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14 changes: 7 additions & 7 deletions src/Bridges/Constraint/bridges/GeoMeanBridge.jl
Original file line number Diff line number Diff line change
Expand Up @@ -16,15 +16,15 @@ Consider the cone of dimension 4:
```math
t \\le \\sqrt[3]{x_1 x_2 x_3}
```
This can be rewritten as ``\\exists y \\ge 0`` such that:
This can be rewritten as ``\\exists y_1 \\ge 0`` such that:
```math
\\begin{align*}
t & \\le y,\\\\
y^4 & \\le x_1 x_2 x_3 y.
t & \\le y_1,\\\\
y_1^4 & \\le x_1 x_2 x_3 y_1.
\\end{align*}
```
Note that we need to create ``y`` and not use ``t^4`` directly because ``t`` is
not allowed to be negative.
Note that we need to create ``y_1`` and not use ``t^4`` directly because ``t`` is
allowed to be negative.

This is equivalent to:
```math
Expand All @@ -33,12 +33,12 @@ 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_1 & \\ge 0.
\\end{align*}
```

More generally, you can show how the geometric mean code is recursively expanded
into a set of new variables ``y`` in [`MOI.Nonnegatives`](@ref), a set of
into a set of new variables ``y_i`` in [`MOI.Nonnegatives`](@ref), a set of
[`MOI.RotatedSecondOrderCone`](@ref) constraints, and a [`MOI.LessThan`](@ref)
constraint between ``t`` and ``y_1``.

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