plot_sparsity (generic function with 1 method)

Difference-of-Convex Algorithm with Frank-Wolfe

This example shows the optimization of a difference-of-convex problem of the form:

\[min\limits_{x \in \mathcal{X}} \phi(x) = f(x) - g(x)\]

with $f$, $g$ convex functions with access to subgradients and $f$ smooth.

The DCA-FW algorithm constructs local convex models of $\phi$ by linearizing $g$ and approximately optimizes them with FW. It is a local method that converges to a stationary point

using FrankWolfe
using LinearAlgebra
using Random
using SparseArrays
using StableRNGs
using Random

The convex functions $f$, $g$ will be generated as random convex quadratics:

\[\begin{align} f(x) & = \frac12 x^\top A x + a^\top x + c \\ g(x) & = \frac12 x^\top B x + b^\top x + d \end{align}\]

Setting up the problem functions and data

const n = 500  # Reduced dimension

# Generate random positive definite matrices to ensure convexity
function generate_problem_data(rng)
    A_raw = randn(rng, n, n)
    A_raw ./= opnorm(A_raw)
    A = A_raw' * A_raw + 0.1 * I

    B_raw = randn(rng, n, n)
    B_raw ./= opnorm(B_raw)
    B = B_raw' * B_raw + 0.1 * I

    a = randn(rng, n)
    b = randn(rng, n)

    c = randn(rng)
    d = -359434.0

    return A, B, a, b, c, d
end

const rng = StableRNGs.StableRNG(1)
const A, B, a, b, c, d = generate_problem_data(rng)

([0.36647164615458416 0.007434201971305435 … 0.004451785375884825 0.0155144115270228; 0.007434201971305435 0.3482281394741435 … 0.01365430884095603 -0.02591048687804639; … ; 0.004451785375884825 0.01365430884095603 … 0.3806261404722968 0.01189691094814741; 0.0155144115270228 -0.02591048687804639 … 0.01189691094814741 0.3429933792454244], [0.3685823592850498 0.0036477368657639 … -0.014747118492049544 -0.00939075884232427; 0.0036477368657639 0.36832277324228313 … -0.004708423922308705 0.019319726914150823; … ; -0.014747118492049544 -0.004708423922308705 … 0.3718244447067386 -0.00705898045087301; -0.00939075884232427 0.019319726914150823 … -0.00705898045087301 0.36976970853592106], [-0.5056985849854947, 1.8254695540945816, 1.1686948924955953, 0.025865043318803068, 0.7618207158940428, -1.9599078996573032, -0.28701522778989075, -1.210555966368952, -0.3476555815275666, 0.36622748810084865  …  -1.2418975260665193, -2.042717239142079, -0.977247840514734, -0.9480533072182336, 0.2354971732415276, 1.1489207372892907, 0.26713737924876685, -0.11111819389928967, 1.2106291210589706, -0.36135542009200855], [-1.7352231531269595, -1.0889348569606767, -1.9696069675033783, -0.6944538146281424, 0.12594479576421444, 2.43890963984164, -0.3332102207771672, -1.598587279152474, 0.8970898814075347, 0.7441204116527911  …  -0.07885951683533911, 0.921491279700674, 0.8755353895807455, 0.4142785039140312, 1.0607172005644177, -0.9845905380360871, 0.7603797972500105, 1.0314248598279463, -1.0727076372535254, -0.1591576537893938], -0.6622940864130684, -359434.0)

We can now define the two functions

function f(x)
    return 0.5 * FrankWolfe.fast_dot(x, A, x) + dot(a, x) + c
end

function grad_f!(storage, x)
    mul!(storage, A, x)
    storage .+= a
    return nothing
end

function g(x)
    return 0.5 * FrankWolfe.fast_dot(x, B, x) + dot(b, x) + d
end

function grad_g!(storage, x)
    mul!(storage, B, x)
    storage .+= b
    return nothing
end

# True objective function for verification
# It is not needed by the solver
function phi(x)
    return f(x) - g(x)
end

lmo = FrankWolfe.KSparseLMO(5, 1000.0)

x0 = FrankWolfe.compute_extreme_point(lmo, randn(n))


res_dca = FrankWolfe.dca_fw(
    f,
    grad_f!,
    g,
    grad_g!,
    lmo,
    copy(x0),
    max_iteration=200, # Outer iterations
    max_inner_iteration=10000, # Inner iterations
    epsilon=1e-5, # Tolerance for DCA gap
    line_search=FrankWolfe.Secant(),
    verbose=true,
    trajectory=true,
    verbose_inner=false,
    print_iter=10,
    use_corrective_fw=true,
    use_dca_early_stopping=true,
    grad_f_workspace=collect(x0),
    grad_g_workspace=collect(x0),
)


res_boost = FrankWolfe.dca_fw(
    f,
    grad_f!,
    g,
    grad_g!,
    lmo,
    copy(x0),
    boosted=true,
    max_iteration=200, # Outer iterations
    max_inner_iteration=10000, # Inner iterations
    epsilon=1e-5, # Tolerance for DCA gap
    line_search=FrankWolfe.Secant(),
    verbose=true,
    trajectory=true,
    verbose_inner=false,
    print_iter=10,
    use_corrective_fw=true,
    use_dca_early_stopping=true,
    grad_f_workspace=collect(x0),
    grad_g_workspace=collect(x0),
)

(x =   [85 ]  =  -509.838
  [284]  =  776.723
  [313]  =  -1000.0
  [348]  =  950.205
  [369]  =  -763.234
  [486]  =  -1000.0, primal = 10296.751941625495, dca_gap = 1.1963787846034393, iterations = 200, status = FrankWolfe.STATUS_MAXITER, traj_data = Any[(1, 264614.2536032264, 204027.19099800388, 60587.0626052225, 0.000240031), (2, 234560.45679882803, 205112.5347270093, 29447.922071818728, 0.001440397), (3, 215661.18822616362, 194139.53713707873, 21521.651089084895, 0.002232757), (4, 199608.34426811023, 184451.26899180614, 15157.075276304087, 0.003017418), (5, 181666.64329622732, 167345.57446816773, 14321.068828059588, 0.003902691), (6, 164467.76997024578, 145263.59723625114, 19204.17273399464, 0.004723491), (7, 147519.88242445485, 128938.3990400656, 18581.48338438924, 0.005456781), (8, 142633.07845726394, 137054.0216567468, 5579.05680051713, 0.006404406), (9, 140842.11026702984, 138856.99094544022, 1985.1193215896155, 0.007606497), (10, 139256.548428729, 137519.6617537215, 1736.8866750075013, 0.008713996)  …  (192, 10312.104968349566, 10311.003372264575, 1.101596084990888, 0.08365252), (193, 10309.986357317073, 10308.917063072426, 1.0692942446476081, 0.083958909), (194, 10307.928818107233, 10306.836905637576, 1.0919124696577294, 0.084296296), (195, 10305.93003422697, 10304.8980989243, 1.0319353026716271, 0.084611164), (196, 10303.988017893396, 10303.005414434458, 0.9826034589386836, 0.08491607), (197, 10302.100970423548, 10301.144649333555, 0.956321089992084, 0.085224841), (198, 10300.267265954986, 10299.220612810901, 1.0466531440849522, 0.08556182), (199, 10298.483963829232, 10296.813319648936, 1.6706441802962217, 0.08585659), (200, 10296.751941625495, 10295.555562840891, 1.1963787846034393, 0.086143253), (200, 10296.751941625495, 10295.555562840891, 1.1963787846034393, 0.086648098)])

Plotting the resulting trajectory

We modify the y axis to highlight that we are plotting the DCA gap, not the FW gap

data = [res_dca.traj_data, res_boost.traj_data]
label = ["DCA-FW", "DCA-FW-B"]
p_res = plot_trajectories(data, label, marker_shapes=[:o, :x])
ylabel!(p_res.subplots[3], "DCA gap")

display(p_res)

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