \in \mathbb{R}^{d} : t \ge \max_i \lvert x_i \rvert \}`` |
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| [`RelativeEntropyCone(d)`](@ref) | ``\{ (u, v, w) \in \mathbb{R}^{d} : u \ge \sum_i w_i \log (\frac{w_i}{v_i}), v_i \ge 0, w_i \ge 0 \}`` |
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| [`HyperRectangle(l, u)`](@ref) | ``\{ x \in \bar{\mathbb{R}}^d: x_i \in [l_i, u_i] \forall i=1,\ldots,d \}`` |
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| [`NormCone(p, d)`](@ref) | ``\{ (t,x) \in \mathbb{R}^{d} : t \ge \left(\sum\limits_i \lvert x_i \rvert^p\right)^{\frac{1}{p}} \}`` |
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| [`VectorNonlinearOracle`](@ref) | ``\{x \in \mathbb{R}^{dimension}: l \le f(x) \le u \}`` |
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## Matrix cones
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The matrix-valued set types implemented in MathOptInterface.jl are:
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| Set | Description |
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| :--------------- | :----------- |
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| [`RootDetConeTriangle(d)`](@ref MathOptInterface.RootDetConeTriangle) | ``\{ (t,X) \in \mathbb{R}^{1+d(1+d)/2} : t \le \det(X)^{1/d}, X \mbox{ is the upper triangle of a PSD matrix} \}`` |
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| [`RootDetConeSquare(d)`](@ref MathOptInterface.RootDetConeSquare) | ``\{ (t,X) \in \mathbb{R}^{1+d^2} : t \le \det(X)^{1/d}, X \mbox{ is a PSD matrix} \}`` |
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| [`PositiveSemidefiniteConeTriangle(d)`](@ref MathOptInterface.PositiveSemidefiniteConeTriangle) | ``\{ X \in \mathbb{R}^{d(d+1)/2} : X \mbox{ is the upper triangle of a PSD matrix} \}`` |
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| [`PositiveSemidefiniteConeSquare(d)`](@ref MathOptInterface.PositiveSemidefiniteConeSquare) | ``\{ X \in \mathbb{R}^{d^2} : X \mbox{ is a PSD matrix} \}`` |
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| [`LogDetConeTriangle(d)`](@ref MathOptInterface.LogDetConeTriangle) | ``\{ (t,u,X) \in \mathbb{R}^{2+d(1+d)/2} : t \le u\log(\det(X/u)), X \mbox{ is the upper triangle of a PSD matrix}, u > 0 \}`` |
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| [`LogDetConeSquare(d)`](@ref MathOptInterface.LogDetConeSquare) | ``\{ (t,u,X) \in \mathbb{R}^{2+d^2} : t \le u \log(\det(X/u)), X \mbox{ is a PSD matrix}, u > 0 \}`` |
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| [`NormSpectralCone(r, c)`](@ref MathOptInterface.NormSpectralCone) | ``\{ (t, X) \in \mathbb{R}^{1 + r \times c} : t \ge \sigma_1(X), X \mbox{ is a } r\times c\mbox{ matrix} \}``
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| [`NormNuclearCone(r, c)`](@ref MathOptInterface.NormNuclearCone) | ``\{ (t, X) \in \mathbb{R}^{1 + r \times c} : t \ge \sum_i \sigma_i(X), X \mbox{ is a } r\times c\mbox{ matrix} \}`` |
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| [`HermitianPositiveSemidefiniteConeTriangle(d)`](@ref MathOptInterface.HermitianPositiveSemidefiniteConeTriangle) | The cone of Hermitian positive semidefinite matrices, with
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`side_dimension` rows and columns. |
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| [`Scaled(S)`](@ref MathOptInterface.Scaled) | The set `S` scaled so that [`Utilities.set_dot`](@ref MathOptInterface.Utilities.set_dot) corresponds to `LinearAlgebra.dot` |
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| Set | Description |
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| :-------------------------------------------- | :------------------------------------------------------------------------------------------------------------------------------ |
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| [`RootDetConeTriangle(d)`](@ref) | ``\{ (t,X) \in \mathbb{R}^{1+d(1+d)/2} : t \le \det(X)^{1/d}, X \mbox{ is the upper triangle of a PSD matrix} \}`` |
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| [`RootDetConeSquare(d)`](@ref) | ``\{ (t,X) \in \mathbb{R}^{1+d^2} : t \le \det(X)^{1/d}, X \mbox{ is a PSD matrix} \}`` |
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| [`PositiveSemidefiniteConeTriangle(d)`](@ref) | ``\{ X \in \mathbb{R}^{d(d+1)/2} : X \mbox{ is the upper triangle of a PSD matrix} \}`` |
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| [`PositiveSemidefiniteConeSquare(d)`](@ref) | ``\{ X \in \mathbb{R}^{d^2} : X \mbox{ is a PSD matrix} \}`` |
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| [`LogDetConeTriangle(d)`](@ref) | ``\{ (t,u,X) \in \mathbb{R}^{2+d(1+d)/2} : t \le u\log(\det(X/u)), X \mbox{ is the upper triangle of a PSD matrix}, u > 0 \}`` |
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| [`LogDetConeSquare(d)`](@ref) | ``\{ (t,u,X) \in \mathbb{R}^{2+d^2} : t \le u \log(\det(X/u)), X \mbox{ is a PSD matrix}, u > 0 \}`` |
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| [`NormSpectralCone(r, c)`](@ref) | ``\{ (t, X) \in \mathbb{R}^{1 + r \times c} : t \ge \sigma_1(X), X \mbox{ is a } r\times c\mbox{ matrix} \}`` |
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| [`NormNuclearCone(r, c)`](@ref) | ``\{ (t, X) \in \mathbb{R}^{1 + r \times c} : t \ge \sum_i \sigma_i(X), X \mbox{ is a } r\times c\mbox{ matrix} \}`` |
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| [`HermitianPositiveSemidefiniteConeTriangle(d)`](@ref) | The cone of Hermitian positive semidefinite matrices, with `side_dimension` rows and columns. |
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| [`Scaled(S)`](@ref) | The set `S` scaled so that [`Utilities.set_dot`](@ref) corresponds to `LinearAlgebra.dot` |
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Some of these cones can take two forms: `XXXConeTriangle` and `XXXConeSquare`.
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Other sets are vector-valued, with a particular combinatorial structure. Read
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their docstrings for more information on how to interpret them.
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| Set | Description |
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| :------------------------- | :---------- |
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| [`SOS1`](@ref) | A Special Ordered Set (SOS) of Type I |
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| [`SOS2`](@ref) | A Special Ordered Set (SOS) of Type II |
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| [`Indicator`](@ref) | A set to specify an indicator constraint |
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| Set | Description |
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| :------------------------- | :-------------------------------------------------- |
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| [`SOS1`](@ref) | A Special Ordered Set (SOS) of Type I |
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| [`SOS2`](@ref) | A Special Ordered Set (SOS) of Type II |
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| [`Indicator`](@ref) | A set to specify an indicator constraint |
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| [`Complements`](@ref) | A set to specify a mixed complementarity constraint |
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| [`AllDifferent`](@ref) | The `all_different` global constraint |
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| [`BinPacking`](@ref) | The `bin_packing` global constraint |
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| [`Circuit`](@ref) | The `circuit` global constraint |
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| [`CountAtLeast`](@ref) | The `at_least` global constraint |
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| [`CountBelongs`](@ref) | The `nvalue` global constraint |
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| [`CountDistinct`](@ref) | The `distinct` global constraint |
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| [`CountGreaterThan`](@ref) | The `count_gt` global constraint |
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| [`Cumulative`](@ref) | The `cumulative` global constraint |
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| [`Path`](@ref) | The `path` global constraint |
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| [`Table`](@ref) | The `table` global constraint |
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| [`AllDifferent`](@ref) | The `all_different` global constraint |
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| [`BinPacking`](@ref) | The `bin_packing` global constraint |
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| [`Circuit`](@ref) | The `circuit` global constraint |
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| [`CountAtLeast`](@ref) | The `at_least` global constraint |
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| [`CountBelongs`](@ref) | The `nvalue` global constraint |
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| [`CountDistinct`](@ref) | The `distinct` global constraint |
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| [`CountGreaterThan`](@ref) | The `count_gt` global constraint |
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| [`Cumulative`](@ref) | The `cumulative` global constraint |
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| [`Path`](@ref) | The `path` global constraint |
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| [`Table`](@ref) | The `table` global constraint |

‎docs/src/submodules/FileFormats/overview.md‎

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## Validating MOF files
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MathOptFormat files are governed by a schema. Use [JSONSchema.jl](https://github.com/fredo-dedup/JSONSchema.jl)
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MathOptFormat files are governed by a schema. Use [JSONSchema.jl](https://github.com/JuliaIO/JSONSchema.jl)
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to check if a `.mof.json` file satisfies the schema.
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First, construct the schema object as follows:

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