diff --git a/docs/src/apireference.md b/docs/src/apireference.md
index e48e2e1e6d..62088f67d6 100644
--- a/docs/src/apireference.md
+++ b/docs/src/apireference.md
@@ -555,6 +555,41 @@ constraints of different types. There are two important concepts to distinguish:
   allow introspection into the bridge selection rationale of
   [`Bridges.LazyBridgeOptimizer`](@ref).
 
+Most bridges are added by default in [`Bridges.full_bridge_optimizer`](@ref).
+However, for technical reasons, some bridges are not added by default, for instance:
+[`Bridges.Constraint.SOCtoPSDBridge`](@ref), [`Bridges.Constraint.SOCtoNonConvexQuadBridge`](@ref)
+and [`Bridges.Constraint.RSOCtoNonConvexQuadBridge`](@ref). See the docs of those bridges
+for more information.
+
+It is possible to add those bridges and also user defined bridges,
+following one of the two methods. We present the examples for:
+[`Bridges.Constraint.SOCtoNonConvexQuadBridge`](@ref).
+
+The first option is to add the specific bridges to a
+`bridged_model` optimizer, with coefficient type `T`. The `bridged_model`
+optimizer itself must have been constructed with a
+[`Bridges.LazyBridgeOptimizer`](@ref). Once such a optimizer is available, we
+can proceed using using [`Bridges.add_bridge`](@ref):
+
+```julia
+MOIB.add_bridge(bridged_model, SOCtoNonConvexQuadBridge{T})
+```
+
+Alternatively, it is possible to create a [`Bridges.Constraint.SingleBridgeOptimizer`](@ref)
+and wrap an existing `model` with it:
+
+```julia
+const SOCtoNonConvexQuad{T, OT<:ModelLike} = Bridges.Constraint.SingleBridgeOptimizer{Bridges.Constraint.SOCtoNonConvexQuadBridge{T}, OT}
+bridged_model = SOCtoNonConvexQuad{Float64}(model)
+```
+
+Those procedures could be applied to user define bridges. For the
+bridges defined in MathOptInterface, the [`Bridges.Constraint.SingleBridgeOptimizer`](@ref)'s are already created, therefore, for the case of [`Bridges.Constraint.SOCtoNonConvexQuadBridge`](@ref), one could simply use the existing optimizer:
+
+```julia
+bridged_model = Bridges.Constraint.SOCtoNonConvexQuad{Float64}(model)
+```
+
 ```@docs
 Bridges.AbstractBridge
 Bridges.AbstractBridgeOptimizer
@@ -769,6 +804,8 @@ Bridges.Constraint.SplitIntervalBridge
 Bridges.Constraint.RSOCBridge
 Bridges.Constraint.SOCRBridge
 Bridges.Constraint.QuadtoSOCBridge
+Bridges.Constraint.SOCtoNonConvexQuadBridge
+Bridges.Constraint.RSOCtoNonConvexQuadBridge
 Bridges.Constraint.NormInfinityBridge
 Bridges.Constraint.NormOneBridge
 Bridges.Constraint.GeoMeanBridge
diff --git a/src/Bridges/Constraint/Constraint.jl b/src/Bridges/Constraint/Constraint.jl
index 26ab889631..3490fa9e9b 100644
--- a/src/Bridges/Constraint/Constraint.jl
+++ b/src/Bridges/Constraint/Constraint.jl
@@ -41,6 +41,9 @@ include("interval.jl")
 const SplitInterval{T, OT<:MOI.ModelLike} = SingleBridgeOptimizer{SplitIntervalBridge{T}, OT}
 include("quad_to_soc.jl")
 const QuadtoSOC{T, OT<:MOI.ModelLike} = SingleBridgeOptimizer{QuadtoSOCBridge{T}, OT}
+include("soc_to_nonconvex_quad.jl") # do not add these bridges by default
+const SOCtoNonConvexQuad{T, OT<:MOI.ModelLike} = SingleBridgeOptimizer{SOCtoNonConvexQuadBridge{T}, OT}
+const RSOCtoNonConvexQuad{T, OT<:MOI.ModelLike} = SingleBridgeOptimizer{RSOCtoNonConvexQuadBridge{T}, OT}
 include("norm_to_lp.jl")
 const NormInfinity{T, OT<:MOI.ModelLike} = SingleBridgeOptimizer{NormInfinityBridge{T}, OT}
 const NormOne{T, OT<:MOI.ModelLike} = SingleBridgeOptimizer{NormOneBridge{T}, OT}
@@ -84,6 +87,9 @@ function add_all_bridges(bridged_model, ::Type{T}) where {T}
     MOIB.add_bridge(bridged_model, VectorFunctionizeBridge{T})
     MOIB.add_bridge(bridged_model, SplitIntervalBridge{T})
     MOIB.add_bridge(bridged_model, QuadtoSOCBridge{T})
+    # We do not add `(R)SOCtoNonConvexQuad` because it starts with a convex
+    # conic constraint and generate a non-convex constraint (in the QCP
+    # interpretation).
     MOIB.add_bridge(bridged_model, NormInfinityBridge{T})
     MOIB.add_bridge(bridged_model, NormOneBridge{T})
     MOIB.add_bridge(bridged_model, GeoMeanBridge{T})
diff --git a/src/Bridges/Constraint/soc_to_nonconvex_quad.jl b/src/Bridges/Constraint/soc_to_nonconvex_quad.jl
new file mode 100644
index 0000000000..6f6c143f1b
--- /dev/null
+++ b/src/Bridges/Constraint/soc_to_nonconvex_quad.jl
@@ -0,0 +1,199 @@
+abstract type AbstractSOCtoNonConvexQuadBridge{T} <: AbstractBridge end
+
+"""
+    SOCtoNonConvexQuadBridge{T}
+
+Constraints of the form `VectorOfVariables`-in-`SecondOrderCone` can be
+transformed into a `ScalarQuadraticFunction`-in-`LessThan` and a
+`ScalarAffineFunction`-in-`GreaterThan`. Indeed, the definition of the
+second-order cone
+```math
+t \\ge \\lVert x \\rVert_2 \\  (1)
+```
+is equivalent to
+```math
+\\sum x_i^2 \\le t^2  (2)
+```
+with ``t \\ge 0``.  (3)
+
+*WARNING* This transformation starts from a convex constraint (1) and creates a
+non-convex constraint (2), because the Q matrix associated with the constraint 2
+has one negative eigenvalue. This might be wrongly interpreted by a solver.
+Some solvers can look at (2) and understand that it is a second order cone, but
+this is not a general rule.
+For these reasons this bridge is not automatically added by [`MOI.Bridges.full_bridge_optimizer`](@ref).
+Care is recommended when adding this bridge to a optimizer.
+"""
+struct SOCtoNonConvexQuadBridge{T} <: AbstractSOCtoNonConvexQuadBridge{T}
+    quad::CI{MOI.ScalarQuadraticFunction{T}, MOI.LessThan{T}}
+    var_pos::Vector{CI{MOI.ScalarAffineFunction{T}, MOI.GreaterThan{T}}}
+    vars::Vector{MOI.VariableIndex}
+end
+function bridge_constraint(::Type{SOCtoNonConvexQuadBridge{T}}, model,
+                           func::MOI.VectorOfVariables,
+                           set::MOI.SecondOrderCone) where T
+
+    vis = func.variables
+
+    t = vis[1]
+    x = vis[2:end]
+    a_terms = MOI.ScalarAffineTerm{T}[]
+    q_terms = MOI.ScalarQuadraticTerm{T}[]
+    push!(q_terms, MOI.ScalarQuadraticTerm(-T(2), t, t))
+    for var in x
+        push!(q_terms, MOI.ScalarQuadraticTerm(T(2), var, var))
+    end
+
+    fq = MOI.ScalarQuadraticFunction(a_terms, q_terms, zero(T))
+    quad = MOI.add_constraint(model, fq, MOI.LessThan(zero(T)))
+    # ScalarAffineFunction's are added instead of SingleVariable's
+    # because models can only have one SingleVariable per variable.
+    # Hence, adding a SingleVariable constraint here could conflict with
+    # a user defined SingleVariable
+    fp = convert(MOI.ScalarAffineFunction{T}, MOI.SingleVariable(t))
+    var_pos = MOI.add_constraint(model, fp, MOI.GreaterThan(zero(T)))
+
+    return SOCtoNonConvexQuadBridge(quad, [var_pos], vis)
+end
+
+"""
+    RSOCtoNonConvexQuadBridge{T}
+
+Constraints of the form `VectorOfVariables`-in-`SecondOrderCone` can be
+transformed into a `ScalarQuadraticFunction`-in-`LessThan` and a
+`ScalarAffineFunction`-in-`GreaterThan`. Indeed, the definition of the
+second-order cone
+```math
+2tu \\ge \\lVert x \\rVert_2^2, t,u \\ge 0  (1)
+```
+is equivalent to
+```math
+\\sum x_i^2 \\le 2tu  (2)
+```
+with ``t,u \\ge 0``.  (3)
+
+*WARNING* This transformation starts from a convex constraint (1) and creates a
+non-convex constraint (2), because the Q matrix associated with the constraint 2
+has two negative eigenvalues. This might be wrongly interpreted by a solver.
+Some solvers can look at (2) and understand that it is a rotated second order cone, but
+this is not a general rule.
+For these reasons, this bridge is not automatically added by [`MOI.Bridges.full_bridge_optimizer`](@ref).
+Care is recommended when adding this bridge to an optimizer.
+"""
+struct RSOCtoNonConvexQuadBridge{T} <: AbstractSOCtoNonConvexQuadBridge{T}
+    quad::CI{MOI.ScalarQuadraticFunction{T}, MOI.LessThan{T}}
+    var_pos::Vector{CI{MOI.ScalarAffineFunction{T}, MOI.GreaterThan{T}}}
+    vars::Vector{MOI.VariableIndex}
+end
+function bridge_constraint(::Type{RSOCtoNonConvexQuadBridge{T}}, model,
+                           func::MOI.VectorOfVariables,
+                           set::MOI.RotatedSecondOrderCone) where T
+
+    vis = func.variables
+
+    t = vis[1]
+    u = vis[2]
+    x = vis[3:end]
+    a_terms = MOI.ScalarAffineTerm{T}[]
+    q_terms = MOI.ScalarQuadraticTerm{T}[]
+    push!(q_terms, MOI.ScalarQuadraticTerm(-T(2), t, u))
+    for var in x
+        push!(q_terms, MOI.ScalarQuadraticTerm(T(2), var, var))
+    end
+
+    fq = MOI.ScalarQuadraticFunction(a_terms, q_terms, zero(T))
+    quad = MOI.add_constraint(model, fq, MOI.LessThan(zero(T)))
+    # ScalarAffineFunction's are added instead of SingleVariable's
+    # because models can only have one SingleVariable per variable.
+    # Hence, adding a SingleVariable constraint here could conflict with
+    # a user defined SingleVariable
+    fp1 = convert(MOI.ScalarAffineFunction{T}, MOI.SingleVariable(t))
+    var_pos1 = MOI.add_constraint(model, fp1, MOI.GreaterThan(zero(T)))
+    fp2 = convert(MOI.ScalarAffineFunction{T}, MOI.SingleVariable(u))
+    var_pos2 = MOI.add_constraint(model, fp2, MOI.GreaterThan(zero(T)))
+
+    return RSOCtoNonConvexQuadBridge(quad, [var_pos1, var_pos2], vis)
+end
+
+function MOI.supports_constraint(::Type{SOCtoNonConvexQuadBridge{T}},
+                                 ::Type{MOI.VectorOfVariables},
+                                 ::Type{MOI.SecondOrderCone}) where T
+    return true
+end
+function MOI.supports_constraint(::Type{RSOCtoNonConvexQuadBridge{T}},
+                                 ::Type{MOI.VectorOfVariables},
+                                 ::Type{MOI.RotatedSecondOrderCone}) where T
+    return true
+end
+
+MOIB.added_constrained_variable_types(::Type{<:AbstractSOCtoNonConvexQuadBridge}) = Tuple{DataType}[]
+function MOIB.added_constraint_types(::Type{<:AbstractSOCtoNonConvexQuadBridge{T}}) where T
+    return [
+        (MOI.ScalarQuadraticFunction{T}, MOI.LessThan{T}),
+        (MOI.ScalarAffineFunction{T}, MOI.GreaterThan{T}),
+        ]
+end
+
+function concrete_bridge_type(::Type{SOCtoNonConvexQuadBridge{T}},
+                              ::Type{MOI.VectorOfVariables},
+                              ::Type{MOI.SecondOrderCone}) where T
+    return SOCtoNonConvexQuadBridge{T}
+end
+function concrete_bridge_type(::Type{RSOCtoNonConvexQuadBridge{T}},
+                              ::Type{MOI.VectorOfVariables},
+                              ::Type{MOI.RotatedSecondOrderCone}) where T
+    return RSOCtoNonConvexQuadBridge{T}
+end
+
+# Attributes, Bridge acting as a model
+function MOI.get(::AbstractSOCtoNonConvexQuadBridge{T},
+                 ::MOI.NumberOfConstraints{MOI.ScalarQuadraticFunction{T},
+                                           MOI.LessThan{T}}) where T
+    return 1
+end
+
+function MOI.get(bridge::AbstractSOCtoNonConvexQuadBridge{T},
+                       ::MOI.ListOfConstraintIndices{
+                           MOI.ScalarQuadraticFunction{T},
+                           MOI.LessThan{T}}) where T
+    return [bridge.quad]
+end
+
+function MOI.get(bridge::AbstractSOCtoNonConvexQuadBridge{T},
+                 ::MOI.NumberOfConstraints{MOI.ScalarAffineFunction{T},
+                                           MOI.GreaterThan{T}}) where T
+    return length(bridge.var_pos)
+end
+
+function MOI.get(bridge::AbstractSOCtoNonConvexQuadBridge{T},
+                 ::MOI.ListOfConstraintIndices{MOI.ScalarAffineFunction{T},
+                                               MOI.GreaterThan{T}}) where T
+    return bridge.var_pos
+end
+
+# References
+function MOI.delete(model::MOI.ModelLike, bridge::AbstractSOCtoNonConvexQuadBridge)
+    MOI.delete(model, bridge.quad)
+    MOI.delete.(model, bridge.var_pos)
+end
+
+# Attributes, Bridge acting as a constraint
+function MOI.get(model::MOI.ModelLike, attr::MOI.ConstraintPrimal,
+                 bridge::AbstractSOCtoNonConvexQuadBridge)
+    vals = MOI.get.(model, MOI.VariablePrimal(attr.N), bridge.vars)
+    return vals
+end
+
+function MOI.get(model::MOI.ModelLike, attr::MOI.ConstraintSet,
+                 b::SOCtoNonConvexQuadBridge{T}) where T
+    return MOI.SecondOrderCone(length(b.vars))
+end
+function MOI.get(model::MOI.ModelLike, attr::MOI.ConstraintSet,
+                 b::RSOCtoNonConvexQuadBridge{T}) where T
+    return MOI.RotatedSecondOrderCone(length(b.vars))
+end
+
+function MOI.get(model::MOI.ModelLike, attr::MOI.ConstraintFunction,
+                 b::AbstractSOCtoNonConvexQuadBridge{T}) where T
+    return MOI.VectorOfVariables(b.vars)
+end
diff --git a/test/Bridges/Constraint/soc_to_nonconvex_quad.jl b/test/Bridges/Constraint/soc_to_nonconvex_quad.jl
new file mode 100644
index 0000000000..483cd3b577
--- /dev/null
+++ b/test/Bridges/Constraint/soc_to_nonconvex_quad.jl
@@ -0,0 +1,83 @@
+using Test
+
+using MathOptInterface
+const MOI = MathOptInterface
+const MOIT = MathOptInterface.Test
+const MOIU = MathOptInterface.Utilities
+const MOIB = MathOptInterface.Bridges
+const MOIBC = MathOptInterface.Bridges.Constraint
+
+include("../utilities.jl")
+
+mock = MOIU.MockOptimizer(MOIU.UniversalFallback(MOIU.Model{Float64}()))
+config = MOIT.TestConfig(duals = false)
+
+@testset "RSOCtoNonConvexQuad" begin
+
+    @test MOIBC.RSOCtoNonConvexQuadBridge{Float64} == MOIBC.concrete_bridge_type(
+        MOIBC.RSOCtoNonConvexQuadBridge{Float64},
+        MOI.VectorOfVariables,
+        MOI.RotatedSecondOrderCone)
+    @test MOI.supports_constraint(MOIBC.RSOCtoNonConvexQuadBridge{Float64},
+                                  MOI.VectorOfVariables,
+                                  MOI.RotatedSecondOrderCone)
+    @test !MOI.supports_constraint(MOIBC.RSOCtoNonConvexQuadBridge{Float64},
+                                   MOI.ScalarAffineFunction{Float64},
+                                   MOI.RotatedSecondOrderCone)
+
+    bridged_mock = MOIB.Constraint.RSOCtoNonConvexQuad{Float64}(mock)
+
+    MOIT.basic_constraint_tests(
+        bridged_mock, config,
+        include = [(F, S)
+                   for F in [MOI.VectorOfVariables]
+                   for S in [MOI.RotatedSecondOrderCone]])
+
+    mock.optimize! = (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, [0.5, 1.0, 1/√2, 1/√2])
+    MOIT.rotatedsoc1vtest(bridged_mock, config)
+
+    ci = first(MOI.get(bridged_mock,
+        MOI.ListOfConstraintIndices{MOI.VectorOfVariables,
+                                    MOI.RotatedSecondOrderCone}()))
+
+    test_delete_bridge(bridged_mock, ci, 4,
+                    (
+                        (MOI.ScalarQuadraticFunction{Float64}, MOI.LessThan{Float64}, 0),
+                        (MOI.ScalarAffineFunction{Float64}, MOI.GreaterThan{Float64}, 0),
+                    ))
+end
+
+@testset "SOCtoNonConvexQuad" begin
+
+    @test MOIBC.SOCtoNonConvexQuadBridge{Float64} == MOIBC.concrete_bridge_type(
+        MOIBC.SOCtoNonConvexQuadBridge{Float64},
+        MOI.VectorOfVariables,
+        MOI.SecondOrderCone)
+    @test MOI.supports_constraint(MOIBC.SOCtoNonConvexQuadBridge{Float64},
+                                  MOI.VectorOfVariables,
+                                  MOI.SecondOrderCone)
+    @test !MOI.supports_constraint(MOIBC.SOCtoNonConvexQuadBridge{Float64},
+                                   MOI.ScalarAffineFunction{Float64},
+                                   MOI.SecondOrderCone)
+
+    bridged_mock = MOIB.Constraint.SOCtoNonConvexQuad{Float64}(mock)
+
+    MOIT.basic_constraint_tests(
+        bridged_mock, config,
+        include = [(F, S)
+                   for F in [MOI.VectorOfVariables]
+                   for S in [MOI.SecondOrderCone]])
+
+    mock.optimize! = (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, [1.0, 1/√2, 1/√2])
+    MOIT.soc1vtest(bridged_mock, config)
+
+    ci = first(MOI.get(bridged_mock,
+        MOI.ListOfConstraintIndices{MOI.VectorOfVariables,
+                                    MOI.SecondOrderCone}()))
+
+    test_delete_bridge(bridged_mock, ci, 3,
+                       (
+                           (MOI.ScalarQuadraticFunction{Float64}, MOI.LessThan{Float64}, 0),
+                           (MOI.ScalarAffineFunction{Float64}, MOI.GreaterThan{Float64}, 0),
+                       ))
+end