diff --git a/src/Utilities/penalty_relaxation.jl b/src/Utilities/penalty_relaxation.jl
index 9c23a796cc..4de8868f7e 100644
--- a/src/Utilities/penalty_relaxation.jl
+++ b/src/Utilities/penalty_relaxation.jl
@@ -187,9 +187,51 @@ function _relax_constraint(
     return [x]
 end
 
+function _slack_vector(set::MOI.AbstractVectorSet)
+    ret = Pair{Int,Int}[]
+    for i in 1:MOI.dimension(set)
+        push!(ret, i => +1)
+        push!(ret, i => -1)
+    end
+    return ret
+end
+
+_slack_vector(set::MOI.Nonnegatives) = [i => +1 for i in 1:MOI.dimension(set)]
+_slack_vector(set::MOI.Nonpositives) = [i => -1 for i in 1:MOI.dimension(set)]
+_slack_vector(::MOI.SecondOrderCone) = [1 => +1]
+_slack_vector(::MOI.RotatedSecondOrderCone) = [1 => +1, 2 => +1]
+_slack_vector(::MOI.NormOneCone) = [1 => +1]
+_slack_vector(::MOI.NormInfinityCone) = [1 => +1]
+_slack_vector(::MOI.NormCone) = [1 => +1]
+_slack_vector(::MOI.NormSpectralCone) = [1 => +1]
+_slack_vector(::MOI.NormNuclearCone) = [1 => +1]
+_slack_vector(::MOI.RootDetConeTriangle) = [1 => -1]
+_slack_vector(::MOI.LogDetConeTriangle) = [1 => -1, 2 => +1]
+
+function _slack_vector(set::MOI.GeometricMeanCone)
+    ret = [i => +1 for i in 1:MOI.dimension(set)]
+    ret[1] = (1 => -1)
+    return ret
+end
+
+function _relax_constraint(
+    ::Type{T},
+    model::MOI.ModelLike,
+    ci::MOI.ConstraintIndex{F,<:MOI.AbstractVectorSet},
+) where {T,F<:Union{MOI.VectorAffineFunction{T},MOI.VectorQuadraticFunction{T}}}
+    set = MOI.get(model, MOI.ConstraintSet(), ci)
+    slack = _slack_vector(set)
+    y, _ = MOI.add_constrained_variables(model, MOI.Nonnegatives(length(slack)))
+    for (i, (row, weight)) in enumerate(slack)
+        change = MOI.MultirowChange(y[i], [(Int64(row), weight * one(T))])
+        MOI.modify(model, ci, change)
+    end
+    return y
+end
+
 function MOI.modify(
     model::MOI.ModelLike,
-    ci::MOI.ConstraintIndex{F,<:MOI.AbstractScalarSet},
+    ci::MOI.ConstraintIndex{F},
     relax::ScalarPenaltyRelaxation{T},
 ) where {
     T,
@@ -197,6 +239,8 @@ function MOI.modify(
         MOI.ScalarAffineFunction{T},
         MOI.ScalarQuadraticFunction{T},
         MOI.ScalarNonlinearFunction,
+        MOI.VectorAffineFunction{T},
+        MOI.VectorQuadraticFunction{T},
     },
 }
     x = _relax_constraint(T, model, ci)
diff --git a/test/Utilities/test_penalty_relaxation.jl b/test/Utilities/test_penalty_relaxation.jl
index 4b19485347..7a91e7c779 100644
--- a/test/Utilities/test_penalty_relaxation.jl
+++ b/test/Utilities/test_penalty_relaxation.jl
@@ -475,6 +475,258 @@ function test_scalar_penalty_relaxation_vector_objective()
     return
 end
 
+function test_relax_vector_epigraph_cones()
+    _test_roundtrip(
+        """
+        variables: x, y, z
+        c1: [1.0 * x, y, z] in SecondOrderCone(3)
+        """,
+        """
+        variables: x, y, z, a
+        minobjective: 1.0 * a
+        c1: [1.0 * x + 1.0 * a, y, z] in SecondOrderCone(3)
+        [a] in Nonnegatives(1)
+        """,
+    )
+    _test_roundtrip(
+        """
+        variables: x, y, z
+        c1: [1.0 * x, y, z] in RotatedSecondOrderCone(3)
+        """,
+        """
+        variables: x, y, z, a, b
+        minobjective: 1.0 * a + 1.0 * b
+        c1: [1.0 * x + 1.0 * a, y + 1.0 * b, z] in RotatedSecondOrderCone(3)
+        [a, b] in Nonnegatives(2)
+        """,
+    )
+    _test_roundtrip(
+        """
+        variables: x, y, z
+        c1: [1.0 * x, y, z] in NormOneCone(3)
+        """,
+        """
+        variables: x, y, z, a
+        minobjective: 1.0 * a
+        c1: [1.0 * x + 1.0 * a, y, z] in NormOneCone(3)
+        [a] in Nonnegatives(1)
+        """,
+    )
+    _test_roundtrip(
+        """
+        variables: x, y, z
+        c1: [1.0 * x, y, z] in NormInfinityCone(3)
+        """,
+        """
+        variables: x, y, z, a
+        minobjective: 1.0 * a
+        c1: [1.0 * x + 1.0 * a, y, z] in NormInfinityCone(3)
+        [a] in Nonnegatives(1)
+        """,
+    )
+    _test_roundtrip(
+        """
+        variables: x, y, z
+        c1: [1.0 * x, y, z] in NormCone(1.5, 3)
+        """,
+        """
+        variables: x, y, z, a
+        minobjective: 1.0 * a
+        c1: [1.0 * x + 1.0 * a, y, z] in NormCone(1.5, 3)
+        [a] in Nonnegatives(1)
+        """,
+    )
+    _test_roundtrip(
+        """
+        variables: x, y, z
+        c1: [1.0 * x, y, z] in NormSpectralCone(1, 2)
+        """,
+        """
+        variables: x, y, z, a
+        minobjective: 1.0 * a
+        c1: [1.0 * x + 1.0 * a, y, z] in NormSpectralCone(1, 2)
+        [a] in Nonnegatives(1)
+        """,
+    )
+    _test_roundtrip(
+        """
+        variables: x, y, z
+        c1: [1.0 * x, y, z] in NormNuclearCone(1, 2)
+        """,
+        """
+        variables: x, y, z, a
+        minobjective: 1.0 * a
+        c1: [1.0 * x + 1.0 * a, y, z] in NormNuclearCone(1, 2)
+        [a] in Nonnegatives(1)
+        """,
+    )
+    _test_roundtrip(
+        """
+        variables: x, y, z
+        c1: [1.0 * x * x, 1.0 * y, 1.0 * z] in NormCone(1.5, 3)
+        """,
+        """
+        variables: x, y, z, a
+        minobjective: 1.0 * a
+        c1: [1.0 * x * x + 1.0 * a, 1.0 * y, 1.0 * z] in NormCone(1.5, 3)
+        [a] in Nonnegatives(1)
+        """,
+    )
+    return
+end
+
+function test_relax_vector_hypograph_cones()
+    _test_roundtrip(
+        """
+        variables: t, x, y, z
+        c1: [1.0 * t, x, y, z] in RootDetConeTriangle(2)
+        """,
+        """
+        variables: t, x, y, z, a
+        minobjective: 1.0 * a
+        c1: [1.0 * t + -1.0 * a, x, y, z] in RootDetConeTriangle(2)
+        [a] in Nonnegatives(1)
+        """,
+    )
+    _test_roundtrip(
+        """
+        variables: t, u, x, y, z
+        c1: [1.0 * t, u, x, y, z] in LogDetConeTriangle(2)
+        """,
+        """
+        variables: t, u, x, y, z, a, b
+        minobjective: 1.0 * a + 1.0 * b
+        c1: [1.0 * t + -1.0 * a, u + 1.0 * b, x, y, z] in LogDetConeTriangle(2)
+        [a, b] in Nonnegatives(2)
+        """,
+    )
+    return
+end
+
+function test_relax_nonnegatives()
+    _test_roundtrip(
+        """
+        variables: x, y
+        c1: [1.0 * x, y] in Nonnegatives(2)
+        """,
+        """
+        variables: x, y, a, b
+        minobjective: 1.0 * a + 1.0 * b
+        c1: [1.0 * x + 1.0 * a, y + 1.0 * b] in Nonnegatives(2)
+        [a, b] in Nonnegatives(2)
+        """,
+    )
+    return
+end
+
+function test_relax_nonpositives()
+    _test_roundtrip(
+        """
+        variables: x, y
+        c1: [1.0 * x, y] in Nonpositives(2)
+        """,
+        """
+        variables: x, y, a, b
+        minobjective: 1.0 * a + 1.0 * b
+        c1: [1.0 * x + -1.0 * a, y + -1.0 * b] in Nonpositives(2)
+        [a, b] in Nonnegatives(2)
+        """,
+    )
+    return
+end
+
+function test_relax_geometric_mean_cone()
+    _test_roundtrip(
+        """
+        variables: x, y, z
+        c1: [1.0 * x, y, z] in GeometricMeanCone(3)
+        """,
+        """
+        variables: x, y, z, a, b, c
+        minobjective: 1.0 * a + 1.0 * b + 1.0 * c
+        c1: [1.0 * x + -1.0 * a, y + 1.0 * b, z + 1.0 * c] in GeometricMeanCone(3)
+        [a, b, c] in Nonnegatives(3)
+        """,
+    )
+    return
+end
+
+function test_relax_both_sides()
+    _test_roundtrip(
+        """
+        variables: x, y
+        c1: [1.0 * x, y] in Zeros(2)
+        """,
+        """
+        variables: x, y, a, b, c, d
+        minobjective: 1.0 * a + 1.0 * b + 1.0 * c + 1.0 * d
+        c1: [1.0 * x + 1.0 * a + -1.0 * b, y + 1.0 * c + -1.0 * d] in Zeros(2)
+        [a, b, c, d] in Nonnegatives(4)
+        """,
+    )
+    _test_roundtrip(
+        """
+        variables: x, y, z
+        c1: [1.0 * x, y, z] in ExponentialCone()
+        """,
+        """
+        variables: x, y, z, a, b, c, d, e, f
+        minobjective: 1.0 * a + 1.0 * b + 1.0 * c + 1.0 * d + 1.0 * e + 1.0 * f
+        c1: [1.0 * x + 1.0 * a + -1.0 * b, y + 1.0 * c + -1.0 * d, z + 1.0 * e + -1.0 * f] in ExponentialCone()
+        [a, b, c, d, e, f] in Nonnegatives(6)
+        """,
+    )
+    _test_roundtrip(
+        """
+        variables: x, y, z
+        c1: [1.0 * x, y, z] in DualExponentialCone()
+        """,
+        """
+        variables: x, y, z, a, b, c, d, e, f
+        minobjective: 1.0 * a + 1.0 * b + 1.0 * c + 1.0 * d + 1.0 * e + 1.0 * f
+        c1: [1.0 * x + 1.0 * a + -1.0 * b, y + 1.0 * c + -1.0 * d, z + 1.0 * e + -1.0 * f] in DualExponentialCone()
+        [a, b, c, d, e, f] in Nonnegatives(6)
+        """,
+    )
+    _test_roundtrip(
+        """
+        variables: x, y, z
+        c1: [1.0 * x, y, z] in PowerCone(0.5)
+        """,
+        """
+        variables: x, y, z, a, b, c, d, e, f
+        minobjective: 1.0 * a + 1.0 * b + 1.0 * c + 1.0 * d + 1.0 * e + 1.0 * f
+        c1: [1.0 * x + 1.0 * a + -1.0 * b, y + 1.0 * c + -1.0 * d, z + 1.0 * e + -1.0 * f] in PowerCone(0.5)
+        [a, b, c, d, e, f] in Nonnegatives(6)
+        """,
+    )
+    _test_roundtrip(
+        """
+        variables: x, y, z
+        c1: [1.0 * x, y, z] in DualPowerCone(0.5)
+        """,
+        """
+        variables: x, y, z, a, b, c, d, e, f
+        minobjective: 1.0 * a + 1.0 * b + 1.0 * c + 1.0 * d + 1.0 * e + 1.0 * f
+        c1: [1.0 * x + 1.0 * a + -1.0 * b, y + 1.0 * c + -1.0 * d, z + 1.0 * e + -1.0 * f] in DualPowerCone(0.5)
+        [a, b, c, d, e, f] in Nonnegatives(6)
+        """,
+    )
+    _test_roundtrip(
+        """
+        variables: x, y, z
+        c1: [1.0 * x, y, z] in HyperRectangle([0.0, 0.0, 0.0], [1.0, 1.0, 1.0])
+        """,
+        """
+        variables: x, y, z, a, b, c, d, e, f
+        minobjective: 1.0 * a + 1.0 * b + 1.0 * c + 1.0 * d + 1.0 * e + 1.0 * f
+        c1: [1.0 * x + 1.0 * a + -1.0 * b, y + 1.0 * c + -1.0 * d, z + 1.0 * e + -1.0 * f] in HyperRectangle([0.0, 0.0, 0.0], [1.0, 1.0, 1.0])
+        [a, b, c, d, e, f] in Nonnegatives(6)
+        """,
+    )
+    return
+end
+
 end  # module
 
 TestPenaltyRelaxation.runtests()