Fix

Author: blegat

src/RelativeEntropy/bridges/age.jl

@@ -1,8 +1,52 @@
+"""
+    AGEBridge{T,F,G,H} <: MOI.Bridges.Constraint.AbstractBridge
+
+The nonnegativity of `x ≥ 0` in
+```
+∑ ci x^αi ≥ -c0 x^α0
+```
+can be reformulated as
+```
+∑ ci exp(αi'y) ≥ -β exp(α0'y)
+```
+In any case, it is shown to be equivalent to
+```
+∃ ν ≥ 0 s.t. D(ν, exp(1)*c) ≤ β, ∑ νi αi = α0 ∑ νi [CP16, (3)]
+```
+where `N(ν, λ) = ∑ νj log(νj/λj)` is the relative entropy function.
+The constant `exp(1)` can also be moved out of `D` into
+```
+∃ ν ≥ 0 s.t. D(ν, c) - ∑ νi ≤ β, ∑ νi αi = α0 ∑ νi [MCW21, (2)]
+```
+The relative entropy cone in MOI is `(u, v, w)` such that `D(w, v) ≥ u`.
+Therefore, we can either encode `(β, exp(1)*c, ν)` [CP16, (3)] or
+`(β + ∑ νi, c, ν)` [MCW21, (2)].
+In this bridge, we use the second formulation.
+
+!!! note
+    A direct application of the Arithmetic-Geometric mean inequality gives
+    ```
+    ∃ ν ≥ 0 s.t. D(ν, exp(1)*c) ≤ -log(-β), ∑ νi αi = α0, ∑ νi = 1 [CP16, (4)]
+    ```
+    which is not jointly convex in (ν, c, β).
+    The key to get the convex formulation [CP16, (3)] or [MCW21, (2)] instead is to
+    use the convex conjugacy between the exponential and the negative entropy functions [CP16, (iv)].
+
+[CP16] Chandrasekaran, Venkat, and Parikshit Shah.
+"Relative entropy relaxations for signomial optimization."
+SIAM Journal on Optimization 26.2 (2016): 1147-1173.
+[MCW21] Murray, Riley, Venkat Chandrasekaran, and Adam Wierman.
+"Signomial and polynomial optimization via relative entropy and partial dualization."
+Mathematical Programming Computation 13 (2021): 257-295.
+https://arxiv.org/pdf/1907.00814.pdf
+
+
+"""
 struct AGEBridge{T,F,G,H} <: MOI.Bridges.Constraint.AbstractBridge
     k::Int
     equality_constraints::Vector{MOI.ConstraintIndex{F,MOI.EqualTo{T}}}
     relative_entropy_constraint::MOI.ConstraintIndex{G,MOI.RelativeEntropyCone}
-end # See https://arxiv.org/pdf/1907.00814.pdf
+end
 
 function MOI.Bridges.Constraint.bridge_constraint(
     ::Type{AGEBridge{T,F,G,H}},
@@ -14,19 +58,17 @@ function MOI.Bridges.Constraint.bridge_constraint(
     ν = MOI.add_variables(model, m - 1)
     sumν = MOI.ScalarAffineFunction(MOI.ScalarAffineTerm.(one(T), ν), zero(T))
     ceq = map(1:size(set.α, 2)) do var
-        f = -sumν
+        f = zero(typeof(sumν))
         j = 0
         for i in 1:m
-            if i != set.k
+            if i == set.k
+                MA.sub_mul!!(f, convert(T, set.α[i, var]), sumν)
+            else
                 j += 1
                 MA.add_mul!!(f, convert(T, set.α[i, var]), ν[j])
             end
         end
-        return MOI.Utilities.normalize_and_add_constraint(
-            model,
-            f,
-            MOI.EqualTo(zero(T)),
-        )
+        return MOI.add_constraint(model, f, MOI.EqualTo(zero(T)))
     end
     scalars = MOI.Utilities.eachscalar(func)
     f = MOI.Utilities.operate(