Add RootDetAtom (#605)

Author: odow

docs/src/manual/operations.md

@@ -113,6 +113,7 @@ solver (including SCS and Mosek).
 | `opnorm(x, 2)` (`operatornorm(x)`)| max of singular values of $x$      | convex       | not monotonic |                        |
 | `eigmax(x)`                       | max eigenvalue of $x$              | convex       | not monotonic |                        |
 | `eigmin(x)`                       | min eigenvalue of $x$              | concave      | not monotonic |                        |
+| `rootdet(X)`                      | n-th root-determinant of the $n$-by-$n$ matrix X, that is $det(X)^{1/n}$ | concave      | not monotonic |                        |
 | `matrixfrac(x, P)`                | $x^TP^{-1}x$                       | convex       | not monotonic | IC: P is positive semidefinite |
 | `sumlargesteigs(x, k)`            | sum of top $k$ eigenvalues of $x$  | convex       | not monotonic | IC: P symmetric        |
 | `T in GeomMeanHypoCone(A, B, t)`  | $T \preceq A \#_t B = A^{1/2} (A^{-1/2} B A^{-1/2})^t A^{1/2}$ | concave | increasing    | IC: $A \succeq 0$, $B \succeq 0$, $t \in [0,1]$ |

src/Convex.jl

@@ -52,6 +52,7 @@ export conv,
     trace_logm,
     trace_mpower,
     lieb_ando,
+    rootdet,
     # Constraints
     Constraint,
     ⪰,

src/atoms/sdp_cone/RootDetAtom.jl

@@ -0,0 +1,29 @@
+mutable struct RootDetAtom <: AbstractExpr
+    children::Tuple{AbstractExpr}
+    size::Tuple{Int,Int}
+
+    RootDetAtom(x::AbstractExpr) = new((x,), (1, 1))
+end
+
+head(io::IO, ::RootDetAtom) = print(io, "rootdet")
+
+Base.sign(::RootDetAtom) = NoSign()
+
+monotonicity(::RootDetAtom) = (NoMonotonicity(),)
+
+curvature(::RootDetAtom) = ConcaveVexity()
+
+function evaluate(x::RootDetAtom)
+    n = size(only(AbstractTrees.children(x)), 1)
+    return LinearAlgebra.det(evaluate(x.children[1]))^(1 / n)
+end
+
+function new_conic_form!(context::Context{T}, x::RootDetAtom) where {T}
+    t = conic_form!(context, Variable())
+    A = only(AbstractTrees.children(x))
+    f = operate(vcat, T, sign(x), t, conic_form!(context, A))
+    MOI_add_constraint(context.model, f, MOI.RootDetConeSquare(size(A, 1)))
+    return t
+end
+
+rootdet(x::AbstractExpr) = RootDetAtom(x)

test/test_atoms.jl

@@ -1439,6 +1439,24 @@ end
 
 # TODO
 
+### sdp_cone/RootDetAtom
+
+function test_RootDetAtom()
+    target = """
+    variables: t, x11, x12, x21, x22
+    minobjective: 1.0 * t + 0.0
+    [1.0*t, 1.0*x11, 1.0 *x12, 1.0*x21, 1.0*x22] in RootDetConeSquare(2)
+    """
+    _test_atom(target) do context
+        return rootdet(Variable(2, 2))
+    end
+    x = Variable(2, 2)
+    x.value = [2.0 -1.5; -1.5 3.0]
+    atom = rootdet(x)
+    @test evaluate(atom) ≈ LinearAlgebra.det(x.value)^(1 / 2)
+    return
+end
+
 ### second_order_cone/EuclideanNormAtom
 
 function test_EuclideanNormAtom()