Compartmental OED
Estimate three parameters by observing one state. We optimize a relaxed selection of six measurements from 19 candidate times, following the compartmental OED benchmark. The nominal vector is p = [θ₁, θ₂, θ₃]. All three parameters are estimated but fixed as design inputs; only measurement weights are optimized. We use no physical control or shooting resets, hence no shooting matching equalities.
ComponentArrays supplies packing; Optimization/OptimizationMOI and Ipopt solve the continuous relaxation; CairoMakie plots event weights. A forward OED solve itself needs none of those optional optimization/plotting packages.
using CorleoneOED, LuxCore, Random, OrdinaryDiffEqTsit5, LinearAlgebra
using Optimization, OptimizationMOI, Ipopt, ComponentArrays, CairoMakie
using SymbolicIndexingInterface: parameter_values_at_time, parameter_index
function compartmental(du, u, p, t)
θ1, θ2, θ3 = p
du[1] = θ3 * (-θ1 * exp(-θ1 * t) + θ2 * exp(-θ2 * t))
return nothing
end
problem = ODEProblem(compartmental, [0.0], (0.0, 50.0), [0.05884, 4.298, 21.8])
measurement_points = [
0.0, 0.166, 0.333, 0.5, 0.666, 1.0, 1.5, 2.0, 2.5, 3.0,
4.0, 5.0, 6.0, 8.0, 10.0, 12.0, 24.0, 30.0, 48.0,
]
oed = OEDLayer(
problem, Tsit5(); params = [1, 2, 3], measurements = (
DiscreteMeasurement(:sample, measurement_points, (x, p, t) -> x[1]),
)
)
ps, st = LuxCore.setup(Xoshiro(1), oed);Solve the relaxed design
Physical parameters remain fixed. Each candidate time has a weight in [0,1]; the total weight is bounded by six. Fractional weights are allowed in this solve. Unlike a continuous density, this budget counts event weights rather than time. The first point at 0.0 is supported even though this model's state sensitivity is zero there; endpoint measurements are not implicitly removed. DCriterion returns 1/det(F), without a logarithm or dimension normalization.
optprob = OptimizationProblem(oed, DCriterion(); ps, st, M = [6.0])
result = solve(
optprob, Ipopt.Optimizer(); tol = 1.0e-8,
hessian_approximation = "limited-memory", max_iter = 100, print_level = 0
);
@assert CorleoneOED.SciMLBase.successful_retcode(result)
@assert isfinite(result.objective)
_, reconstruct = pack(default_vectorizer(), ps)
optimal_ps = reconstruct(result.u)
trajectory, _ = oed(nothing, optimal_ps, st)
F = fisher_information(oed, trajectory)
optimal_budget = only(sampling_sums(oed, trajectory))
@assert isposdef(Symmetric(F))
@assert optimal_budget <= 6.0 + 1.0e-5
@assert isapprox(result.objective, DCriterion()(F); rtol = 1.0e-5)
(; termination = result.retcode, objective = result.objective, budget = optimal_budget)(termination = SciMLBase.ReturnCode.Success, objective = 3.73296717652639e-6, budget = 5.999995943190418)Evaluate weights from the current parameter schedule at actual event times. The saved augmented states contain sensitivities/information, not weight histories.
weights = [
parameter_values_at_time(trajectory, trajectory, t)[parameter_index(trajectory, :sample)]
for t in measurement_points
]
figure = Figure()
axis = CairoMakie.Axis(figure[1, 1]; xlabel = "Time", ylabel = "Measurement weight")
scatter!(axis, measurement_points, weights)
figure
Optimized relaxed weights at the 19 candidate times. A point near one is a heavily weighted event; fractional values are not rounded here. Successful local solver termination and feasible budgets do not prove a globally optimal integer design.
Integer sampling decisions
To require binary selections, construct the problem below and choose an integer-capable optimizer. Ipopt solves the continuous relaxation only.
integer_problem = OptimizationProblem(oed, DCriterion(); ps, st, M = [6.0], integer_weights = true);
@assert sum(integer_problem.int) == length(measurement_points)This creates a binary-weight problem description; it does not solve it. Passing the integer mask to a continuous solver does not enforce binary choices.
Appendix
This example was built using these direct dependencies,
using Pkg
Pkg.status()Status `~/work/Corleone.jl/Corleone.jl/docs/Project.toml`
[13f3f980] CairoMakie v0.15.17
[b0b7db55] ComponentArrays v0.15.50
[2751e0db] Corleone v2.0.0 `~/work/Corleone.jl/Corleone.jl`
[6590a4b1] CorleoneOED v2.0.0 `../lib/CorleoneOED`
[e30172f5] Documenter v1.19.0
[daee34ce] DocumenterCitations v1.5.0
[f6369f11] ForwardDiff v1.4.6
[b6b21f68] Ipopt v1.16.0
[98b081ad] Literate v2.21.0
[bb33d45b] LuxCore v1.5.4
[961ee093] ModelingToolkit v11.46.1
[d147904a] OptimalControlBenchmarks v0.1.0 `../lib/OptimalControlBenchmarks`
[7f7a1694] Optimization v5.9.1
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[fd9f6733] OptimizationMOI v1.6.1
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[ddb6d928] YAML v0.4.17
[2a0f44e3] Base64 v1.11.0
[9a3f8284] Random v1.11.0
and using this machine and Julia version.
using InteractiveUtils
InteractiveUtils.versioninfo()Julia Version 1.13.3
Commit 67ac47b9cd8 (2026-10-25 15:28 UTC)
Build Info:
Official https://julialang.org release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 4 × AMD EPYC 7763 64-Core Processor
WORD_SIZE: 64
LLVM: libLLVM-20.1.8 (ORCJIT, znver3)
GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 4 virtual cores)
A more complete overview of all dependencies and their versions is also provided.
using Pkg
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[30392449] Pixman_jll v0.46.4+0
[f50d1b31] Rmath_jll v0.5.2+0
[319450e9] SPRAL_jll v2025.9.18+1
[02c8fc9c] XML2_jll v2.15.4+0
[ffd25f8a] XZ_jll v5.8.4+0
[4f6342f7] Xorg_libX11_jll v1.8.14+0
[0c0b7dd1] Xorg_libXau_jll v1.0.13+0
[a3789734] Xorg_libXdmcp_jll v1.1.6+0
[1082639a] Xorg_libXext_jll v1.3.8+0
[d091e8ba] Xorg_libXfixes_jll v6.0.2+0
[ea2f1a96] Xorg_libXrender_jll v0.9.13+0
[a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
[c7cfdc94] Xorg_libxcb_jll v1.17.1+0
[c5fb5394] Xorg_xtrans_jll v1.6.0+0
[9a68df92] isoband_jll v0.2.3+0
[a4ae2306] libaom_jll v3.15.3+0
[0ac62f75] libass_jll v0.17.5+0
[46979653] libdeflate_jll v1.26.0+1
[8e53e030] libdrm_jll v2.4.134+0
⌅ [f638f0a6] libfdk_aac_jll v2.0.4+0
[b53b4c65] libpng_jll v1.6.59+1
[075b6546] libsixel_jll v1.80.702+0
[9a156e7d] libva_jll v2.24.1+0
[f27f6e37] libvorbis_jll v1.3.8+0
[c5f90fcd] libwebp_jll v1.6.1+0
[1317d2d5] oneTBB_jll v2023.1.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
⌅ [dfaa095f] x265_jll v4.1.0+0
[0dad84c5] ArgTools v1.1.2
[56f22d72] Artifacts v1.11.0
[2a0f44e3] Base64 v1.11.0
[8bf52ea8] CRC32c v1.11.0
[ade2ca70] Dates v1.11.0
[8ba89e20] Distributed v1.11.0
[f43a241f] Downloads v1.7.0
[7b1f6079] FileWatching v1.11.0
[9fa8497b] Future v1.11.0
[b77e0a4c] InteractiveUtils v1.11.0
[ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
[4af54fe1] LazyArtifacts v1.11.0
[b27032c2] LibCURL v1.0.0
[76f85450] LibGit2 v1.11.0
[8f399da3] Libdl v1.11.0
[37e2e46d] LinearAlgebra v1.13.0
[56ddb016] Logging v1.11.0
[d6f4376e] Markdown v1.11.0
[a63ad114] Mmap v1.11.0
[ca575930] NetworkOptions v1.3.0
[44cfe95a] Pkg v1.13.0
[de0858da] Printf v1.11.0
[3fa0cd96] REPL v1.11.0
[9a3f8284] Random v1.11.0
[ea8e919c] SHA v1.0.0
[9e88b42a] Serialization v1.11.0
[1a1011a3] SharedArrays v1.11.0
[6462fe0b] Sockets v1.11.0
[2f01184e] SparseArrays v1.13.0
[f489334b] StyledStrings v1.11.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test v1.11.0
[cf7118a7] UUIDs v1.11.0
[4ec0a83e] Unicode v1.11.0
[e66e0078] CompilerSupportLibraries_jll v1.5.5+2
[deac9b47] LibCURL_jll v8.22.0+0
[e37daf67] LibGit2_jll v1.9.7+0
[29816b5a] LibSSH2_jll v1.11.104+0
[14a3606d] MozillaCACerts_jll v2026.10.20
[4536629a] OpenBLAS_jll v0.3.30+0
[05823500] OpenLibm_jll v0.8.7+0
[458c3c95] OpenSSL_jll v3.5.8+0
[efcefdf7] PCRE2_jll v10.47.0+0
[bea87d4a] SuiteSparse_jll v7.10.1+0
[83775a58] Zlib_jll v1.3.2+0
[3161d3a3] Zstd_jll v1.5.7+1
[8e850b90] libblastrampoline_jll v5.15.0+0
[8e850ede] nghttp2_jll v1.70.0+0
[3f19e933] p7zip_jll v17.8.3+0
Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`
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