[Bridges] fix typos in bridge docstrings (#1894)

Author: odow

src/Bridges/Constraint/bridges/det.jl

@@ -67,7 +67,7 @@ that
     X & Δ\\\\
     Δ^\\top & \\mathrm{Diag}(Δ)
   \\end{pmatrix} & \\succeq 0\\\\
-  t - \\sum_{i=1}^n u \\log(Δ_{ii}/u) & \\le 0
+  t - \\sum_{i=1}^n u \\log\\left(\\frac{Δ_{ii}}{u}\\right) & \\le 0
 \\end{align*}
 ```
 Which we reformulate further into
@@ -77,7 +77,7 @@ Which we reformulate further into
     X & Δ\\\\
     Δ^\\top & \\mathrm{Diag}(Δ)
   \\end{pmatrix} & \\succeq 0\\\\
-  (l_i, u , Δ_{ii}) in ExponentialCone & \\forall i \\\\
+  (l_i, u , Δ_{ii}) & \\in ExponentialCone\\quad \\forall i \\\\
   t - \\sum_{i=1}^n l_i & \\le 0
 \\end{align*}
 ```
@@ -310,7 +310,7 @@ matrix ``Δ`` such that:
     X & Δ\\\\
     Δ^\\top & \\mathrm{Diag}(Δ)
   \\end{pmatrix} & \\succeq 0\\\\
-  (t, \\mathrm{Diag}(Δ)) in GeometricMeanCone
+  (t, \\mathrm{Diag}(Δ)) & \\in GeometricMeanCone
 \\end{align*}
 ```
 

src/Bridges/Constraint/bridges/geomean.jl

@@ -33,7 +33,7 @@ This is equivalent to:
     y_1^2 & \\le 2y_2 y_3,\\\\
     y_2^2 & \\le 2x_1 x_2, \\\\
     y_3^2 & \\le 2x_3(y_1/\\sqrt{4}) \\\\
-    y \\ge 0.
+    y     & \\ge 0.
 \\end{align*}
 ```
 

src/Bridges/Constraint/bridges/indicator_activate_on_zero.jl

@@ -9,8 +9,8 @@
 
 `IndicatorActiveOnFalseBridge` implements the following reformulation:
 
-  * ``!z \\implies {f(x) \\in S}`` into ``y \\implies {f(x) \\in S}``,
-    ``z + y == 1``, and ``y \\in \\{0, \\}``
+  * ``\\neg z \\implies {f(x) \\in S}`` into ``y \\implies {f(x) \\in S}``,
+    ``z + y = 1``, and ``y \\in \\{0, 1\\}``
 
 ## Source node
 

src/Bridges/Constraint/bridges/norm_spec_nuc_to_psd.jl

@@ -10,7 +10,7 @@
 `NormSpectralBridge` implements the following reformulation:
 
   * ``t \\ge \\sigma_1(X)`` into
-    ``\\left[\\begin{array}{c c}t\\mathbf{I} & X^\\top \\\\ X & t \\mathbf{I}\\end{array}\\right] \\succ 0``
+    ``\\left[\\begin{array}{c c}t\\mathbf{I} & X^\\top \\\\ X & t \\mathbf{I}\\end{array}\\right] \\succeq 0``
 
 ## Source node
 
@@ -221,8 +221,8 @@ end
 `NormNuclearBridge` implements the following reformulation:
 
   * ``t \\ge \\sum_i \\sigma_i (X)`` into
-    ``\\left[\\begin{array}{c c}U & X^\\top \\\\ X & V\\end{array}\\right] \\succ 0``
-    and ``t \\ge (tr(U) + tr(V)) / 2``.
+    ``\\left[\\begin{array}{c c}U & X^\\top \\\\ X & V\\end{array}\\right] \\succeq 0``
+    and ``2t \\ge tr(U) + tr(V)``.
 
 ## Source node
 

src/Bridges/Constraint/bridges/relentr_to_exp.jl

@@ -11,7 +11,7 @@
 [`MOI.RelativeEntropyCone`](@ref) into an [`MOI.ExponentialCone`](@ref):
 
   * ``u \\ge \\sum_{i=1}^n w_i \\log \\left(\\frac{w_i}{v_i}\\right)`` into
-    ``y_i \\ge 0``, ``\\sum_{i=1}^n y_i``, and
+    ``y_i \\ge 0``, ``u \\ge \\sum_{i=1}^n y_i``, and
     ``(-y_i, w_i, v_i) \\in ExponentialCone``.
 
 ## Source node

src/Bridges/Constraint/bridges/rsoc_soc.jl

@@ -9,10 +9,8 @@
 
 `RSOCtoSOCBridge` implements the following reformulation:
 
-  * ``||x||_2^2 \\le 2tu`` into ``||v|| \\le w ``, with the substitution rules
-    ``t = \\frac{w}{\\sqrt 2} + \\frac{v_1}{\\sqrt 2}``,
-    ``u = \\frac{w}{\\sqrt 2} - \\frac{v_1}{\\sqrt 2}``, and
-    ``x = (v_2,\\ldots,v_N)``.
+  * ``||x||_2^2 \\le 2tu`` into
+    ``||\\frac{t - u}{\\sqrt 2}, x||_2 \\le  \\frac{t + u}{\\sqrt 2}``
 
 ## Source node
 

src/Bridges/Constraint/bridges/semi_to_binary.jl

@@ -12,17 +12,17 @@
   * ``x \\in \\{0\\} \\cup [l, u]`` into
     ```math
     \\begin{aligned}
-    x \\leq z \\times u \\
-    x \\geq z \\times l \\
+    x \\leq z u \\\\
+    x \\geq z l \\\\
     z \\in \\{0, 1\\}
     \\end{aligned}
     ```
   * ``x \\in \\{0\\} \\cup \\{l, \\ldots, u\\}`` into
     ```math
     \\begin{aligned}
-    x \\leq z \\times u \\
-    x \\geq z \\times l \\
-    z \\in \\{0, 1\\} \\
+    x \\leq z u \\\\
+    x \\geq z l \\\\
+    z \\in \\{0, 1\\} \\\\
     x \\in \\mathbb{Z}
     \\end{aligned}
     ```

src/Bridges/Constraint/bridges/soc_rsoc.jl

@@ -9,9 +9,7 @@
 
 `SOCtoRSOCBridge` implements the following reformulation:
 
-  * ``||x||_2 \\le t`` into ``2uv \\ge ||w||_2^2``, with the substitution rules
-    ``t = \\frac{u}{\\sqrt 2} + \\frac{v}{\\sqrt 2}``,
-    ``x = (\\frac{u}{\\sqrt 2} - \\frac{v}{\\sqrt 2}, w)``.
+  * ``||x||_2 \\le t`` into ``(t+x_1)(t-x_1)\\ge ||(x_2\\ldots,x_N)||_2^2``
 
 ## Assumptions
 

src/Bridges/Constraint/bridges/soc_to_nonconvex_quad.jl

@@ -168,7 +168,7 @@ end
 
 `RSOCtoNonConvexQuadBridge` implements the following reformulations:
 
-  * ``||x||_2 \\le t \\cdot u`` into ``\\sum x^2 - t\\cdot u \\le 0``,
+  * ``||x||_2^2 \\le 2tu`` into ``\\sum x^2 - 2tu \\le 0``,
     ``1t + 0 \\ge 0``, and ``1u + 0 \\ge 0``.
 
 The [`MOI.ScalarAffineFunction`](@ref)s ``1t + 0`` and ``1u + 0`` are used

src/Bridges/Constraint/bridges/soc_to_psd.jl

@@ -44,7 +44,7 @@ end
 `SOCtoPSDBridge` implements the following reformulation:
 
   * ``||x||_2 \\le t`` into
-    ``\\left[\\begin{array}{c c}t & x^\\top \\\\ x & t \\mathbf{I}\\end{array}\\right]``
+    ``\\left[\\begin{array}{c c}t & x^\\top \\\\ x & t \\mathbf{I}\\end{array}\\right]\\succeq 0``
 
 !!! warning
     This bridge is not added by default by [`MOI.Bridges.full_bridge_optimizer`](@ref)
@@ -155,8 +155,8 @@ end
 
 `RSOCtoPSDBridge` implements the following reformulation:
 
-  * ``||x||_2 \\le t\\cdot u`` into
-    ``\\left[\\begin{array}{c c}t & x^\\top \\\\ x & 2tu \\mathbf{I}\\end{array}\\right]``
+  * ``||x||_2^2 \\le 2t\\cdot u`` into
+    ``\\left[\\begin{array}{c c}t & x^\\top \\\\ x & 2tu \\mathbf{I}\\end{array}\\right]\\succeq 0``
 
 ## Source node