diff --git a/src/Bridges/Bridges.jl b/src/Bridges/Bridges.jl
index 83900237b7..2e21fe5a33 100644
--- a/src/Bridges/Bridges.jl
+++ b/src/Bridges/Bridges.jl
@@ -43,6 +43,7 @@ Returns a `LazyBridgeOptimizer` bridging `model` for every bridge defined in thi
 function fullbridgeoptimizer(model::MOI.ModelLike, ::Type{T}) where T
     bridgedmodel = MOIB.LazyBridgeOptimizer(model, AllBridgedConstraints{T}())
     add_bridge(bridgedmodel, MOIB.SplitIntervalBridge{T})
+    add_bridge(bridgedmodel, MOIB.QuadtoSOCBridge{T})
     add_bridge(bridgedmodel, MOIB.GeoMeanBridge{T})
     add_bridge(bridgedmodel, MOIB.SquarePSDBridge{T})
     add_bridge(bridgedmodel, MOIB.LogDetBridge{T})
@@ -57,6 +58,8 @@ include("intervalbridge.jl")
 @bridge SplitInterval SplitIntervalBridge () (MOI.Interval,) () () (MOI.SingleVariable,) (MOI.ScalarAffineFunction, MOI.ScalarQuadraticFunction) () ()
 include("rsocbridge.jl")
 @bridge RSOC RSOCBridge () () (MOI.RotatedSecondOrderCone,) () () () (MOI.VectorOfVariables,) (MOI.VectorAffineFunction,)
+include("quadtosocbridge.jl")
+@bridge QuadtoSOC QuadtoSOCBridge () (MOI.LessThan, MOI.GreaterThan) () () () (MOI.ScalarQuadraticFunction,) () ()
 include("geomeanbridge.jl")
 @bridge GeoMean GeoMeanBridge () () (MOI.GeometricMeanCone,) () () () (MOI.VectorOfVariables,) (MOI.VectorAffineFunction,)
 include("squarepsdbridge.jl")
diff --git a/src/Bridges/quadtosocbridge.jl b/src/Bridges/quadtosocbridge.jl
new file mode 100644
index 0000000000..35278db383
--- /dev/null
+++ b/src/Bridges/quadtosocbridge.jl
@@ -0,0 +1,202 @@
+using Compat.LinearAlgebra, Compat.SparseArrays
+
+"""
+    QuadtoSOCBridge{T}
+
+The set of points `x` satisfying the constraint
+```math
+\\frac{1}{2}x^T Q x + a^T x + b \\le 0
+```
+is a convex set if `Q` is positive semidefinite and is the union of two convex
+cones if `a` and `b` are zero (i.e. *homogeneous* case) and `Q` has only one
+negative eigenvalue. Currently, only the non-homogeneous transformation
+is implemented, see the [Note](@ref) section for more details.
+
+## Non-homogeneous case
+
+If `Q` is positive semidefinite, there exists `U` such that ``Q = U^T U``, the
+inequality can then be rewritten as
+```math
+\\|U x\\|_2 \\le 2 (a^T x + b)
+```
+which is equivalent to the membership of `(1, a^T x + b, Ux)` to the rotated
+second-order cone.
+
+## Homogeneous case
+
+If `Q` has only one negative eigenvalue, the set of `x` such that ``x^T Q x \\le
+0`` is the union of a convex cone and its opposite. We can choose which one to
+model by checking the existence of bounds on variables as shown below.
+
+### Second-order cone
+
+If `Q` is diagonal and has eigenvalues `(1, 1, -1)`, the inequality
+``x^2 + x^2 \\le z^2`` combined with ``z \\ge 0`` defines the Lorenz cone (i.e.
+the second-order cone) but when combined with ``z \\le 0``, it gives the
+opposite of the second order cone. Therefore, we need to check if the variable
+`z` has a lower bound 0 or an upper bound 0 in order to determine which cone is
+
+### Rotated second-order cone
+
+The matrix `Q` corresponding to the inequality ``x^2 \\le 2yz`` has one
+eigenvalue 1 with eigenvectors `(1, 0, 0)` and `(0, 1, -1)` and one eigenvalue
+`-1` corresponding to the eigenvector `(0, 1, 1)`. Hence if we intersect this
+union of two convex cone with the halfspace ``x + y \\ge 0``, we get the rotated
+second-order cone and if we intersect it with the halfspace ``x + y \\le 0`` we
+get the opposite of the rotated second-order cone. Note that `y` and `z` have
+the same sign since `yz` is nonnegative hence ``x + y \\ge 0`` is equivalent to
+``x \\ge 0`` and ``y \\ge 0``.
+
+### Note
+
+The check for existence of bound can be implemented (but inefficiently) with the
+current interface but if bound is removed or transformed (e.g. `≤ 0` transformed
+into `≥ 0`) then the bridge is no longer valid. For this reason the homogeneous
+version of the bridge is not implemented yet.
+"""
+struct QuadtoSOCBridge{T} <: AbstractBridge
+    soc::CI{MOI.VectorAffineFunction{T}, MOI.RotatedSecondOrderCone}
+    dimension::Int  # dimension of the SOC constraint
+    less_than::Bool # whether the constraint was ≤ or ≥
+    set_constant::T # the constant that was on the set
+end
+function QuadtoSOCBridge{T}(model, func::MOI.ScalarQuadraticFunction{T},
+                            set::Union{MOI.LessThan{T},
+                                       MOI.GreaterThan{T}}) where T
+    set_constant = MOIU.getconstant(set)
+    less_than = set isa MOI.LessThan
+    if !less_than
+        set_constant = -set_constant
+    end
+    Q, index_to_variable_map = matrix_from_quadratic_terms(func.quadratic_terms)
+    if !less_than
+        Compat.rmul!(Q, -1)
+    end
+    # We have L × L' ≈ Q[p, p]
+    L, p = try
+        @static if VERSION >= v"0.7-"
+            F = cholesky(Symmetric(Q))
+            sparse(F.L), F.p
+        else
+            F = cholfact(Symmetric(Q))
+            sparse(F[:L]), F[:p]
+        end
+    catch err
+        if err isa PosDefException
+            error("The optimizer supports second-order cone constraints and",
+                  " not quadratic constraints but you entered a quadratic",
+                  " constraint of type: `$(typeof(func))`-in-`$(typeof(set))`.",
+                  " A bridge attempted to transform the quadratic constraint",
+                  " to a second order cone constraint but the constraint is",
+                  " not strongly convex, i.e., the symmetric matrix of",
+                  " quadratic coefficients is not positive definite. Convex",
+                  " constraints that are not strongly convex, i.e. the matrix is",
+                  " positive semidefinite but not positive definite, are not",
+                  " supported yet.")
+        else
+            rethrow(err)
+        end
+    end
+    Ux_terms = matrix_to_vector_affine_terms(L, p, index_to_variable_map)
+    Ux = MOI.VectorAffineFunction(Ux_terms, zeros(T, size(L, 2)))
+    t = MOI.ScalarAffineFunction(less_than ? MOIU.operate_terms(-, func.affine_terms) : func.affine_terms,
+                                 less_than ? set_constant - func.constant : func.constant - set_constant)
+    f = MOIU.operate(vcat, T, one(T), t, Ux)
+    dimension = MOI.output_dimension(f)
+    soc = MOI.add_constraint(model, f,
+                             MOI.RotatedSecondOrderCone(dimension))
+    return QuadtoSOCBridge(soc, dimension, less_than, set_constant)
+end
+
+function matrix_from_quadratic_terms(terms::Vector{MOI.ScalarQuadraticTerm{T}}) where T
+    variable_to_index_map = Dict{VI, Int}()
+    index_to_variable_map = Dict{Int, VI}()
+    n = 0
+    for term in terms
+        for variable in (term.variable_index_1, term.variable_index_2)
+            if !(variable in keys(variable_to_index_map))
+                n += 1
+                variable_to_index_map[variable] = n
+                index_to_variable_map[n] = variable
+            end
+        end
+    end
+    I = Int[]
+    J = Int[]
+    V = T[]
+    for term in terms
+        i = variable_to_index_map[term.variable_index_1]
+        j = variable_to_index_map[term.variable_index_2]
+        push!(I, i)
+        push!(J, j)
+        push!(V, term.coefficient)
+        if i != j
+            push!(I, i)
+            push!(J, j)
+            push!(V, term.coefficient)
+        end
+    end
+    # Duplicate terms are summed together in `sparse`
+    Q = sparse(I, J, V, n, n)
+    return Q, index_to_variable_map
+end
+
+function matrix_to_vector_affine_terms(L::SparseMatrixCSC{T},
+                                       p::Vector,
+                                       index_to_variable_map::Dict{Int, VI}) where T
+    # We know that L × L' ≈ Q[p, p] hence (L × L')[i, :] ≈ Q[p[i], p]
+    # We precompute the map to avoid having to do a dictionary lookup for every
+    # term
+    variable = map(i -> index_to_variable_map[p[i]], 1:size(L, 1))
+    function term(i::Integer, j::Integer, v::T)
+        return MOI.VectorAffineTerm(j, MOI.ScalarAffineTerm(v, variable[i]))
+    end
+    return map(ijv -> term(ijv...), zip(findnz(L)...))
+end
+
+function MOI.supports_constraint(::Type{QuadtoSOCBridge{T}},
+                                 ::Type{MOI.ScalarQuadraticFunction{T}},
+                                 ::Type{<:Union{MOI.LessThan{T},
+                                                MOI.GreaterThan{T}}}) where T
+    return true
+end
+function added_constraint_types(::Type{QuadtoSOCBridge{T}}) where T
+    return [(MOI.VectorAffineFunction{T}, MOI.RotatedSecondOrderCone)]
+end
+function concrete_bridge_type(::Type{<:QuadtoSOCBridge{T}},
+                              ::Type{MOI.ScalarQuadraticFunction{T}},
+                              ::Type{<:Union{MOI.LessThan{T},
+                                               MOI.GreaterThan{T}}}) where T
+    return QuadtoSOCBridge{T}
+end
+
+# Attributes, Bridge acting as an model
+function MOI.get(::QuadtoSOCBridge{T},
+                 ::MOI.NumberOfConstraints{MOI.VectorAffineFunction{T},
+                                           MOI.RotatedSecondOrderCone}) where T
+    return 1
+end
+function MOI.get(bridge::QuadtoSOCBridge{T},
+                 ::MOI.ListOfConstraintIndices{MOI.VectorAffineFunction{T},
+                                               MOI.RotatedSecondOrderCone}) where T
+    return [bridge.soc]
+end
+
+# References
+function MOI.delete(model::MOI.ModelLike, bridge::QuadtoSOCBridge)
+    MOI.delete(model, bridge.soc)
+end
+
+# Attributes, Bridge acting as a constraint
+function MOI.get(model::MOI.ModelLike, attr::MOI.ConstraintPrimal,
+                 bridge::QuadtoSOCBridge)
+    soc = MOI.get(model, attr, bridge.soc)
+    output = sum(soc[i]^2 for i in 3:bridge.dimension)
+    output /= 2
+    output -= soc[1] * soc[2]
+    if !bridge.less_than
+        output = -output
+    end
+    output += bridge.set_constant
+    return output
+end
diff --git a/test/bridge.jl b/test/bridge.jl
index 90da0bff4d..b39acfa656 100644
--- a/test/bridge.jl
+++ b/test/bridge.jl
@@ -255,6 +255,40 @@ end
         test_delete_bridge(bridgedmock, ci, 2, ((MOI.VectorAffineFunction{Float64}, MOI.SecondOrderCone, 0),))
     end
 
+    @testset "QuadtoSOC" begin
+        bridgedmock = MOIB.QuadtoSOC{Float64}(mock)
+        @testset "Error for non-convex quadratic constraints" begin
+            x = MOI.add_variable(bridgedmock)
+            @test_throws ErrorException begin
+                MOI.add_constraint(bridgedmock,
+                                   MOI.ScalarQuadraticFunction(MOI.ScalarAffineTerm{Float64}[],
+                                                               [MOI.ScalarQuadraticTerm(1.0, x, x)],
+                                                               0.0),
+                                   MOI.GreaterThan(0.0))
+            end
+            @test_throws ErrorException begin
+                MOI.add_constraint(bridgedmock,
+                                   MOI.ScalarQuadraticFunction(MOI.ScalarAffineTerm{Float64}[],
+                                                               [MOI.ScalarQuadraticTerm(-1.0, x, x)],
+                                                               0.0),
+                                   MOI.LessThan(0.0))
+            end
+        end
+        MOIU.set_mock_optimize!(mock,
+            (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, [1/2, 7/4], MOI.FeasiblePoint))
+        MOIT.qcp1test(bridgedmock, config)
+        MOIU.set_mock_optimize!(mock,
+            (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, [√2], MOI.FeasiblePoint))
+        MOIT.qcp2test(bridgedmock, config)
+        MOIU.set_mock_optimize!(mock,
+            (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, [√2], MOI.FeasiblePoint))
+        MOIT.qcp3test(bridgedmock, config)
+        ci = first(MOI.get(bridgedmock,
+                           MOI.ListOfConstraintIndices{MOI.ScalarQuadraticFunction{Float64},
+                                                       MOI.LessThan{Float64}}()))
+        test_delete_bridge(bridgedmock, ci, 1, ((MOI.VectorAffineFunction{Float64}, MOI.RotatedSecondOrderCone, 0),))
+    end
+
     @testset "SquarePSD" begin
         bridgedmock = MOIB.SquarePSD{Float64}(mock)
         mock.optimize! = (mock::MOIU.MockOptimizer) -> MOIU.mock_optimize!(mock, ones(4),