Constrained optimization (IPNewton?) for big numbers

[olugovoy]

Hello, Following the #647 and the fix #648 (which works - thanks!). Are the any available options of constrained problems for BigFloat? Any plans to add methods for bigs to IPNewton or other options? Thanks again!

[anriseth]

Any plans to add methods for bigs to IPNewton or other options

Have you tried if it works already? If it doesn't, I believe it can be enabled fairly easily by relaxing the number type specifications in the relevant IPNewton code.

I have not planned to do so myself, but it should not be too difficult so please have a go and make a PR :)

[pkofod]

That shouldn’t be a problem to get to work, if it doesn’t already. If BigFloats don’t work, I’d consider it a bug that should be fixed.

[olugovoy]

Good to know it should work in general! And that the fix should be easy. Here is the working example which I'm trying to replicate http://julianlsolvers.github.io/Optim.jl/stable/#examples/generated/ipnewton_basics/#plain-program with big numbers, but have the "no method" error:

julia> using Optim, NLSolversBase #hide

julia> fun(x) =  (1.0 - x[1])^2 + 100.0 * (x[2] - x[1]^2)^2
fun (generic function with 1 method)

julia> function fun_grad!(g, x)
           g[1] = -2.0 * (1.0 - x[1]) - 400.0 * (x[2] - x[1]^2) * x[1]
           g[2] = 200.0 * (x[2] - x[1]^2)
       end
fun_grad! (generic function with 1 method)

julia> function fun_hess!(h, x)
           h[1, 1] = 2.0 - 400.0 * x[2] + 1200.0 * x[1]^2
           h[1, 2] = -400.0 * x[1]
           h[2, 1] = -400.0 * x[1]
           h[2, 2] = 200.0
       end;

julia> x0 = big.([0.0, 0.0])
2-element Array{BigFloat,1}:
 0.000000000000000000000000000000000000000000000000000000000000000000000000000000
 0.000000000000000000000000000000000000000000000000000000000000000000000000000000

julia> df = TwiceDifferentiable(fun, fun_grad!, fun_hess!, x0)
NLSolversBase.TwiceDifferentiable{BigFloat,Array{BigFloat,1},Array{BigFloat,2},Array{BigFloat,1}}(fun, fun_grad!, NLSolversBase.fg!, fun_hess!, 0.000000000000000000000000000000000000000000000000000000000000000000000000000000, BigFloat[#undef, #undef], BigFloat[BigFloat(NaN, 256) BigFloat(NaN, 256); BigFloat(NaN, 256) BigFloat(NaN, 256)], BigFloat[BigFloat(NaN, 256), BigFloat(NaN, 256)], BigFloat[BigFloat(NaN, 256), BigFloat(NaN, 256)], BigFloat[BigFloat(NaN, 256), BigFloat(NaN, 256)], [0], [0], [0])

julia> lx = big.([-0.5, -0.5])
2-element Array{BigFloat,1}:
 -5.000000000000000000000000000000000000000000000000000000000000000000000000000000e-01
 -5.000000000000000000000000000000000000000000000000000000000000000000000000000000e-01

julia> ux = big.([0.5, 0.5])
2-element Array{BigFloat,1}:
 5.000000000000000000000000000000000000000000000000000000000000000000000000000000e-01
 5.000000000000000000000000000000000000000000000000000000000000000000000000000000e-01

julia> dfc = TwiceDifferentiableConstraints(lx, ux)
NLSolversBase.TwiceDifferentiableConstraints{NLSolversBase.##90#93,NLSolversBase.##91#94,NLSolversBase.##92#95,BigFloat}(NLSolversBase.#90, NLSolversBase.#91, NLSolversBase.#92, ConstraintBounds:
  Variables:
    x[1]≥-5.000000000000000000000000000000000000000000000000000000000000000000000000000000e-01, x[1]≤5.000000000000000000000000000000000000000000000000000000000000000000000000000000e-01, x[2]≥-5.000000000000000000000000000000000000000000000000000000000000000000000000000000e-01, x[2]≤5.000000000000000000000000000000000000000000000000000000000000000000000000000000e-01
  Linear/nonlinear constraints:)

julia> res = optimize(df, dfc, x0, IPNewton())
ERROR: MethodError: no method matching getindex(::Base.LinAlg.CholeskyPivoted{BigFloat,Array{BigFloat,2}}, ::Symbol)
Stacktrace:
 [1] convert at ./linalg/cholesky.jl:408 [inlined]
 [2] convert at ./linalg/cholesky.jl:411 [inlined]
 [3] full(::Base.LinAlg.CholeskyPivoted{BigFloat,Array{BigFloat,2}}) at ./linalg/cholesky.jl:414
 [4] solve_step!(::Optim.IPNewtonState{BigFloat,Array{BigFloat,1}}, ::NLSolversBase.TwiceDifferentiableConstraints{NLSolversBase.##90#93,NLSolversBase.##91#94,NLSolversBase.##92#95,BigFloat}, ::Optim.Options{Float64,Void}, ::Bool) at /Users/OL/.julia/v0.6/Optim/src/multivariate/solvers/constrained/ipnewton/ipnewton.jl:302
 [5] update_state!(::NLSolversBase.TwiceDifferentiable{BigFloat,Array{BigFloat,1},Array{BigFloat,2},Array{BigFloat,1}}, ::NLSolversBase.TwiceDifferentiableConstraints{NLSolversBase.##90#93,NLSolversBase.##91#94,NLSolversBase.##92#95,BigFloat}, ::Optim.IPNewtonState{BigFloat,Array{BigFloat,1}}, ::Optim.IPNewton{Optim.#backtrack_constrained_grad,Symbol}, ::Optim.Options{Float64,Void}) at /Users/OL/.julia/v0.6/Optim/src/multivariate/solvers/constrained/ipnewton/ipnewton.jl:234
 [6] optimize(::NLSolversBase.TwiceDifferentiable{BigFloat,Array{BigFloat,1},Array{BigFloat,2},Array{BigFloat,1}}, ::NLSolversBase.TwiceDifferentiableConstraints{NLSolversBase.##90#93,NLSolversBase.##91#94,NLSolversBase.##92#95,BigFloat}, ::Array{BigFloat,1}, ::Optim.IPNewton{Optim.#backtrack_constrained_grad,Symbol}, ::Optim.Options{Float64,Void}, ::Optim.IPNewtonState{BigFloat,Array{BigFloat,1}}) at /Users/OL/.julia/v0.6/Optim/src/multivariate/solvers/constrained/ipnewton/interior.jl:216
 [7] optimize(::NLSolversBase.TwiceDifferentiable{BigFloat,Array{BigFloat,1},Array{BigFloat,2},Array{BigFloat,1}}, ::NLSolversBase.TwiceDifferentiableConstraints{NLSolversBase.##90#93,NLSolversBase.##91#94,NLSolversBase.##92#95,BigFloat}, ::Array{BigFloat,1}, ::Optim.IPNewton{Optim.#backtrack_constrained_grad,Symbol}) at /Users/OL/.julia/v0.6/Optim/src/multivariate/solvers/constrained/ipnewton/interior.jl:196

The rest of the code with big.() is below, but basically it will show the same "no method" issue in optimize():

ux = big.(fill(Inf, 2))
dfc = TwiceDifferentiableConstraints(lx, ux)

clear!(df)
res = optimize(df, dfc, x0, IPNewton())

lx =  big.(fill(-Inf, 2)); ux =  big.(fill(Inf, 2))
dfc = TwiceDifferentiableConstraints(lx, ux)

clear!(df)
res = optimize(df, dfc, x0, IPNewton())

lx =  big.(Float64[]); ux =  big.(Float64[])
dfc = TwiceDifferentiableConstraints(lx, ux)

clear!(df)
res = optimize(df, dfc, x0, IPNewton())

con_c!(c, x) = (c[1] = x[1]^2 + x[2]^2; c)
function con_jacobian!(J, x)
    J[1,1] = 2*x[1]
    J[1,2] = 2*x[2]
    J
end
function con_h!(h, x, λ)
    h[1,1] += λ[1]*2
    h[2,2] += λ[1]*2
end;

lx =  big.(Float64[]); ux =  big.(Float64[])
lc =  big.([-Inf]); uc =  big.([0.5^2])
dfc = TwiceDifferentiableConstraints(con_c!, con_jacobian!, con_h!,
                                     lx, ux, lc, uc)
res = optimize(df, dfc, x0, IPNewton())

lc =  big.([0.1^2])
dfc = TwiceDifferentiableConstraints(con_c!, con_jacobian!, con_h!,
                                     lx, ux, lc, uc)
res = optimize(df, dfc, x0, IPNewton())

function con2_c!(c, x)
    c[1] = x[1]^2 + x[2]^2     ## First constraint
    c[2] = x[2]*sin(x[1])-x[1] ## Second constraint
    c
end
function con2_jacobian!(J, x)
    # First constraint
    J[1,1] = 2*x[1]
    J[1,2] = 2*x[2]
    # Second constraint
    J[2,1] = x[2]*cos(x[1])-1.0
    J[2,2] = sin(x[1])
    J
end
function con2_h!(h, x, λ)
    # First constraint
    h[1,1] += λ[1]*2
    h[2,2] += λ[1]*2
    # Second constraint
    h[1,1] += λ[2]*x[2]*-sin(x[1])
    h[1,2] += λ[2]*cos(x[1])
    # Symmetrize h
    h[2,1]  = h[1,2]
    h
end;

x0 =  big.([0.25, 0.25])
lc =  big.([-Inf, 0.0]); uc =  big.([0.5^2, 0.0])
dfc = TwiceDifferentiableConstraints(con2_c!, con2_jacobian!, con2_h!,
                                     lx, ux, lc, uc)
res = optimize(df, dfc, x0, IPNewton())

[pkofod]

This is positivefactorizations again? I’ll look into it, thanks for bringing it up

[anriseth]

This is positivefactorizations again?

I think this goes deeper than that. The error appears when IPNewton calls full(state.Htilde).

On Julia 0.6, you can create pivoted Cholesky factorization for BigFloat matrices, but they have only implemented getindex, which is called in full, for BlasFloat types.

On Julia 0.7, pivoted Cholesky factorization has not been implemented for non-BLAS types:

julia> A = Matrix(big(1.0)I, 3,3)
3×3 Array{BigFloat,2}:
 1.0  0.0  0.0
 0.0  1.0  0.0
 0.0  0.0  1.0

julia> F = cholesky(A, Val(true))
ERROR: ArgumentError: generic pivoted Cholesky factorization is not implemented yet
Stacktrace:
 [1] #cholesky!#93(::Float64, ::Bool, ::Function, ::Hermitian{BigFloat,Array{BigFloat,2}}, ::Val{true}) at /scratch/riseth/julia-dev/usr/share/julia/stdlib/v0.7/LinearAlgebra/src/cholesky.jl:201
 [2] (::getfield(LinearAlgebra, Symbol("#kw##cholesky!")))(::NamedTuple{(:tol, :check),Tuple{Float64,Bool}}, ::typeof(cholesky!), ::Hermitian{BigFloat,Array{BigFloat,2}}, ::Val{true}) at ./none:0
 [3] #cholesky!#94(::Float64, ::Bool, ::Function, ::Array{BigFloat,2}, ::Val{true}) at /scratch/riseth/julia-dev/usr/share/julia/stdlib/v0.7/LinearAlgebra/src/cholesky.jl:221
 [4] #cholesky! at ./none:0 [inlined]
 [5] #cholesky#96(::Float64, ::Bool, ::Function, ::Array{BigFloat,2}, ::Val{true}) at /scratch/riseth/julia-dev/usr/share/julia/stdlib/v0.7/LinearAlgebra/src/cholesky.jl:296
 [6] cholesky(::Array{BigFloat,2}, ::Val{true}) at /scratch/riseth/julia-dev/usr/share/julia/stdlib/v0.7/LinearAlgebra/src/cholesky.jl:296
 [7] top-level scope at none:0

[anriseth]

I guess we have ~two~three options:

1. Implement a more generic linear system solver to IPNewton
2. Only apply pivoted Cholesky for BlasFloats, and otherwise revert back to the default Cholesky (I don't know why Tim wanted Pivoted Cholesky in the first place, so this may cause instabilities)
3. File an issue upstream to see whether they could loosen the Cholesky codes that currently only work with BlasFloat to Union{BigFloat, BlasFloat}. (This won't fix the problem for arbitrary precision, however)

[pkofod]

Hmm, I’ll have to be at a computer to understand the issue

[anriseth]

Sorry, I didn't explain the issue very well. Let me know if I should expand on what I mean after you've had the opportunity to look at it on a computer ;)