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ported cvxquad: matrix log, geometric mean, quantum entropy (#418)
* ported cvxquad: matrix log, geometric mean, quantum entropy * comments about vexity * throw DimensionMismatch for wrong size matrices * throw DomainError instead of `error` * added @test_throw * refactored range assert * added function overloads taking Integer, because that doesn't auto-convert to Rational * more tests * fix relative_entropy_epicone comment * document relative_entropy_epicone * made RelativeEntropyEpiCone a constraint rather than a function * minor changes and more coverage * fix diagm calls for old julia * fix for entropy with real matrices * RelativeEntropyEpiCone test * trace_logm tests * added cvxquad license * fix
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| Original file line number | Diff line number | Diff line change |
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| ############################################################################# | ||
| # geom_mean_epicone.jl | ||
| # Constrains T to | ||
| # A #_t B ⪯ T | ||
| # where: | ||
| # * A #_t B is the t-weighted geometric mean of A and B: | ||
| # A^{1/2} (A^{-1/2} B A^{-1/2})^t A^{1/2} | ||
| # Parameter t should be in [-1, 0] or [1, 2]. | ||
| # * Constraints A ⪰ 0, B ⪰ 0 are added. | ||
| # All expressions and atoms are subtypes of AbstractExpr. | ||
| # Please read expressions.jl first. | ||
| # | ||
| # Note on fullhyp: GeomMeanEpiCone will always return a full epigraph cone | ||
| # (unlike GeomMeanHypoCone) and so this parameter is not really used. It is | ||
| # here just for consistency with the GeomMeanHypoCone function. | ||
| # | ||
| #REFERENCE | ||
| # Ported from CVXQUAD which is based on the paper: "Lieb's concavity | ||
| # theorem, matrix geometric means and semidefinite optimization" by Hamza | ||
| # Fawzi and James Saunderson (arXiv:1512.03401) | ||
| ############################################################################# | ||
|
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| struct GeomMeanEpiCone | ||
| A::AbstractExpr | ||
| B::AbstractExpr | ||
| t::Rational | ||
| size::Tuple{Int, Int} | ||
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| function GeomMeanEpiCone(A::AbstractExpr, B::AbstractExpr, t::Rational, fullhyp::Bool=true) | ||
| if size(A) != size(B) | ||
| throw(DimensionMismatch("A and B must be the same size")) | ||
| end | ||
| n = size(A)[1] | ||
| if size(A) != (n, n) | ||
| throw(DimensionMismatch("A and B must be square")) | ||
| end | ||
| if t < -1 || (t > 0 && t < 1) || t > 2 | ||
| throw(DomainError(t, "t must be in the range [-1, 0] or [1, 2]")) | ||
| end | ||
| return new(A, B, t, (n, n)) | ||
| end | ||
|
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||
| GeomMeanEpiCone(A::Value, B::AbstractExpr, t::Rational, fullhyp::Bool=true) = GeomMeanEpiCone(Constant(A), B, t, fullhyp) | ||
| GeomMeanEpiCone(A::AbstractExpr, B::Value, t::Rational, fullhyp::Bool=true) = GeomMeanEpiCone(A, Constant(B), t, fullhyp) | ||
| GeomMeanEpiCone(A::Value, B::Value, t::Rational, fullhyp::Bool=true) = GeomMeanEpiCone(Constant(A), Constant(B), t, fullhyp) | ||
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| GeomMeanEpiCone(A::AbstractExprOrValue, B::AbstractExprOrValue, t::Integer, fullhyp::Bool=true) = GeomMeanEpiCone(A, B, t//1, fullhyp) | ||
| end | ||
|
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| struct GeomMeanEpiConeConstraint <: Constraint | ||
| head::Symbol | ||
| id_hash::UInt64 | ||
| T::AbstractExpr | ||
| cone::GeomMeanEpiCone | ||
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| function GeomMeanEpiConeConstraint(T::AbstractExpr, cone::GeomMeanEpiCone) | ||
| if size(T) != cone.size | ||
| throw(DimensionMismatch("T must be size $(cone.size)")) | ||
| end | ||
| id_hash = hash((cone.A, cone.B, cone.t, :GeomMeanEpiCone)) | ||
| return new(:GeomMeanEpiCone, id_hash, T, cone) | ||
| end | ||
|
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| GeomMeanEpiConeConstraint(T::Value, cone::GeomMeanEpiCone) = GeomMeanEpiConeConstraint(Constant(T), cone) | ||
| end | ||
|
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| in(T, cone::GeomMeanEpiCone) = GeomMeanEpiConeConstraint(T, cone) | ||
|
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| function AbstractTrees.children(constraint::GeomMeanEpiConeConstraint) | ||
| return (constraint.T, constraint.cone.A, constraint.cone.B, "t=$(constraint.cone.t)") | ||
| end | ||
|
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| # For t ∈ [-1, 0] ∪ [1, 2] the t-weighted matrix geometric mean is matrix convex (arxiv:1512.03401). | ||
| # So if A and B are convex sets, then T ⪰ A #_t B will be a convex set. | ||
| function vexity(constraint::GeomMeanEpiConeConstraint) | ||
| A = vexity(constraint.cone.A) | ||
| B = vexity(constraint.cone.B) | ||
| T = vexity(constraint.T) | ||
|
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| # NOTE: can't say A == NotDcp() because the NotDcp constructor prints a warning message. | ||
| if typeof(A) == ConcaveVexity || typeof(A) == NotDcp | ||
| return NotDcp() | ||
| end | ||
| if typeof(B) == ConcaveVexity || typeof(B) == NotDcp | ||
| return NotDcp() | ||
| end | ||
| # Copied from vexity(c::GtConstraint) | ||
| vex = ConvexVexity() + (-T) | ||
| if vex == ConcaveVexity() | ||
| vex = NotDcp() | ||
| end | ||
| return vex | ||
| end | ||
|
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| function conic_form!(constraint::GeomMeanEpiConeConstraint, unique_conic_forms::UniqueConicForms) | ||
| if !has_conic_form(unique_conic_forms, constraint) | ||
| A = constraint.cone.A | ||
| B = constraint.cone.B | ||
| t = constraint.cone.t | ||
| T = constraint.T | ||
| is_complex = sign(A) == ComplexSign() || sign(B) == ComplexSign() || sign(T) == ComplexSign() | ||
| if is_complex | ||
| make_temporary = () -> HermitianSemidefinite(size(A)[1]) | ||
| else | ||
| make_temporary = () -> Semidefinite(size(A)[1]) | ||
| end | ||
|
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| Z = make_temporary() | ||
|
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| if t <= 0 | ||
| conic_form!([T A; A Z] ⪰ 0, unique_conic_forms) | ||
| conic_form!(Z in GeomMeanHypoCone(A, B, -t, false), unique_conic_forms) | ||
| else | ||
| @assert t >= 1 # range of t checked in GeomMeanEpiCone constructor | ||
| conic_form!([T B; B Z] ⪰ 0, unique_conic_forms) | ||
| conic_form!(Z in GeomMeanHypoCone(A, B, 2-t, false), unique_conic_forms) | ||
| end | ||
|
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| cache_conic_form!(unique_conic_forms, constraint, Array{Convex.ConicConstr,1}()) | ||
| end | ||
| return get_conic_form(unique_conic_forms, constraint) | ||
| end |
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