[Utilities] add distance_to_set for SecondOrderCone (#2060)

src/Utilities/distance_to_set.jl

@@ -193,3 +193,23 @@ function distance_to_set(
     _check_dimension(x, set)
     return zero(T)
 end
+
+function distance_to_set(
+    ::ProjectionUpperBoundDistance,
+    x::AbstractVector{T},
+    set::MOI.SecondOrderCone,
+) where {T<:Real}
+    _check_dimension(x, set)
+    # Projections come from:
+    # Parikh, N., & Boyd, S. (2014). Proximal algorithms. Foundations and
+    # trends in Optimization, page 184, section 6.3.2.
+    t, rhs = x[1], LinearAlgebra.norm(@views x[2:end])
+    if t >= rhs
+        return zero(T) # The point is feasible!
+    end
+    if rhs <= -t  # Projection to the point (0, [0])
+        return LinearAlgebra.norm(x)
+    end
+    # Projection to the point (t, x) + 0.5 * (|x|_2 - t, (t/|x|_2 - 1) * x)
+    return sqrt(2) / 2 * abs(t - rhs)
+end

test/Utilities/distance_to_set.jl

@@ -8,6 +8,7 @@ module TestFeasibilityChecker
 
 using Test
 
+import LinearAlgebra
 import MathOptInterface
 
 const MOI = MathOptInterface
@@ -127,6 +128,21 @@ function test_zeros()
     return
 end
 
+function test_secondordercone()
+    @test_throws(
+        DimensionMismatch,
+        MOI.Utilities.distance_to_set([-1.0, 1.0], MOI.SecondOrderCone(3))
+    )
+    set = MOI.SecondOrderCone(3)
+    @test MOI.Utilities.distance_to_set([sqrt(2), 1, 1], set) ≈ 0.0
+    @test MOI.Utilities.distance_to_set([-sqrt(2), 1, 1], set) ≈ 2
+    @test MOI.Utilities.distance_to_set([-2, 1, 1], set) ≈ sqrt(6)
+    # According to Boyd, (t, x) = (1, [1, 1]), projects to:
+    d = ((1 / 2) * (1 + 1 / √2) * [√2, 1, 1]) .- [1, 1, 1]
+    @test MOI.Utilities.distance_to_set([1, 1, 1], set) ≈ LinearAlgebra.norm(d)
+    return
+end
+
 end
 
 TestFeasibilityChecker.runtests()