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feature/re
...
fs11_01
| Author | SHA1 | Date | |
|---|---|---|---|
| aa11db99a2 | |||
| a4ff65275e | |||
| 295e29c351 | |||
| 67e706d727 | |||
| 407312e129 | |||
| e2e69415c1 | |||
| 9713873a34 | |||
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| 8edd031bbe | |||
| 0a0d133161 | |||
| 0b5ec4740e | |||
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| a9f1b2c394 | |||
| 2ea0043c03 | |||
| 9ac2f74856 | |||
| 672bb220c1 | |||
| 20a7cfb42d | |||
| b6ba75c3dc | |||
| a5a3690bb6 | |||
| e5a2550c21 |
@@ -1,11 +1,12 @@
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||||
name = "MIPLearn"
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||||
uuid = "2b1277c3-b477-4c49-a15e-7ba350325c68"
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||||
authors = ["Alinson S Xavier <git@axavier.org>"]
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||||
version = "0.4.0"
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version = "0.4.2"
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||||
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[deps]
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Conda = "8f4d0f93-b110-5947-807f-2305c1781a2d"
|
||||
DataStructures = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8"
|
||||
Gurobi = "2e9cd046-0924-5485-92f1-d5272153d98b"
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||||
HDF5 = "f67ccb44-e63f-5c2f-98bd-6dc0ccc4ba2f"
|
||||
HiGHS = "87dc4568-4c63-4d18-b0c0-bb2238e4078b"
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JLD2 = "033835bb-8acc-5ee8-8aae-3f567f8a3819"
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@@ -28,6 +29,7 @@ TimerOutputs = "a759f4b9-e2f1-59dc-863e-4aeb61b1ea8f"
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[compat]
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Conda = "1"
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DataStructures = "0.18"
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Gurobi = "1.7.5"
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HDF5 = "0.16"
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HiGHS = "1"
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JLD2 = "0.4"
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@@ -36,8 +38,10 @@ JuMP = "1"
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KLU = "0.4"
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MathOptInterface = "1"
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OrderedCollections = "1"
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PrecompileTools = "1"
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PyCall = "1"
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Requires = "1"
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SCIP = "0.12"
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Statistics = "1"
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TimerOutputs = "0.5"
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julia = "1"
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2
deps/build.jl
vendored
2
deps/build.jl
vendored
@@ -5,7 +5,7 @@ function install_miplearn()
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Conda.update()
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pip = joinpath(dirname(pyimport("sys").executable), "pip")
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isfile(pip) || error("$pip: invalid path")
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run(`$pip install miplearn==0.4.0`)
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run(`$pip install miplearn==0.4.4`)
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end
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install_miplearn()
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@@ -6,7 +6,7 @@ using Printf
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function print_progress_header()
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@printf(
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"%8s %9s %9s %13s %13s %9s %6s %13s %6s %-24s %9s %9s %6s %6s",
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"%8s %9s %9s %13s %13s %9s %9s %13s %9s %-24s %9s %9s %6s %6s",
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"time",
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"processed",
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"pending",
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@@ -46,7 +46,7 @@ function print_progress(
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branch_ub = @sprintf("%9.2f", last(node.branch_ub))
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end
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@printf(
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"%8.2f %9d %9d %13.6e %13.6e %9.2e %6d %13.6e %6s %-24s %9s %9s %6d %6d",
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"%8.2f %9d %9d %13.6e %13.6e %9.2e %9d %13.6e %9s %-24s %9s %9s %6d %6d",
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time_elapsed,
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pool.processed,
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length(pool.processing) + length(pool.pending),
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@@ -134,7 +134,11 @@ function _get_int_variables(
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var_ub = constr.upper
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MOI.delete(optimizer, _upper_bound_index(var))
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end
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MOI.add_constraint(optimizer, var, MOI.Interval(var_lb, var_ub))
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MOI.add_constraint(
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optimizer,
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MOI.VariableIndex(var.index),
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MOI.Interval(var_lb, var_ub),
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)
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end
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push!(vars, var)
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push!(lb, var_lb)
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@@ -185,7 +185,7 @@ function collect_gmi(
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)
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end
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function select_gmi_rows(data, basis, x; max_rows = 10, atol = 1e-4)
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function select_gmi_rows(data, basis, x; max_rows = 10, atol = 0.001)
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candidate_rows = [
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r for r = 1:length(basis.var_basic) if (
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(data.var_types[basis.var_basic[r]] != 'C') &&
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@@ -199,23 +199,82 @@ function select_gmi_rows(data, basis, x; max_rows = 10, atol = 1e-4)
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return [candidate_rows[perm[i]] for i = 1:min(length(perm), max_rows)]
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end
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# function compute_gmi(data::ProblemData, tableau::Tableau)::ConstraintSet
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# @timeit "Initialization" begin
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# nrows, ncols = size(tableau.lhs)
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# ub = Float64[Inf for _ = 1:nrows]
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# lb = Float64[0.999 for _ = 1:nrows]
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# tableau_I, tableau_J, tableau_V = findnz(tableau.lhs)
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# lhs_I = Int[]
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# lhs_J = Int[]
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# lhs_V = Float64[]
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# end
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# @timeit "Compute coefficients" begin
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# for k = 1:nnz(tableau.lhs)
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# i::Int = tableau_I[k]
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# j::Int = tableau_J[k]
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# v::Float64 = 0.0
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# frac_alpha_j = frac(tableau_V[k])
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# alpha_j = tableau_V[k]
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# beta = frac(tableau.rhs[i])
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# if data.var_types[j] == 'C'
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# if alpha_j >= 0
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# v = alpha_j / beta
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# else
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# v = -alpha_j / (1 - beta)
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# end
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# else
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# if frac_alpha_j < beta
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# v = frac_alpha_j / beta
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# else
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# v = (1 - frac_alpha_j) / (1 - beta)
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# end
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# end
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# if abs(v) > 1e-8
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# push!(lhs_I, i)
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# push!(lhs_J, tableau_J[k])
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# push!(lhs_V, v)
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# end
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# end
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# end
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# @timeit "Convert to ConstraintSet" begin
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# lhs = sparse(lhs_I, lhs_J, lhs_V, nrows, ncols)
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# cs = ConstraintSet(; lhs, ub, lb)
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# end
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# return cs
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# end
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function compute_gmi(data::ProblemData, tableau::Tableau)::ConstraintSet
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nrows, ncols = size(tableau.lhs)
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ub = Float64[Inf for _ = 1:nrows]
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lb = Float64[0.9999 for _ = 1:nrows]
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tableau_I, tableau_J, tableau_V = findnz(tableau.lhs)
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lhs_I = Int[]
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lhs_J = Int[]
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lhs_V = Float64[]
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@timeit "Initialization" begin
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nrows::Int, ncols::Int = size(tableau.lhs)
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var_types::Vector{Char} = data.var_types
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tableau_rhs::Vector{Float64} = tableau.rhs
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tableau_I::Vector{Int}, tableau_J::Vector{Int}, tableau_V::Vector{Float64} = findnz(tableau.lhs)
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end
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@timeit "Pre-allocation" begin
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cut_ub::Vector{Float64} = fill(Inf, nrows)
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cut_lb::Vector{Float64} = fill(0.999, nrows)
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nnz_tableau::Int = length(tableau_I)
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cut_lhs_I::Vector{Int} = Vector{Int}(undef, nnz_tableau)
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cut_lhs_J::Vector{Int} = Vector{Int}(undef, nnz_tableau)
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cut_lhs_V::Vector{Float64} = Vector{Float64}(undef, nnz_tableau)
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cut_hash::Vector{UInt64} = zeros(UInt64, nrows)
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nnz_count::Int = 0
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end
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@timeit "Compute coefficients" begin
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for k = 1:nnz(tableau.lhs)
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@inbounds for k = 1:nnz_tableau
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i::Int = tableau_I[k]
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j::Int = tableau_J[k]
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v::Float64 = 0.0
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frac_alpha_j = frac(tableau_V[k])
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alpha_j = tableau_V[k]
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beta = frac(tableau.rhs[i])
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if data.var_types[j] == 'C'
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alpha_j::Float64 = tableau_V[k]
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frac_alpha_j::Float64 = alpha_j - floor(alpha_j)
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beta_i::Float64 = tableau_rhs[i]
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beta::Float64 = beta_i - floor(beta_i)
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v::Float64 = 0
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# Compute coefficient
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if var_types[j] == 'C'
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if alpha_j >= 0
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v = alpha_j / beta
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else
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@@ -228,16 +287,34 @@ function compute_gmi(data::ProblemData, tableau::Tableau)::ConstraintSet
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v = (1 - frac_alpha_j) / (1 - beta)
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end
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end
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# Store if significant
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if abs(v) > 1e-8
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push!(lhs_I, i)
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push!(lhs_J, tableau_J[k])
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push!(lhs_V, v)
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nnz_count += 1
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cut_lhs_I[nnz_count] = i
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cut_lhs_J[nnz_count] = j
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cut_lhs_V[nnz_count] = v
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cut_hash[i] = hash(j, cut_hash[i])
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cut_hash[i] = hash(v, cut_hash[i])
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end
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end
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lhs = sparse(lhs_I, lhs_J, lhs_V, nrows, ncols)
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end
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return ConstraintSet(; lhs, ub, lb)
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@timeit "Resize arrays to actual size" begin
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resize!(cut_lhs_I, nnz_count)
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resize!(cut_lhs_J, nnz_count)
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resize!(cut_lhs_V, nnz_count)
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end
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# TODO: Build cut in compressed row format instead of converting
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@timeit "Convert to ConstraintSet" begin
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cut_lhs::SparseMatrixCSC = sparse(cut_lhs_I, cut_lhs_J, cut_lhs_V, nrows, ncols)
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cs::ConstraintSet = ConstraintSet(; lhs=cut_lhs, ub=cut_ub, lb=cut_lb, hash=cut_hash)
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end
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return cs
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end
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export compute_gmi,
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frac, select_gmi_rows, assert_cuts_off, assert_does_not_cut_off, collect_gmi
|
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File diff suppressed because it is too large
Load Diff
@@ -27,25 +27,26 @@ function assert_eq(a, b; atol = 1e-4)
|
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end
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function assert_cuts_off(cuts::ConstraintSet, x::Vector{Float64}, tol = 1e-6)
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vals = cuts.lhs * x
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for i = 1:length(cuts.lb)
|
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val = cuts.lhs[i, :]' * x
|
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if (val <= cuts.ub[i] - tol) && (val >= cuts.lb[i] + tol)
|
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throw(ErrorException("inequality fails to cut off fractional solution"))
|
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if (vals[i] <= cuts.ub[i] - tol) && (vals[i] >= cuts.lb[i] + tol)
|
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throw(ErrorException("inequality $i fails to cut off fractional solution: $(cuts.lb[i]) <= $(vals[i]) <= $(cuts.ub[i])"))
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end
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end
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end
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function assert_does_not_cut_off(cuts::ConstraintSet, x::Vector{Float64}; tol = 1e-6)
|
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vals = cuts.lhs * x
|
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for i = 1:length(cuts.lb)
|
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val = cuts.lhs[i, :]' * x
|
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ub = cuts.ub[i]
|
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lb = cuts.lb[i]
|
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if (val >= ub) || (val <= lb)
|
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throw(
|
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ErrorException(
|
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"inequality $i cuts off integer solution ($lb <= $val <= $ub)",
|
||||
),
|
||||
)
|
||||
if (vals[i] >= cuts.ub[i]) || (vals[i] <= cuts.lb[i])
|
||||
throw(ErrorException("inequality $i cuts off integer solution: $(cuts.lb[i]) <= $(vals[i]) <= $(cuts.ub[i])"))
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
function assert_int(x::Float64, tol=1e-5)
|
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fx = frac(x)
|
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if min(fx, 1 - fx) >= tol
|
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throw(ErrorException("Number must be integer: $x"))
|
||||
end
|
||||
end
|
||||
|
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@@ -18,10 +18,10 @@ Base.@kwdef mutable struct ProblemData
|
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end
|
||||
|
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Base.@kwdef mutable struct Tableau
|
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obj::Any
|
||||
lhs::Any
|
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rhs::Any
|
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z::Any
|
||||
obj::Vector{Float64}
|
||||
lhs::SparseMatrixCSC
|
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rhs::Vector{Float64}
|
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z::Float64
|
||||
end
|
||||
|
||||
Base.@kwdef mutable struct Basis
|
||||
@@ -35,6 +35,7 @@ Base.@kwdef mutable struct ConstraintSet
|
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lhs::SparseMatrixCSC
|
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ub::Vector{Float64}
|
||||
lb::Vector{Float64}
|
||||
hash::Union{Nothing,Vector{UInt64}} = nothing
|
||||
end
|
||||
|
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export ProblemData, Tableau, Basis, ConstraintSet
|
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|
||||
@@ -4,48 +4,161 @@
|
||||
|
||||
using KLU
|
||||
using TimerOutputs
|
||||
using Gurobi
|
||||
|
||||
function get_basis(model::JuMP.Model)::Basis
|
||||
var_basic = Int[]
|
||||
var_nonbasic = Int[]
|
||||
constr_basic = Int[]
|
||||
constr_nonbasic = Int[]
|
||||
|
||||
# Variables
|
||||
for (i, var) in enumerate(all_variables(model))
|
||||
bstatus = MOI.get(model, MOI.VariableBasisStatus(), var)
|
||||
if bstatus == MOI.BASIC
|
||||
push!(var_basic, i)
|
||||
elseif bstatus == MOI.NONBASIC_AT_LOWER
|
||||
push!(var_nonbasic, i)
|
||||
else
|
||||
error("Unknown basis status: $bstatus")
|
||||
end
|
||||
if isa(unsafe_backend(model), Gurobi.Optimizer)
|
||||
return get_basis_gurobi(model)
|
||||
end
|
||||
|
||||
# Constraints
|
||||
constr_index = 1
|
||||
for (ftype, stype) in list_of_constraint_types(model)
|
||||
for constr in all_constraints(model, ftype, stype)
|
||||
if ftype == VariableRef
|
||||
# nop
|
||||
elseif ftype == AffExpr
|
||||
bstatus = MOI.get(model, MOI.ConstraintBasisStatus(), constr)
|
||||
if bstatus == MOI.BASIC
|
||||
push!(constr_basic, constr_index)
|
||||
elseif bstatus == MOI.NONBASIC
|
||||
push!(constr_nonbasic, constr_index)
|
||||
else
|
||||
error("Unknown basis status: $bstatus")
|
||||
end
|
||||
constr_index += 1
|
||||
@timeit "Initialization" begin
|
||||
var_basic = Int[]
|
||||
var_nonbasic = Int[]
|
||||
constr_basic = Int[]
|
||||
constr_nonbasic = Int[]
|
||||
nvars = num_variables(model)
|
||||
sizehint!(var_basic, nvars)
|
||||
sizehint!(var_nonbasic, nvars)
|
||||
end
|
||||
|
||||
@timeit "Query variables" begin
|
||||
for (i, var) in enumerate(all_variables(model))
|
||||
bstatus = MOI.get(model, MOI.VariableBasisStatus(), var)
|
||||
if bstatus == MOI.BASIC
|
||||
push!(var_basic, i)
|
||||
elseif bstatus == MOI.NONBASIC_AT_LOWER
|
||||
push!(var_nonbasic, i)
|
||||
else
|
||||
error("Unsupported constraint type: ($ftype, $stype)")
|
||||
error("Unknown basis status: $bstatus")
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
return Basis(; var_basic, var_nonbasic, constr_basic, constr_nonbasic)
|
||||
@timeit "Query constraints" begin
|
||||
constr_index = 1
|
||||
for (ftype, stype) in list_of_constraint_types(model)
|
||||
for constr in all_constraints(model, ftype, stype)
|
||||
if ftype == VariableRef
|
||||
# nop
|
||||
elseif ftype == AffExpr
|
||||
bstatus = MOI.get(model, MOI.ConstraintBasisStatus(), constr)
|
||||
if bstatus == MOI.BASIC
|
||||
push!(constr_basic, constr_index)
|
||||
elseif bstatus == MOI.NONBASIC
|
||||
push!(constr_nonbasic, constr_index)
|
||||
else
|
||||
error("Unknown basis status: $bstatus")
|
||||
end
|
||||
constr_index += 1
|
||||
else
|
||||
error("Unsupported constraint type: ($ftype, $stype)")
|
||||
end
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
@timeit "Build basis struct" begin
|
||||
basis = Basis(; var_basic, var_nonbasic, constr_basic, constr_nonbasic)
|
||||
end
|
||||
|
||||
return basis
|
||||
end
|
||||
|
||||
function set_basis(model::JuMP.Model, basis::Basis)
|
||||
if isa(unsafe_backend(model), Gurobi.Optimizer)
|
||||
# NOP
|
||||
return
|
||||
end
|
||||
|
||||
@timeit "Initialization" begin
|
||||
nvars = num_variables(model)
|
||||
gurobi_model = unsafe_backend(model).inner
|
||||
end
|
||||
|
||||
@timeit "Set variable basis" begin
|
||||
var_basis_statuses = Vector{Cint}(undef, nvars)
|
||||
fill!(var_basis_statuses, -1) # Default to GRB_NONBASIC_LOWER
|
||||
|
||||
for var_idx in basis.var_basic
|
||||
var_basis_statuses[var_idx] = 0 # GRB_BASIC
|
||||
end
|
||||
|
||||
ret = GRBsetintattrarray(gurobi_model, "VBasis", 0, nvars, var_basis_statuses)
|
||||
if ret != 0
|
||||
error("Failed to set variable basis statuses in Gurobi: error code $ret")
|
||||
end
|
||||
end
|
||||
|
||||
@timeit "Set constraint basis" begin
|
||||
nconstr = num_constraints(model, AffExpr, MOI.EqualTo{Float64})
|
||||
constr_basis_statuses = Vector{Cint}(undef, nconstr)
|
||||
fill!(constr_basis_statuses, -1) # Default to GRB_NONBASIC
|
||||
|
||||
for constr_idx in basis.constr_basic
|
||||
constr_basis_statuses[constr_idx] = 0 # GRB_BASIC
|
||||
end
|
||||
|
||||
ret = GRBsetintattrarray(gurobi_model, "CBasis", 0, nconstr, constr_basis_statuses)
|
||||
if ret != 0
|
||||
error("Failed to set constraint basis statuses in Gurobi: error code $ret")
|
||||
end
|
||||
end
|
||||
|
||||
return nothing
|
||||
end
|
||||
|
||||
function get_basis_gurobi(model::JuMP.Model)::Basis
|
||||
@timeit "Initialization" begin
|
||||
var_basic = Int[]
|
||||
var_nonbasic = Int[]
|
||||
constr_basic = Int[]
|
||||
constr_nonbasic = Int[]
|
||||
nvars = num_variables(model)
|
||||
sizehint!(var_basic, nvars)
|
||||
sizehint!(var_nonbasic, nvars)
|
||||
gurobi_model = unsafe_backend(model).inner
|
||||
end
|
||||
|
||||
@timeit "Query variables" begin
|
||||
var_basis_statuses = Vector{Cint}(undef, nvars)
|
||||
ret = GRBgetintattrarray(gurobi_model, "VBasis", 0, nvars, var_basis_statuses)
|
||||
if ret != 0
|
||||
error("Failed to get variable basis statuses from Gurobi: error code $ret")
|
||||
end
|
||||
for i in 1:nvars
|
||||
if var_basis_statuses[i] == 0 # GRB_BASIC
|
||||
push!(var_basic, i)
|
||||
elseif var_basis_statuses[i] == -1 # GRB_NONBASIC_LOWER
|
||||
push!(var_nonbasic, i)
|
||||
else
|
||||
error("Unknown variable basis status: $(var_basis_statuses[i])")
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
@timeit "Query constraints" begin
|
||||
nconstr = num_constraints(model, AffExpr, MOI.EqualTo{Float64})
|
||||
constr_basis_statuses = Vector{Cint}(undef, nconstr)
|
||||
ret = GRBgetintattrarray(gurobi_model, "CBasis", 0, nconstr, constr_basis_statuses)
|
||||
if ret != 0
|
||||
error("Failed to get constraint basis statuses from Gurobi: error code $ret")
|
||||
end
|
||||
for i in 1:nconstr
|
||||
if constr_basis_statuses[i] == 0 # GRB_BASIC
|
||||
push!(constr_basic, i)
|
||||
elseif constr_basis_statuses[i] == -1 # GRB_NONBASIC
|
||||
push!(constr_nonbasic, i)
|
||||
else
|
||||
error("Unknown constraint basis status: $(constr_basis_statuses[i])")
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
@timeit "Build basis struct" begin
|
||||
basis = Basis(; var_basic, var_nonbasic, constr_basic, constr_nonbasic)
|
||||
end
|
||||
|
||||
return basis
|
||||
end
|
||||
|
||||
function get_x(model::JuMP.Model)
|
||||
@@ -58,7 +171,12 @@ function compute_tableau(
|
||||
x::Union{Nothing,Vector{Float64}} = nothing,
|
||||
rows::Union{Vector{Int},Nothing} = nothing,
|
||||
tol = 1e-8,
|
||||
estimated_density = 0.10,
|
||||
)::Tableau
|
||||
if isnan(estimated_density) || estimated_density <= 0
|
||||
estimated_density = 0.10
|
||||
end
|
||||
|
||||
@timeit "Split data" begin
|
||||
nrows, ncols = size(data.constr_lhs)
|
||||
lhs_slacks = sparse(I, nrows, nrows)
|
||||
@@ -73,35 +191,71 @@ function compute_tableau(
|
||||
factor = klu(sparse(lhs_b'))
|
||||
end
|
||||
|
||||
@timeit "Compute tableau" begin
|
||||
@timeit "Initialize" begin
|
||||
tableau_rhs = zeros(length(rows))
|
||||
tableau_lhs = zeros(length(rows), ncols)
|
||||
end
|
||||
for k in eachindex(1:length(rows))
|
||||
@timeit "Prepare inputs" begin
|
||||
i = rows[k]
|
||||
e = zeros(nrows)
|
||||
e[i] = 1.0
|
||||
end
|
||||
@timeit "Initialize arrays" begin
|
||||
num_rows = length(rows)
|
||||
tableau_rhs::Array{Float64} = zeros(num_rows)
|
||||
tableau_rowptr::Array{Int} = zeros(Int, num_rows + 1)
|
||||
tableau_colval::Array{Int} = Int[]
|
||||
tableau_nzval::Array{Float64} = Float64[]
|
||||
estimated_nnz::Int = round(num_rows * ncols * estimated_density)
|
||||
sizehint!(tableau_colval, estimated_nnz)
|
||||
sizehint!(tableau_nzval, estimated_nnz)
|
||||
e::Array{Float64} = zeros(nrows)
|
||||
sol::Array{Float64} = zeros(nrows)
|
||||
tableau_row::Array{Float64} = zeros(ncols)
|
||||
end
|
||||
|
||||
A = data.constr_lhs'
|
||||
b = data.constr_ub
|
||||
tableau_rowptr[1] = 1
|
||||
|
||||
@timeit "Process rows" begin
|
||||
for k in eachindex(rows)
|
||||
@timeit "Solve" begin
|
||||
sol = factor \ e
|
||||
fill!(e, 0.0)
|
||||
e[rows[k]] = 1.0
|
||||
ldiv!(sol, factor, e)
|
||||
end
|
||||
@timeit "Multiply" begin
|
||||
tableau_lhs[k, :] = sol' * data.constr_lhs
|
||||
tableau_rhs[k] = sol' * data.constr_ub
|
||||
@timeit "Compute row" begin
|
||||
mul!(tableau_row, A, sol)
|
||||
tableau_rhs[k] = dot(sol, b)
|
||||
end
|
||||
needed_space = length(tableau_colval) + ncols
|
||||
if needed_space > estimated_nnz
|
||||
@timeit "Grow arrays" begin
|
||||
estimated_nnz *= 2
|
||||
sizehint!(tableau_colval, estimated_nnz)
|
||||
sizehint!(tableau_nzval, estimated_nnz)
|
||||
end
|
||||
end
|
||||
@timeit "Collect nonzeros for row" begin
|
||||
for j in 1:ncols
|
||||
val = tableau_row[j]
|
||||
if abs(val) > tol
|
||||
push!(tableau_colval, j)
|
||||
push!(tableau_nzval, val)
|
||||
end
|
||||
end
|
||||
end
|
||||
tableau_rowptr[k + 1] = length(tableau_colval) + 1
|
||||
end
|
||||
@timeit "Sparsify" begin
|
||||
tableau_lhs[abs.(tableau_lhs) .<= tol] .= 0
|
||||
tableau_lhs = sparse(tableau_lhs)
|
||||
end
|
||||
end
|
||||
|
||||
@timeit "Shrink arrays" begin
|
||||
sizehint!(tableau_colval, length(tableau_colval))
|
||||
sizehint!(tableau_nzval, length(tableau_nzval))
|
||||
end
|
||||
|
||||
@timeit "Build sparse matrix" begin
|
||||
tableau_lhs_transposed = SparseMatrixCSC(ncols, num_rows, tableau_rowptr, tableau_colval, tableau_nzval)
|
||||
tableau_lhs = transpose(tableau_lhs_transposed)
|
||||
end
|
||||
|
||||
@timeit "Compute tableau objective row" begin
|
||||
sol = factor \ obj_b
|
||||
tableau_obj = -data.obj' + sol' * data.constr_lhs
|
||||
tableau_obj[abs.(tableau_obj).<tol] .= 0
|
||||
tableau_obj = Array(tableau_obj')
|
||||
end
|
||||
|
||||
# Compute z if solution is provided
|
||||
@@ -113,4 +267,4 @@ function compute_tableau(
|
||||
return Tableau(obj = tableau_obj, lhs = tableau_lhs, rhs = tableau_rhs, z = z)
|
||||
end
|
||||
|
||||
export get_basis, get_x, compute_tableau
|
||||
export get_basis, get_basis_gurobi, set_basis, get_x, compute_tableau
|
||||
|
||||
@@ -96,46 +96,70 @@ Base.@kwdef mutable struct AddSlackVariables <: Transform
|
||||
end
|
||||
|
||||
function forward!(t::AddSlackVariables, data::ProblemData)
|
||||
nrows, ncols = size(data.constr_lhs)
|
||||
isequality = abs.(data.constr_ub .- data.constr_lb) .< 1e-6
|
||||
eq = [i for i = 1:nrows if isequality[i]]
|
||||
ge = [i for i = 1:nrows if isfinite(data.constr_lb[i]) && !isequality[i]]
|
||||
le = [i for i = 1:nrows if isfinite(data.constr_ub[i]) && !isequality[i]]
|
||||
EQ, GE, LE = length(eq), length(ge), length(le)
|
||||
|
||||
t.M1 = [
|
||||
I spzeros(ncols, GE + LE)
|
||||
data.constr_lhs[ge, :] spzeros(GE, GE + LE)
|
||||
-data.constr_lhs[le, :] spzeros(LE, GE + LE)
|
||||
]
|
||||
t.M2 = [
|
||||
zeros(ncols)
|
||||
data.constr_lb[ge]
|
||||
-data.constr_ub[le]
|
||||
]
|
||||
t.ncols_orig = ncols
|
||||
t.GE, t.LE = GE, LE
|
||||
t.lhs_ge = data.constr_lhs[ge, :]
|
||||
t.lhs_le = data.constr_lhs[le, :]
|
||||
t.rhs_ge = data.constr_lb[ge]
|
||||
t.rhs_le = data.constr_ub[le]
|
||||
|
||||
data.constr_lhs = [
|
||||
data.constr_lhs[eq, :] spzeros(EQ, GE) spzeros(EQ, LE)
|
||||
data.constr_lhs[ge, :] -I spzeros(GE, LE)
|
||||
data.constr_lhs[le, :] spzeros(LE, GE) I
|
||||
]
|
||||
data.obj = [data.obj; zeros(GE + LE)]
|
||||
data.var_lb = [data.var_lb; zeros(GE + LE)]
|
||||
data.var_ub = [data.var_ub; [Inf for _ = 1:(GE+LE)]]
|
||||
data.var_names = [data.var_names; ["__s$i" for i = 1:(GE+LE)]]
|
||||
data.var_types = [data.var_types; ['C' for _ = 1:(GE+LE)]]
|
||||
data.constr_lb = [
|
||||
data.constr_lb[eq]
|
||||
data.constr_lb[ge]
|
||||
data.constr_ub[le]
|
||||
]
|
||||
data.constr_ub = copy(data.constr_lb)
|
||||
@timeit "Identify constraint type" begin
|
||||
nrows, ncols = size(data.constr_lhs)
|
||||
isequality = abs.(data.constr_ub .- data.constr_lb) .< 1e-6
|
||||
eq = [i for i = 1:nrows if isequality[i]]
|
||||
ge = [i for i = 1:nrows if isfinite(data.constr_lb[i]) && !isequality[i]]
|
||||
le = [i for i = 1:nrows if isfinite(data.constr_ub[i]) && !isequality[i]]
|
||||
EQ, GE, LE = length(eq), length(ge), length(le)
|
||||
end
|
||||
@timeit "Identify slack type" begin
|
||||
constr_lhs_t = sparse(data.constr_lhs')
|
||||
function is_integral(row_idx, rhs)
|
||||
rhs_is_integer = abs(rhs - round(rhs)) <= 1e-6
|
||||
cols, coeffs = findnz(constr_lhs_t[:, row_idx])[1:2]
|
||||
vars_are_integer = all(j -> data.var_types[j] ∈ ['I', 'B'], cols)
|
||||
coeffs_are_integer = all(v -> abs(v - round(v)) <= 1e-6, coeffs)
|
||||
return rhs_is_integer && vars_are_integer && coeffs_are_integer
|
||||
end
|
||||
slack_types = [
|
||||
[is_integral(ge[i], data.constr_lb[ge[i]]) ? 'I' : 'C' for i = 1:GE];
|
||||
[is_integral(le[i], data.constr_ub[le[i]]) ? 'I' : 'C' for i = 1:LE]
|
||||
]
|
||||
end
|
||||
@timeit "Build M1" begin
|
||||
t.M1 = [
|
||||
I spzeros(ncols, GE + LE)
|
||||
data.constr_lhs[ge, :] spzeros(GE, GE + LE)
|
||||
-data.constr_lhs[le, :] spzeros(LE, GE + LE)
|
||||
]
|
||||
end
|
||||
@timeit "Build M2" begin
|
||||
t.M2 = [
|
||||
zeros(ncols)
|
||||
data.constr_lb[ge]
|
||||
-data.constr_ub[le]
|
||||
]
|
||||
end
|
||||
@timeit "Build t.lhs, t.rhs" begin
|
||||
t.ncols_orig = ncols
|
||||
t.GE, t.LE = GE, LE
|
||||
t.lhs_ge = data.constr_lhs[ge, :]
|
||||
t.lhs_le = data.constr_lhs[le, :]
|
||||
t.rhs_ge = data.constr_lb[ge]
|
||||
t.rhs_le = data.constr_ub[le]
|
||||
end
|
||||
@timeit "Build data.constr_lhs" begin
|
||||
data.constr_lhs = [
|
||||
data.constr_lhs[eq, :] spzeros(EQ, GE) spzeros(EQ, LE)
|
||||
data.constr_lhs[ge, :] -I spzeros(GE, LE)
|
||||
data.constr_lhs[le, :] spzeros(LE, GE) I
|
||||
]
|
||||
end
|
||||
@timeit "Build other data fields" begin
|
||||
data.obj = [data.obj; zeros(GE + LE)]
|
||||
data.var_lb = [data.var_lb; zeros(GE + LE)]
|
||||
data.var_ub = [data.var_ub; [Inf for _ = 1:(GE+LE)]]
|
||||
data.var_names = [data.var_names; ["__s$i" for i = 1:(GE+LE)]]
|
||||
data.var_types = [data.var_types; slack_types]
|
||||
data.constr_lb = [
|
||||
data.constr_lb[eq]
|
||||
data.constr_lb[ge]
|
||||
data.constr_ub[le]
|
||||
]
|
||||
data.constr_ub = copy(data.constr_lb)
|
||||
end
|
||||
end
|
||||
|
||||
function backwards!(t::AddSlackVariables, c::ConstraintSet)
|
||||
@@ -155,71 +179,55 @@ end
|
||||
# -----------------------------------------------------------------------------
|
||||
|
||||
Base.@kwdef mutable struct SplitFreeVars <: Transform
|
||||
F::Int = 0
|
||||
B::Int = 0
|
||||
free::Vector{Int} = []
|
||||
others::Vector{Int} = []
|
||||
ncols::Int = 0
|
||||
is_var_free::Vector{Bool} = []
|
||||
end
|
||||
|
||||
function forward!(t::SplitFreeVars, data::ProblemData)
|
||||
lhs = data.constr_lhs
|
||||
_, ncols = size(lhs)
|
||||
free = [i for i = 1:ncols if !isfinite(data.var_lb[i]) && !isfinite(data.var_ub[i])]
|
||||
others = [i for i = 1:ncols if isfinite(data.var_lb[i]) || isfinite(data.var_ub[i])]
|
||||
t.F = length(free)
|
||||
t.B = length(others)
|
||||
t.free, t.others = free, others
|
||||
is_var_free = [!isfinite(data.var_lb[i]) && !isfinite(data.var_ub[i]) for i = 1:ncols]
|
||||
free_idx = findall(is_var_free)
|
||||
data.obj = [
|
||||
data.obj[others]
|
||||
data.obj[free]
|
||||
-data.obj[free]
|
||||
data.obj
|
||||
[-data.obj[i] for i in free_idx]
|
||||
]
|
||||
data.constr_lhs = [lhs[:, others] lhs[:, free] -lhs[:, free]]
|
||||
data.var_lb = [
|
||||
data.var_lb[others]
|
||||
[0.0 for _ in free]
|
||||
[0.0 for _ in free]
|
||||
[is_var_free[i] ? 0.0 : data.var_lb[i] for i in 1:ncols]
|
||||
[0 for _ in free_idx]
|
||||
]
|
||||
data.var_ub = [
|
||||
data.var_ub[others]
|
||||
[Inf for _ in free]
|
||||
[Inf for _ in free]
|
||||
[is_var_free[i] ? Inf : data.var_ub[i] for i in 1:ncols]
|
||||
[Inf for _ in free_idx]
|
||||
]
|
||||
data.var_types = [
|
||||
data.var_types[others]
|
||||
data.var_types[free]
|
||||
data.var_types[free]
|
||||
data.var_types
|
||||
[data.var_types[i] for i in free_idx]
|
||||
]
|
||||
data.var_names = [
|
||||
data.var_names[others]
|
||||
["$(v)_p" for v in data.var_names[free]]
|
||||
["$(v)_m" for v in data.var_names[free]]
|
||||
data.var_names
|
||||
["$(data.var_names[i])_neg" for i in free_idx]
|
||||
]
|
||||
data.constr_lhs = [lhs -lhs[:, free_idx]]
|
||||
t.is_var_free, t.ncols = is_var_free, ncols
|
||||
end
|
||||
|
||||
function backwards!(t::SplitFreeVars, c::ConstraintSet)
|
||||
# Convert GE constraints into LE
|
||||
nrows, _ = size(c.lhs)
|
||||
ge = [i for i = 1:nrows if isfinite(c.lb[i])]
|
||||
c.ub[ge], c.lb[ge] = -c.lb[ge], -c.ub[ge]
|
||||
c.lhs[ge, :] *= -1
|
||||
ncols, is_var_free = t.ncols, t.is_var_free
|
||||
free_idx = findall(is_var_free)
|
||||
|
||||
# Assert only LE constraints are left (EQ constraints are not supported)
|
||||
@assert all(c.lb .== -Inf)
|
||||
|
||||
# Take minimum (weakest) coefficient
|
||||
B, F = t.B, t.F
|
||||
for i = 1:F
|
||||
c.lhs[:, B+i] = min.(c.lhs[:, B+i], -c.lhs[:, B+F+i])
|
||||
for (offset, var_idx) in enumerate(free_idx)
|
||||
@assert c.lhs[:, var_idx] == -c.lhs[:, ncols+offset]
|
||||
end
|
||||
c.lhs = c.lhs[:, 1:(B+F)]
|
||||
c.lhs = c.lhs[:, 1:ncols]
|
||||
end
|
||||
|
||||
function forward(t::SplitFreeVars, p::Vector{Float64})::Vector{Float64}
|
||||
ncols, is_var_free = t.ncols, t.is_var_free
|
||||
free_idx = findall(is_var_free)
|
||||
return [
|
||||
p[t.others]
|
||||
max.(p[t.free], 0)
|
||||
max.(-p[t.free], 0)
|
||||
[is_var_free[i] ? max(0, p[i]) : p[i] for i in 1:ncols]
|
||||
[max(0, -p[i]) for i in free_idx]
|
||||
]
|
||||
end
|
||||
|
||||
|
||||
@@ -13,6 +13,7 @@ include("collectors.jl")
|
||||
include("components.jl")
|
||||
include("extractors.jl")
|
||||
include("io.jl")
|
||||
include("problems/maxcut.jl")
|
||||
include("problems/setcover.jl")
|
||||
include("problems/stab.jl")
|
||||
include("problems/tsp.jl")
|
||||
@@ -24,6 +25,7 @@ function __init__()
|
||||
__init_components__()
|
||||
__init_extractors__()
|
||||
__init_io__()
|
||||
__init_problems_maxcut__()
|
||||
__init_problems_setcover__()
|
||||
__init_problems_stab__()
|
||||
__init_problems_tsp__()
|
||||
@@ -37,48 +39,48 @@ include("Cuts/Cuts.jl")
|
||||
# Precompilation
|
||||
# =============================================================================
|
||||
|
||||
function __precompile_cuts__()
|
||||
function build_model(mps_filename)
|
||||
model = read_from_file(mps_filename)
|
||||
set_optimizer(model, SCIP.Optimizer)
|
||||
return JumpModel(model)
|
||||
end
|
||||
BASEDIR = dirname(@__FILE__)
|
||||
mps_filename = "$BASEDIR/../test/fixtures/bell5.mps.gz"
|
||||
h5_filename = "$BASEDIR/../test/fixtures/bell5.h5"
|
||||
collect_gmi_dual(
|
||||
mps_filename;
|
||||
optimizer=HiGHS.Optimizer,
|
||||
max_rounds = 10,
|
||||
max_cuts_per_round = 500,
|
||||
)
|
||||
knn = KnnDualGmiComponent(
|
||||
extractor = H5FieldsExtractor(instance_fields = ["static_var_obj_coeffs"]),
|
||||
k = 2,
|
||||
)
|
||||
knn.fit([h5_filename, h5_filename])
|
||||
solver = LearningSolver(
|
||||
components = [
|
||||
ExpertPrimalComponent(action = SetWarmStart()),
|
||||
knn,
|
||||
],
|
||||
skip_lp = true,
|
||||
)
|
||||
solver.optimize(mps_filename, build_model)
|
||||
end
|
||||
# function __precompile_cuts__()
|
||||
# function build_model(mps_filename)
|
||||
# model = read_from_file(mps_filename)
|
||||
# set_optimizer(model, SCIP.Optimizer)
|
||||
# return JumpModel(model)
|
||||
# end
|
||||
# BASEDIR = dirname(@__FILE__)
|
||||
# mps_filename = "$BASEDIR/../test/fixtures/bell5.mps.gz"
|
||||
# h5_filename = "$BASEDIR/../test/fixtures/bell5.h5"
|
||||
# collect_gmi_dual(
|
||||
# mps_filename;
|
||||
# optimizer=HiGHS.Optimizer,
|
||||
# max_rounds = 10,
|
||||
# max_cuts_per_round = 500,
|
||||
# )
|
||||
# knn = KnnDualGmiComponent(
|
||||
# extractor = H5FieldsExtractor(instance_fields = ["static_var_obj_coeffs"]),
|
||||
# k = 2,
|
||||
# )
|
||||
# knn.fit([h5_filename, h5_filename])
|
||||
# solver = LearningSolver(
|
||||
# components = [
|
||||
# ExpertPrimalComponent(action = SetWarmStart()),
|
||||
# knn,
|
||||
# ],
|
||||
# skip_lp = true,
|
||||
# )
|
||||
# solver.optimize(mps_filename, build_model)
|
||||
# end
|
||||
|
||||
@setup_workload begin
|
||||
using SCIP
|
||||
using HiGHS
|
||||
using MIPLearn.Cuts
|
||||
using PrecompileTools: @setup_workload, @compile_workload
|
||||
# @setup_workload begin
|
||||
# using SCIP
|
||||
# using HiGHS
|
||||
# using MIPLearn.Cuts
|
||||
# using PrecompileTools: @setup_workload, @compile_workload
|
||||
|
||||
__init__()
|
||||
Cuts.__init__()
|
||||
# __init__()
|
||||
# Cuts.__init__()
|
||||
|
||||
@compile_workload begin
|
||||
__precompile_cuts__()
|
||||
end
|
||||
end
|
||||
# @compile_workload begin
|
||||
# __precompile_cuts__()
|
||||
# end
|
||||
# end
|
||||
|
||||
end # module
|
||||
|
||||
31
src/problems/maxcut.jl
Normal file
31
src/problems/maxcut.jl
Normal file
@@ -0,0 +1,31 @@
|
||||
# MIPLearn: Extensible Framework for Learning-Enhanced Mixed-Integer Optimization
|
||||
# Copyright (C) 2020-2025, UChicago Argonne, LLC. All rights reserved.
|
||||
# Released under the modified BSD license. See COPYING.md for more details.
|
||||
|
||||
using JuMP
|
||||
|
||||
global MaxCutData = PyNULL()
|
||||
global MaxCutGenerator = PyNULL()
|
||||
|
||||
function __init_problems_maxcut__()
|
||||
copy!(MaxCutData, pyimport("miplearn.problems.maxcut").MaxCutData)
|
||||
copy!(MaxCutGenerator, pyimport("miplearn.problems.maxcut").MaxCutGenerator)
|
||||
end
|
||||
|
||||
function build_maxcut_model_jump(data::Any; optimizer)
|
||||
if data isa String
|
||||
data = read_pkl_gz(data)
|
||||
end
|
||||
nodes = collect(data.graph.nodes())
|
||||
edges = collect(data.graph.edges())
|
||||
model = Model(optimizer)
|
||||
@variable(model, x[nodes], Bin)
|
||||
@objective(
|
||||
model,
|
||||
Min,
|
||||
sum(-data.weights[i] * x[e[1]] * (1 - x[e[2]]) for (i, e) in enumerate(edges))
|
||||
)
|
||||
return JumpModel(model)
|
||||
end
|
||||
|
||||
export MaxCutData, MaxCutGenerator, build_maxcut_model_jump
|
||||
@@ -89,14 +89,27 @@ function _extract_after_load_vars(model::JuMP.Model, h5)
|
||||
for v in vars
|
||||
]
|
||||
types = [JuMP.is_binary(v) ? "B" : JuMP.is_integer(v) ? "I" : "C" for v in vars]
|
||||
obj = objective_function(model, AffExpr)
|
||||
obj_coeffs = [v ∈ keys(obj.terms) ? obj.terms[v] : 0.0 for v in vars]
|
||||
|
||||
# Linear obj terms
|
||||
obj = objective_function(model, QuadExpr)
|
||||
obj_coeffs_linear = [v ∈ keys(obj.aff.terms) ? obj.aff.terms[v] : 0.0 for v in vars]
|
||||
|
||||
# Quadratic obj terms
|
||||
if length(obj.terms) > 0
|
||||
nvars = length(vars)
|
||||
obj_coeffs_quad = zeros(nvars, nvars)
|
||||
for (pair, coeff) in obj.terms
|
||||
obj_coeffs_quad[pair.a.index.value, pair.b.index.value] = coeff
|
||||
end
|
||||
h5.put_array("static_var_obj_coeffs_quad", obj_coeffs_quad)
|
||||
end
|
||||
|
||||
h5.put_array("static_var_names", to_str_array(JuMP.name.(vars)))
|
||||
h5.put_array("static_var_types", to_str_array(types))
|
||||
h5.put_array("static_var_lower_bounds", lb)
|
||||
h5.put_array("static_var_upper_bounds", ub)
|
||||
h5.put_array("static_var_obj_coeffs", obj_coeffs)
|
||||
h5.put_scalar("static_obj_offset", obj.constant)
|
||||
h5.put_array("static_var_obj_coeffs", obj_coeffs_linear)
|
||||
h5.put_scalar("static_obj_offset", obj.aff.constant)
|
||||
end
|
||||
|
||||
function _extract_after_load_constrs(model::JuMP.Model, h5)
|
||||
@@ -143,7 +156,7 @@ function _extract_after_load_constrs(model::JuMP.Model, h5)
|
||||
end
|
||||
end
|
||||
if isempty(names)
|
||||
error("no model constraints found; note that MIPLearn ignores unnamed constraints")
|
||||
return
|
||||
end
|
||||
lhs = sparse(lhs_rows, lhs_cols, lhs_values, length(rhs), JuMP.num_variables(model))
|
||||
h5.put_sparse("static_constr_lhs", lhs)
|
||||
@@ -282,9 +295,11 @@ function _extract_after_mip(model::JuMP.Model, h5)
|
||||
|
||||
# Slacks
|
||||
lhs = h5.get_sparse("static_constr_lhs")
|
||||
rhs = h5.get_array("static_constr_rhs")
|
||||
slacks = abs.(lhs * x - rhs)
|
||||
h5.put_array("mip_constr_slacks", slacks)
|
||||
if lhs !== nothing
|
||||
rhs = h5.get_array("static_constr_rhs")
|
||||
slacks = abs.(lhs * x - rhs)
|
||||
h5.put_array("mip_constr_slacks", slacks)
|
||||
end
|
||||
|
||||
# Cuts and lazy constraints
|
||||
ext = model.ext[:miplearn]
|
||||
|
||||
@@ -24,6 +24,7 @@ include("Cuts/tableau/test_gmi_dual.jl")
|
||||
include("problems/test_setcover.jl")
|
||||
include("problems/test_stab.jl")
|
||||
include("problems/test_tsp.jl")
|
||||
include("problems/test_maxcut.jl")
|
||||
include("solvers/test_jump.jl")
|
||||
include("test_io.jl")
|
||||
include("test_usage.jl")
|
||||
@@ -37,6 +38,7 @@ function runtests()
|
||||
test_problems_setcover()
|
||||
test_problems_stab()
|
||||
test_problems_tsp()
|
||||
test_problems_maxcut()
|
||||
test_solvers_jump()
|
||||
test_usage()
|
||||
test_cuts()
|
||||
|
||||
54
test/src/problems/test_maxcut.jl
Normal file
54
test/src/problems/test_maxcut.jl
Normal file
@@ -0,0 +1,54 @@
|
||||
# MIPLearn: Extensible Framework for Learning-Enhanced Mixed-Integer Optimization
|
||||
# Copyright (C) 2020-2025, UChicago Argonne, LLC. All rights reserved.
|
||||
# Released under the modified BSD license. See COPYING.md for more details.
|
||||
|
||||
using PyCall
|
||||
|
||||
function test_problems_maxcut()
|
||||
np = pyimport("numpy")
|
||||
random = pyimport("random")
|
||||
scipy_stats = pyimport("scipy.stats")
|
||||
randint = scipy_stats.randint
|
||||
uniform = scipy_stats.uniform
|
||||
|
||||
# Set random seed
|
||||
random.seed(42)
|
||||
np.random.seed(42)
|
||||
|
||||
# Build random instance
|
||||
data = MaxCutGenerator(
|
||||
n = randint(low = 10, high = 11),
|
||||
p = uniform(loc = 0.5, scale = 0.0),
|
||||
fix_graph = false,
|
||||
).generate(
|
||||
1,
|
||||
)[1]
|
||||
|
||||
# Build model
|
||||
model = build_maxcut_model_jump(data, optimizer = SCIP.Optimizer)
|
||||
|
||||
# Check static features
|
||||
h5 = H5File(tempname(), "w")
|
||||
model.extract_after_load(h5)
|
||||
obj_linear = h5.get_array("static_var_obj_coeffs")
|
||||
obj_quad = h5.get_array("static_var_obj_coeffs_quad")
|
||||
@test obj_linear == [3.0, 1.0, 3.0, 1.0, -1.0, 0.0, -1.0, 0.0, -1.0, 0.0]
|
||||
@test obj_quad == [
|
||||
0.0 0.0 -1.0 1.0 -1.0 0.0 0.0 0.0 -1.0 -1.0
|
||||
0.0 0.0 1.0 -1.0 0.0 -1.0 -1.0 0.0 0.0 1.0
|
||||
0.0 0.0 0.0 0.0 0.0 -1.0 0.0 0.0 -1.0 -1.0
|
||||
0.0 0.0 0.0 0.0 0.0 -1.0 1.0 -1.0 0.0 0.0
|
||||
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 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 1.0
|
||||
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 -1.0
|
||||
0.0 0.0 0.0 0.0 0.0 0.0 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
|
||||
]
|
||||
|
||||
# Check optimal solution
|
||||
model.optimize()
|
||||
model.extract_after_mip(h5)
|
||||
@test h5.get_scalar("mip_obj_value") == -4
|
||||
h5.close()
|
||||
end
|
||||
Reference in New Issue
Block a user