diff --git a/src/Bridges/Constraint/bridges/DualGeoMeanBridge.jl b/src/Bridges/Constraint/bridges/DualGeoMeanBridge.jl
index 7b69dabcd7..7354ac896c 100644
--- a/src/Bridges/Constraint/bridges/DualGeoMeanBridge.jl
+++ b/src/Bridges/Constraint/bridges/DualGeoMeanBridge.jl
@@ -33,6 +33,9 @@ end
 const DualGeoMean{T,OT<:MOI.ModelLike} =
     SingleBridgeOptimizer{DualGeoMeanBridge{T},OT}
 
+#prevent bridge from being used instead of FunctionConversionBridge
+MOI.Bridges.bridging_cost(::Type{<:DualGeoMeanBridge}) = 2.0
+
 function bridge_constraint(
     ::Type{DualGeoMeanBridge{T,G,H}},
     model::MOI.ModelLike,
diff --git a/src/Bridges/Constraint/bridges/DualRelativeEntropyBridge.jl b/src/Bridges/Constraint/bridges/DualRelativeEntropyBridge.jl
index ffbfaed4ad..570fb55d8d 100644
--- a/src/Bridges/Constraint/bridges/DualRelativeEntropyBridge.jl
+++ b/src/Bridges/Constraint/bridges/DualRelativeEntropyBridge.jl
@@ -32,6 +32,9 @@ end
 const DualRelativeEntropy{T,OT<:MOI.ModelLike} =
     SingleBridgeOptimizer{DualRelativeEntropyBridge{T},OT}
 
+#prevent bridge from being used instead of FunctionConversionBridge
+MOI.Bridges.bridging_cost(::Type{<:DualRelativeEntropyBridge}) = 2.0
+
 function bridge_constraint(
     ::Type{DualRelativeEntropyBridge{T,F,G}},
     model::MOI.ModelLike,