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test_linear_programs.jl
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using Zygote
using DifferentiationInterface: AutoZygote
using ProximalOperators: Linear, IndNonnegative, IndPoint, IndAffine, SlicedSeparableSum
using ProximalAlgorithms
using LinearAlgebra
function assert_lp_solution(c, A, b, x, y, tol)
# Check and print solution quality measures (for some reason the
# returned dual iterate is the negative of the dual LP variable y)
nonneg = -minimum(min.(0.0, x))
@test nonneg <= tol
primal_feasibility = norm(A * x - b)
@test primal_feasibility <= tol
dual_feasibility = maximum(max.(0.0, -A' * y - c))
@test dual_feasibility <= tol
complementarity = abs(dot(c + A' * y, x))
@test complementarity <= tol
end
@testset "Linear programs ($T)" for T in [Float32, Float64]
# Solving LP with AFBA
#
# minimize c'x -> f = <c,.>
# subject to Ax = b -> h = ind{b}, L = A
# x >= 0 -> g = ind{>=0}
#
# Dual LP
#
# maximize b'y
# subject to A'y <= c
#
# Optimality conditions
#
# x >= 0 [primal feasibility 1]
# Ax = b [primal feasibility 2]
# A'y <= c [dual feasibility]
# X'(c - A'y) = 0 [complementarity slackness]
#
n = 10 # primal dimension
m = 8 # dual dimension (i.e. number of linear equalities)
k = 5 # number of active dual constraints (must be 0 <= k <= n)
# primal optimal point
x_star = T[
0.03606099647643202,
0.6641306619990367,
0.14689326835593086,
0.8616058527226432,
0.6518888386753204,
0.0,
0.0,
0.0,
0.0,
0.0,
]
# dual optimal slack variable
s_star = T[
0.0,
0.0,
0.0,
0.0,
0.0,
0.7652088547602296,
0.5474850582278736,
0.6291635905927829,
0.5505791259743739,
0.2787959059899776,
]
# dual optimal point
y_star = T[
-0.18584225497010673,
1.0289179383971105,
-0.5464276767930338,
0.6916826379378143,
0.0052105118567898154,
-1.3874339670318891,
2.6448648322602337,
0.4436510755789434,
]
# equality constraints coefficients
A = T[
0.9670566934980286 0.3533472582831891 -0.2819205403663176 -0.3071751292615649 -0.008197151577028876 1.9968628487924958 -1.3978028828102382 -0.10109443461339453 0.3540170332321368 -0.5983697948494805
-0.08643002247762153 1.0822796020885652 -0.8133412010128359 0.5073579509595517 -0.2793798574957181 -0.3437059320143032 -0.10596751644319548 0.8296541102523696 -0.28406001552384064 0.33566140783820164
-0.6474810505335304 1.2524424221527595 0.7261766049639965 0.6930548839308885 0.9064585656428545 0.5197152355759463 1.3746077918877961 1.5289983684262054 0.9258506062644877 -1.3774011601531342
0.8725430405646543 1.1911309824177332 -0.7221185305116811 0.44324697589700257 -0.15551892138880116 -0.5167033349322372 -1.4774045165687548 -0.8670756465083638 -1.4024841888738206 0.296570674868075
0.5252662593900623 -0.6817132148621747 0.37446015899900237 0.4180282147408251 -0.8405436435394317 -1.742941478155391 -0.06419427972299957 0.43910205564784205 1.0643883425210827 2.3063869854427335
-0.7148913270640012 -0.769028546306448 -1.059257097999333 -0.6795170119545777 -0.0498976886779524 -1.2392107698826862 -0.4415384005606088 -1.058758868936871 -0.108504245219676 -0.5576550366602419
-1.208388774142618 -0.15206129387542855 2.311520055340236 0.8043266793420988 -0.5692874893454578 1.1246423711381501 0.5335942753441769 2.6595405998250876 0.09162292399585106 0.3749905036072034
-0.2300660921924555 -0.7014271654627467 -0.20170532145095504 -0.02503019691724233 -2.2191605023268512 0.9110653907470295 1.8238644805628141 -1.1908921287611471 0.12168786553115268 0.17399181994853638
]
b = A * x_star
c = A' * y_star + s_star
tol = 100 * eps(T)
maxit = 100_000
@testset "AFBA" begin
f = ProximalAlgorithms.AutoDifferentiable(x -> dot(c, x), AutoZygote())
g = IndNonnegative()
h = IndPoint(b)
x0 = zeros(T, n)
x0_backup = copy(x0)
y0 = zeros(T, m)
y0_backup = copy(y0)
solver = ProximalAlgorithms.AFBA(tol = tol, maxit = maxit)
(x, y), it = solver(x0 = x0, y0 = y0, f = f, g = g, h = h, L = A, beta_f = 0)
@test eltype(x) == T
@test eltype(y) == T
@test it <= maxit
assert_lp_solution(c, A, b, x, y, 1000 * tol)
@test x0 == x0_backup
@test y0 == y0_backup
end
@testset "VuCondat" begin
f = ProximalAlgorithms.AutoDifferentiable(x -> dot(c, x), AutoZygote())
g = IndNonnegative()
h = IndPoint(b)
x0 = zeros(T, n)
x0_backup = copy(x0)
y0 = zeros(T, m)
y0_backup = copy(y0)
solver = ProximalAlgorithms.VuCondat(tol = tol, maxit = maxit)
(x, y), it = solver(x0 = x0, y0 = y0, f = f, g = g, h = h, L = A, beta_f = 0)
@test eltype(x) == T
@test eltype(y) == T
@test it <= maxit
assert_lp_solution(c, A, b, x, y, 1000 * tol)
@test x0 == x0_backup
@test y0 == y0_backup
end
@testset "ChambollePock" begin
g = Linear(c)
h = SlicedSeparableSum((IndPoint(b), IndNonnegative()), ((1:m,), (m+1:m+n,)))
x0 = zeros(T, n)
x0_backup = copy(x0)
y0 = zeros(T, m + n)
y0_backup = copy(y0)
solver = ProximalAlgorithms.ChambollePock(tol = tol, maxit = maxit)
(x, y), it = solver(x0 = x0, y0 = y0, g = g, h = h, L = vcat(A, Matrix{T}(I, n, n)))
@test eltype(x) == T
@test eltype(y) == T
@test it <= maxit
assert_lp_solution(c, A, b, x, y[1:m], 1000 * tol)
@test x0 == x0_backup
@test y0 == y0_backup
end
@testset "DavisYin" begin
f = ProximalAlgorithms.AutoDifferentiable(x -> dot(c, x), AutoZygote())
g = IndNonnegative()
h = IndAffine(A, b)
x0 = zeros(T, n)
x0_backup = copy(x0)
solver = ProximalAlgorithms.DavisYin(gamma = T(1), tol = tol, maxit = maxit)
xf, it = solver(x0 = x0, f = f, g = g, h = h)
@test eltype(xf) == T
@test it <= maxit
@assert norm(xf - x_star) <= 1e2 * tol
@test x0 == x0_backup
end
end