Stylistic changes and typo fixes in intro docs

docs/src/background/duality.md

@@ -108,8 +108,8 @@ and similarly, the dual is:
 ```
 
 !!! warning
-    For the LP case is that the signs of the feasible duals depend only on the
-    sense of the inequality and not on the objective sense.
+    For the LP case, the signs of the feasible dual variables depend only on the
+    sense of the corresponding primal inequality and not on the objective sense.
 
 ## Duality and scalar product
 
@@ -133,7 +133,7 @@ Given a problem with quadratic functions:
 & \;\;\text{s.t.} & \frac{1}{2}x^TQ_ix + a_i^T x + b_i & \in \mathcal{C}_i & i = 1 \ldots m
 \end{align*}
 ```
-Consider the Lagrangian function
+with cones ``\mathcal{C}_i \subseteq \mathbb{R}`` for ``i = 1 \ldots m``, consider the Lagrangian function
 ```math
 L(x, y) = \frac{1}{2}x^TQ_0x + a_0^T x + b_0 - \sum_{i = 1}^m y_i (\frac{1}{2}x^TQ_ix + a_i^T x + b_i)
 ```
@@ -152,23 +152,23 @@ A pair of primal-dual variables $(x^\star, y^\star)$ is optimal if
   ```
   That is, for all ``i = 1, \ldots, m``, ``\frac{1}{2}x^TQ_ix + a_i^T x + b_i`` is
   either zero or in the normal cone of ``\mathcal{C}_i^*`` at ``y^\star``.
-  For instance, if ``\mathcal{C}_i`` is ``\{ x \in \mathbb{R} : x \le 0 \}``, it means that
-  if ``\frac{1}{2}x^TQ_ix + a_i^T x + b_i`` is nonzero then ``\lambda_i = 0``,
-  this is the classical complementary slackness condition.
+  For instance, if ``\mathcal{C}_i`` is ``\{ x \in \mathbb{R} : x \le 0 \}``, this means that
+  if ``\frac{1}{2}x^TQ_ix + a_i^T x + b_i`` is nonzero at ``x^\star`` then ``y_i^\star = 0``.
+  This is the classical complementary slackness condition.
 
 If ``\mathcal{C}_i`` is a vector set, the discussion remains valid with
 ``y_i(\frac{1}{2}x^TQ_ix + a_i^T x + b_i)`` replaced with the scalar product
 between ``y_i`` and the vector of scalar-valued quadratic functions.
 
 !!! note
     For quadratic programs with only affine constraints, the optimality condition
-    ``\nabla_x L(x, y^\star) = 0`` can be simplified as follows
+    ``\nabla_x L(x, y^\star) = 0`` can be simplified as follows:
     ```math
     0 = \nabla_x L(x, y^\star) = Q_0x + a_0 - \sum_{i = 1}^m y_i^\star a_i
     ```
     which gives
     ```math
-    Q_0x = \sum_{i = 1}^m y_i^\star a_i - a_0
+    Q_0x = \sum_{i = 1}^m y_i^\star a_i - a_0 .
     ```
     The Lagrangian function
     ```math

docs/src/manual/standard_form.md

@@ -63,9 +63,9 @@ The one-dimensional set types implemented in MathOptInterface.jl are:
 * [`Integer()`](@ref MathOptInterface.Integer): ``\mathbb{Z}``
 * [`ZeroOne()`](@ref MathOptInterface.ZeroOne): ``\{ 0, 1 \}``
 * [`Semicontinuous(lower,upper)`](@ref MathOptInterface.Semicontinuous):
-  ``\{ 0\} \cup [lower,upper]``
+  ``\{ 0\} \cup [\mbox{lower},\mbox{upper}]``
 * [`Semiinteger(lower,upper)`](@ref MathOptInterface.Semiinteger):
-  ``\{ 0\} \cup \{lower,lower+1,\ldots,upper-1,upper\}``
+  ``\{ 0\} \cup \{\mbox{lower},\mbox{lower}+1,\ldots,\mbox{upper}-1,\mbox{upper}\}``
 
 ## Vector cones
 
@@ -85,18 +85,17 @@ The vector-valued set types implemented in MathOptInterface.jl are:
 * [`ExponentialCone()`](@ref MathOptInterface.ExponentialCone):
   ``\{ (x,y,z) \in \mathbb{R}^3 : y \exp (x/y) \le z, y > 0 \}``
 * [`DualExponentialCone()`](@ref MathOptInterface.DualExponentialCone):
-  ``\{ (u,v,w) \in \mathbb{R}^3 : -u \exp (v/u) \le exp(1) w, u < 0 \}``
+  ``\{ (u,v,w) \in \mathbb{R}^3 : -u \exp (v/u) \le \exp(1) w, u < 0 \}``
 * [`GeometricMeanCone(dimension)`](@ref MathOptInterface.GeometricMeanCone):
   ``\{ (t,x) \in \mathbb{R}^{n+1} : x \ge 0, t \le \sqrt[n]{x_1 x_2 \cdots x_n} \}``
-  where ``n`` is ``dimension - 1``
+  where ``n`` is ``\mbox{dimension} - 1``
 * [`PowerCone(exponent)`](@ref MathOptInterface.PowerCone):
   ``\{ (x,y,z) \in \mathbb{R}^3 : x^\mbox{exponent} y^{1-\mbox{exponent}} \ge |z|, x,y \ge 0 \}``
 * [`DualPowerCone(exponent)`](@ref MathOptInterface.DualPowerCone):
   ``\{ (u,v,w) \in \mathbb{R}^3 : \frac{u}{\mbox{exponent}}^\mbox{exponent}\frac{v}{1-\mbox{exponent}}^{1-\mbox{exponent}} \ge |w|, u,v \ge 0 \}``
-* [`NormOneCone(dimension)`](@ref MathOptInterface.NormOneCone):
-``\{ (t,x) \in \mathbb{R}^\mbox{dimension} : t \ge \lVert x \rVert_1 = \sum_i \lvert x_i \rvert \}``
+* [`NormOneCone(dimension)`](@ref MathOptInterface.NormOneCone): ``\{ (t,x) \in \mathbb{R}^\mbox{dimension} : t \ge \lVert x \rVert_1 \}`` where ``\lVert x \rVert_1 = \sum_i \lvert x_i \rvert``
 * [`NormInfinityCone(dimension)`](@ref MathOptInterface.NormInfinityCone):
-  ``\{ (t,x) \in \mathbb{R}^\mbox{dimension} : t \ge \lVert x \rVert_\infty = \max_i \lvert x_i \rvert \}``
+  ``\{ (t,x) \in \mathbb{R}^\mbox{dimension} : t \ge \lVert x \rVert_\infty \}`` where ``\lVert x \rVert_\infty = \max_i \lvert x_i \rvert``.
 * [`RelativeEntropyCone(dimension)`](@ref MathOptInterface.RelativeEntropyCone):
   ``\{ (u, v, w) \in \mathbb{R}^\mbox{dimension} : u \ge \sum_i w_i \log (\frac{w_i}{v_i}), v_i \ge 0, w_i \ge 0 \}``
 
@@ -105,24 +104,24 @@ The vector-valued set types implemented in MathOptInterface.jl are:
 The matrix-valued set types implemented in MathOptInterface.jl are:
 
 * [`RootDetConeTriangle(dimension)`](@ref MathOptInterface.RootDetConeTriangle):
-  ``\{ (t,X) \in \mathbb{R}^{1+\mbox{dimension}(1+\mbox{dimension})/2} : t \le det(X)^{1/\mbox{dimension}}, X \mbox{is the upper triangle of a PSD matrix} \}``
+  ``\{ (t,X) \in \mathbb{R}^{1+\mbox{dimension}(1+\mbox{dimension})/2} : t \le \det(X)^{1/\mbox{dimension}}, X \mbox{ is the upper triangle of a PSD matrix} \}``
 * [`RootDetConeSquare(dimension)`](@ref MathOptInterface.RootDetConeSquare):
-  ``\{ (t,X) \in \mathbb{R}^{1+\mbox{dimension}^2} : t \le \det(X)^{1/\mbox{dimension}}, X \mbox{is a PSD matrix} \}``
+  ``\{ (t,X) \in \mathbb{R}^{1+\mbox{dimension}^2} : t \le \det(X)^{1/\mbox{dimension}}, X \mbox{ is a PSD matrix} \}``
 
 * [`PositiveSemidefiniteConeTriangle(dimension)`](@ref MathOptInterface.PositiveSemidefiniteConeTriangle):
-  ``\{ X \in \mathbb{R}^{\mbox{dimension}(\mbox{dimension}+1)/2} : X \mbox{is the upper triangle of a PSD matrix} \}``
+  ``\{ X \in \mathbb{R}^{\mbox{dimension}(\mbox{dimension}+1)/2} : X \mbox{ is the upper triangle of a PSD matrix} \}``
 * [`PositiveSemidefiniteConeSquare(dimension)`](@ref MathOptInterface.PositiveSemidefiniteConeSquare):
-  ``\{ X \in \mathbb{R}^{\mbox{dimension}^2} : X \mbox{is a PSD matrix} \}``
+  ``\{ X \in \mathbb{R}^{\mbox{dimension}^2} : X \mbox{ is a PSD matrix} \}``
 
 * [`LogDetConeTriangle(dimension)`](@ref MathOptInterface.LogDetConeTriangle):
-  ``\{ (t,u,X) \in \mathbb{R}^{2+\mbox{dimension}(1+\mbox{dimension})/2} : t \le u\log(\det(X/u)), X \mbox{is the upper triangle of a PSD matrix}, u > 0 \}``
+  ``\{ (t,u,X) \in \mathbb{R}^{2+\mbox{dimension}(1+\mbox{dimension})/2} : t \le u\log(\det(X/u)), X \mbox{ is the upper triangle of a PSD matrix}, u > 0 \}``
 * [`LogDetConeSquare(dimension)`](@ref MathOptInterface.LogDetConeSquare):
-  ``\{ (t,u,X) \in \mathbb{R}^{2+\mbox{dimension}^2} : t \le u \log(\det(X/u)), X \mbox{is a PSD matrix}, u > 0 \}``
+  ``\{ (t,u,X) \in \mathbb{R}^{2+\mbox{dimension}^2} : t \le u \log(\det(X/u)), X \mbox{ is a PSD matrix}, u > 0 \}``
 
 * [`NormSpectralCone(row_dim, column_dim)`](@ref MathOptInterface.NormSpectralCone):
-  ``\{ (t, X) \in \mathbb{R}^{1 + \mbox{row_dim} \times \mbox{column_dim}} : t \ge \sigma_1(X), X \mbox{is a matrix with row_dim rows and column_dim columns} \}``
+  ``\{ (t, X) \in \mathbb{R}^{1 + \mbox{row_dim} \times \mbox{column_dim}} : t \ge \sigma_1(X), X \mbox{ is a matrix with row_dim rows and column_dim columns} \}``
 * [`NormNuclearCone(row_dim, column_dim)`](@ref MathOptInterface.NormNuclearCone):
-  ``\{ (t, X) \in \mathbb{R}^{1 + \mbox{row_dim} \times \mbox{column_dim}} : t \ge \sum_i \sigma_i(X), X \mbox{is a matrix with row_dim rows and column_dim columns} \}``
+  ``\{ (t, X) \in \mathbb{R}^{1 + \mbox{row_dim} \times \mbox{column_dim}} : t \ge \sum_i \sigma_i(X), X \mbox{ is a matrix with row_dim rows and column_dim columns} \}``
 
 Some of these cones can take two forms: `XXXConeTriangle` and `XXXConeSquare`.
 
@@ -151,5 +150,5 @@ or solver developers.
   A special ordered set of Type II.
 * [`Indicator(set)`](@ref MathOptInterface.Indicator):
   A set to specify indicator constraints.
-* [`Complements`](@ref MathOptInterface.Complements):
+* [`Complements(dimension)`](@ref MathOptInterface.Complements):
   A set for mixed complementarity constraints.

docs/src/tutorials/example.md

@@ -18,6 +18,12 @@ s.t. \; & w^\top x \le C \\
 \end{aligned}
 ```
 
+Load the MathOptInterface module and define the shorthand `MOI`:
+```julia
+using MathOptInterface
+const MOI = MathOptInterface
+```
+
 As an optimizer, we choose GLPK:
 ```julia
 using GLPK

docs/src/tutorials/implementing.md

@@ -51,7 +51,7 @@ has a good list of solvers, along with the problem classes they support.
 
 ### Create a low-level interface
 
-Before writing a MathOptInterface, you first need to be able to call the solver
+Before writing a MathOptInterface wrapper, you first need to be able to call the solver
 from Julia.
 
 #### Wrapping solvers written in Julia
@@ -77,7 +77,7 @@ easiest way to do this is to copy an existing solver. Good examples to follow
 are the [COIN-OR solvers](https://github.com/JuliaPackaging/Yggdrasil/tree/master/C/Coin-OR).
 
 !!! warning
-    Building the solver via Yggdrasil is non-trivial. please ask the
+    Building the solver via Yggdrasil is non-trivial. Please ask the
     [Developer chatroom](https://gitter.im/JuliaOpt/JuMP-dev) for help.
 
 If the code is commercial or not publicly available, the user will need to
@@ -252,7 +252,7 @@ Now that we have an `Optimizer`, we need to implement a few basic methods.
     It is also very helpful to look at an existing wrapper for a similar solver.
 
 You should also implement `Base.show(::IO, ::Optimizer)` to print a nice string
-when some prints your model. For example
+when someone prints your model. For example
 ```julia
 function Base.show(io::IO, model::Optimizer)
     return print(io, "NewSolver with the pointer $(model.ptr)")
@@ -660,12 +660,12 @@ For example, Gurobi.jl adds attributes for multiobjective optimization by
 struct NumberOfObjectives <: MOI.AbstractModelAttribute end
 
 function MOI.set(model::Optimizer, ::NumberOfObjectives, n::Integer)
-    # Code to set NumberOfOBjectives
+    # Code to set NumberOfObjectives
     return
 end
 
 function MOI.get(model::Optimizer, ::NumberOfObjectives)
-    n = # Code to get NumberOfobjectives
+    n = # Code to get NumberOfObjectives
     return n
 end
 ```