Hessians

Hessians are a crucial ingredient in NewtonSolvers and SimpleSolvers.NewtonOptimizerStates.

using SimpleSolvers
using LinearAlgebra: norm

x = rand(3)
obj = MultivariateObjective(x -> norm(x - vcat(0., 0., 1.))  ^ 2, x)
hes = HessianAutodiff(obj, x)

HessianAutodiff{Float64, Main.var"#2#3", Matrix{Float64}, ForwardDiff.HessianConfig{ForwardDiff.Tag{Main.var"#2#3", Float64}, Float64, 3, Vector{ForwardDiff.Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}, ForwardDiff.Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}, Float64, 3}, 3}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}, Float64, 3}}}}(Main.var"#2#3"(), [NaN NaN NaN; NaN NaN NaN; NaN NaN NaN], ForwardDiff.HessianConfig{ForwardDiff.Tag{Main.var"#2#3", Float64}, Float64, 3, Vector{ForwardDiff.Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}, ForwardDiff.Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}, Float64, 3}, 3}}, Vector{ForwardDiff.Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}, Float64, 3}}}(ForwardDiff.JacobianConfig{ForwardDiff.Tag{Main.var"#2#3", Float64}, Float64, 3, Vector{ForwardDiff.Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}, Float64, 3}}}((Partials(1.0, 0.0, 0.0), Partials(0.0, 1.0, 0.0), Partials(0.0, 0.0, 1.0)), ForwardDiff.Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}, Float64, 3}[Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(6.9213518010037e-310,5.0e-324,6.92134959929263e-310,6.9213505074849e-310), Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(6.92136020234376e-310,6.92135645028355e-310,6.9213560468501e-310,6.92134961408456e-310), Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(6.92134959928156e-310,6.9213565477774e-310,6.9213493487816e-310,6.92134959928156e-310)]), ForwardDiff.GradientConfig{ForwardDiff.Tag{Main.var"#2#3", Float64}, ForwardDiff.Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}, Float64, 3}, 3, Vector{ForwardDiff.Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}, ForwardDiff.Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}, Float64, 3}, 3}}}((Partials(Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(1.0,0.0,0.0,0.0), Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0), Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0)), Partials(Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0), Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(1.0,0.0,0.0,0.0), Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0)), Partials(Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0), Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0), Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(1.0,0.0,0.0,0.0))), ForwardDiff.Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}, ForwardDiff.Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}, Float64, 3}, 3}[Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(5.0e-324,6.92131950084827e-310,6.92131950085143e-310,5.0e-324),Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(2.8476216620468583e-306,6.9213518010037e-310,0.0,0.0),Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0),Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0)), Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0),Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0),Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0),Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0)), Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0),Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0),Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0),Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}}(0.0,0.0,0.0,0.0))])))

An instance of HessianAutodiff stores a Hessian matrix:

hes.H

3×3 Matrix{Float64}:
 NaN  NaN  NaN
 NaN  NaN  NaN
 NaN  NaN  NaN

The instance of HessianAutodiff can be called:

hes(x)

3×3 Matrix{Float64}:
 2.0          0.0           0.0
 0.0          2.0          -1.11022e-16
 5.55112e-17  1.11022e-16   2.0

Or equivalently with:

update!(hes, x)

This updates hes.H:

hes.H

3×3 Matrix{Float64}:
 2.0          0.0           0.0
 0.0          2.0          -1.11022e-16
 5.55112e-17  1.11022e-16   2.0

BFGS Hessian

using SimpleSolvers: initialize!
hes = HessianBFGS(obj, x)
initialize!(hes, x)

HessianBFGS{Float64, Vector{Float64}, Matrix{Float64}, MultivariateObjective{Float64, Vector{Float64}, Main.var"#2#3", GradientAutodiff{Float64, Main.var"#2#3", ForwardDiff.GradientConfig{ForwardDiff.Tag{Main.var"#2#3", Float64}, Float64, 3, Vector{ForwardDiff.Dual{ForwardDiff.Tag{Main.var"#2#3", Float64}, Float64, 3}}}}, Float64, Vector{Float64}}}(MultivariateObjective (for vector-valued quantities only the first component is printed):

    f(x)              = NaN 
    g(x)₁             = 3.82e-01 
    x_f₁              = NaN 
    x_g₁              = 1.91e-01 
    number of f calls = 0 
    number of g calls = 1 
, [NaN, NaN, NaN], [0.19090669902576285, 0.5256623915420473, 0.3905882754313441], [NaN, NaN, NaN], [NaN, NaN, NaN], [0.3818133980515257, 1.0513247830840946, -1.2188234491373118], [NaN, NaN, NaN], [1.0 0.0 0.0; 0.0 1.0 0.0; 0.0 0.0 1.0], [0.0 0.0 0.0; 0.0 0.0 0.0; 0.0 0.0 0.0], [0.0 0.0 0.0; 0.0 0.0 0.0; 0.0 0.0 0.0], [0.0 0.0 0.0; 0.0 0.0 0.0; 0.0 0.0 0.0], [0.0 0.0 0.0; 0.0 0.0 0.0; 0.0 0.0 0.0], [0.0 0.0 0.0; 0.0 0.0 0.0; 0.0 0.0 0.0])

For computational reasons we save the inverse of the Hessian, it can be accessed by calling inv:

inv(hes)

3×3 Matrix{Float64}:
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  1.0

Similarly to HessianAutodiff we can call update!:

update!(hes, x)