2-Dimensional Rosenbrock Function Nonlinear Optimization Benchmark

This is a test of the 2-dimensional Rosenbrock function, a standard nonlinear optimization test case.

# Define the problem to solve
using Optimization, ForwardDiff, Zygote, BenchmarkTools

rosenbrock(x, p) = (p[1] - x[1])^2 + p[2] * (x[2] - x[1]^2)^2
x0 = zeros(2)
_p = [1.0, 100.0]

f = OptimizationFunction(rosenbrock, Optimization.AutoForwardDiff())
l1 = rosenbrock(x0, _p)
prob = OptimizationProblem(f, x0, _p)

OptimizationProblem. In-place: true
u0: 2-element Vector{Float64}:
 0.0
 0.0

Optim.jl Solvers

using OptimizationOptimJL

Start with some derivative-free optimizers

@btime sol = solve(prob, SimulatedAnnealing())

1.931 ms (19059 allocations: 817.86 KiB)
retcode: MaxIters
u: 2-element Vector{Float64}:
 0.8168082272836104
 0.6661817136743902

prob = OptimizationProblem(f, x0, _p, lb = [-1.0, -1.0], ub = [0.8, 0.8])
@btime sol = solve(prob, SAMIN())

1.987 ms (18396 allocations: 895.48 KiB)
retcode: MaxIters
u: 2-element Vector{Float64}:
 0.7745771163469926
 0.6028391220205219

l1 = rosenbrock(x0, _p)
prob = OptimizationProblem(rosenbrock, x0, _p)
@btime sol = solve(prob, NelderMead())

125.479 μs (1106 allocations: 53.80 KiB)
retcode: Success
u: 2-element Vector{Float64}:
 0.9999634355313174
 0.9999315506115275

Now a gradient-based optimizer with forward-mode automatic differentiation

optf = OptimizationFunction(rosenbrock, Optimization.AutoForwardDiff())
prob = OptimizationProblem(optf, x0, _p)
@btime sol = solve(prob, BFGS())

81.489 μs (823 allocations: 44.89 KiB)
retcode: Success
u: 2-element Vector{Float64}:
 1.0000000000681024
 1.0000000001353884

Now a second order optimizer using Hessians generated by forward-mode automatic differentiation

@btime sol = solve(prob, Newton())

184.008 μs (1041 allocations: 72.96 KiB)
retcode: Success
u: 2-element Vector{Float64}:
 0.9999999969390242
 0.9999999938464837

Now a second order Hessian-free optimizer

@btime sol = solve(prob, Optim.KrylovTrustRegion())

157.599 μs (1021 allocations: 64.52 KiB)
retcode: Success
u: 2-element Vector{Float64}:
 0.999999999999108
 0.9999999999981819

Now derivative-based optimizers with various constraints

cons = (res, x, p) -> res .= [x[1]^2 + x[2]^2]
optf = OptimizationFunction(rosenbrock, Optimization.AutoForwardDiff(); cons = cons)

prob = OptimizationProblem(optf, x0, _p, lcons = [-Inf], ucons = [Inf])
@btime sol = solve(prob, IPNewton()) # Note that -Inf < x[1]^2 + x[2]^2 < Inf is always true

919.221 μs (9707 allocations: 430.95 KiB)
retcode: Success
u: 2-element Vector{Float64}:
 0.9999999992619217
 0.9999999985003628

prob = OptimizationProblem(optf, x0, _p, lcons = [-5.0], ucons = [10.0])
@btime sol = solve(prob, IPNewton()) # Again, -5.0 < x[1]^2 + x[2]^2 < 10.0

1.045 ms (12195 allocations: 566.54 KiB)
retcode: Success
u: 2-element Vector{Float64}:
 0.9999999992669327
 0.9999999985109471

prob = OptimizationProblem(optf, x0, _p, lcons = [-Inf], ucons = [Inf],
    lb = [-500.0, -500.0], ub = [50.0, 50.0])
@btime sol = solve(prob, IPNewton())

1.023 ms (11278 allocations: 538.76 KiB)
retcode: Success
u: 2-element Vector{Float64}:
 0.9999999992541948
 0.9999999984843432

prob = OptimizationProblem(optf, x0, _p, lcons = [0.5], ucons = [0.5],
    lb = [-500.0, -500.0], ub = [50.0, 50.0])
@btime sol = solve(prob, IPNewton()) # Notice now that x[1]^2 + x[2]^2 ≈ 0.5:
# cons(sol.u, _p) = 0.49999999999999994

function con_c(res, x, p)
    res .= [x[1]^2 + x[2]^2]
end

optf = OptimizationFunction(rosenbrock, Optimization.AutoForwardDiff(); cons = con_c)
prob = OptimizationProblem(optf, x0, _p, lcons = [-Inf], ucons = [0.25^2])
@btime sol = solve(prob, IPNewton()) # -Inf < cons_circ(sol.u, _p) = 0.25^2

1.729 ms (17316 allocations: 783.57 KiB)
  951.111 μs (10882 allocations: 500.15 KiB)
retcode: Success
u: 2-element Vector{Float64}:
 0.24327905408863862
 0.05757865786675858

function con2_c(res, x, p)
    res .= [x[1]^2 + x[2]^2, x[2] * sin(x[1]) - x[1]]
end

optf = OptimizationFunction(rosenbrock, Optimization.AutoForwardDiff(); cons = con2_c)
prob = OptimizationProblem(optf, x0, _p, lcons = [-Inf, -Inf], ucons = [Inf, Inf])
@btime sol = solve(prob, IPNewton())

988.141 μs (9976 allocations: 469.57 KiB)
retcode: Success
u: 2-element Vector{Float64}:
 0.9999999992619217
 0.9999999985003628

Now let's switch over to OptimizationOptimisers with reverse-mode AD

using OptimizationOptimisers
optf = OptimizationFunction(rosenbrock, Optimization.AutoZygote())
prob = OptimizationProblem(optf, x0, _p)
@btime sol = solve(prob, OptimizationOptimisers.Adam(0.05), maxiters = 1000, progress = false)

8.928 ms (89282 allocations: 5.44 MiB)
retcode: Default
u: 2-element Vector{Float64}:
 0.999957450694706
 0.9999163934196296

Try out CMAEvolutionStrategy.jl's evolutionary methods

using OptimizationCMAEvolutionStrategy
@btime sol = solve(prob, CMAEvolutionStrategyOpt())

661.303 μs (8200 allocations: 422.68 KiB)
retcode: Success
u: 2-element Vector{Float64}:
 1.0000004233740893
 1.0000008683015602

Now try a few NLopt.jl solvers with symbolic differentiation via ModelingToolkit.jl

using OptimizationNLopt, ModelingToolkit
optf = OptimizationFunction(rosenbrock, Optimization.AutoSymbolics())
prob = OptimizationProblem(optf, x0, _p)

@btime sol = solve(prob, Opt(:LN_BOBYQA, 2))

964.341 μs (2485 allocations: 78.31 KiB)
retcode: Success
u: 2-element Vector{Float64}:
 0.9999999999999999
 0.9999999999999998

@btime sol = solve(prob, Opt(:LD_LBFGS, 2))

2.158 ms (5109 allocations: 186.89 KiB)
retcode: Success
u: 2-element Vector{Float64}:
 0.9999999999894374
 0.9999999999844783

Add some box constraints and solve with a few NLopt.jl methods

prob = OptimizationProblem(optf, x0, _p, lb = [-1.0, -1.0], ub = [0.8, 0.8])
@btime sol = solve(prob, Opt(:LD_LBFGS, 2))

2.839 ms (4913 allocations: 179.55 KiB)
retcode: Success
u: 2-element Vector{Float64}:
 0.8
 0.6400000000000001

Evolutionary.jl Solvers

using OptimizationEvolutionary
@btime sol = solve(prob, CMAES(μ = 40, λ = 100), abstol = 1e-15) # -1.0 ≤ x[1], x[2] ≤ 0.8

2.814 ms (67437 allocations: 3.20 MiB)
retcode: Success
u: 2-element Vector{Float64}:
 0.8
 0.8

BlackBoxOptim.jl Solvers

using OptimizationBBO
prob = Optimization.OptimizationProblem(rosenbrock, x0, _p, lb = [-1.0, -1.0],
    ub = [0.8, 0.43])
@btime sol = solve(prob, BBO_adaptive_de_rand_1_bin()) # -1.0 ≤ x[1] ≤ 0.8, 0.2 ≤ x[2] ≤ 0.43

10.635 ms (145482 allocations: 8.10 MiB)
retcode: MaxIters
u: 2-element Vector{Float64}:
 0.6577248385055136
 0.43

Threads.nthreads()

128

Appendix

Appendix

These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.

To locally run this benchmark, do the following commands:

using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/Optimization","2drosenbrock.jmd")

Computer Information:

Julia Version 1.11.9
Commit 53a02c0720c (2026-02-06 00:27 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-16.0.6 (ORCJIT, znver2)
Threads: 128 default, 0 interactive, 64 GC (on 128 virtual cores)
Environment:
  JULIA_DEPOT_PATH = /home/crackauc/github-runners/amdci8-1/.julia
  JULIA_NUM_THREADS = auto

Package Information:

Status `~/github-runners/amdci8-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/Optimization/Project.toml`
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⌃ [3e6eede4] OptimizationBBO v0.4.11
⌃ [bd407f91] OptimizationCMAEvolutionStrategy v0.3.12
⌃ [cb963754] OptimizationEvolutionary v0.4.13
⌃ [4e6fcdb7] OptimizationNLopt v0.3.15
⌃ [36348300] OptimizationOptimJL v0.4.18
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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated`

And the full manifest:

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⌅ [e9f186c6] Libffi_jll v3.4.7+0
  [7e76a0d4] Libglvnd_jll v1.7.1+1
  [94ce4f54] Libiconv_jll v1.18.0+0
  [4b2f31a3] Libmount_jll v2.42.0+0
  [89763e89] Libtiff_jll v4.7.3+0
  [38a345b3] Libuuid_jll v2.42.0+0
  [856f044c] MKL_jll v2025.2.0+0
  [079eb43e] NLopt_jll v2.11.0+0
  [e7412a2a] Ogg_jll v1.3.6+0
  [458c3c95] OpenSSL_jll v3.5.8+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.6.1+0
  [36c8627f] Pango_jll v1.58.2+0
  [30392449] Pixman_jll v0.46.4+0
  [c0090381] Qt6Base_jll v6.10.2+2
  [629bc702] Qt6Declarative_jll v6.10.2+2
  [ce943373] Qt6ShaderTools_jll v6.10.2+1
  [6de9746b] Qt6Svg_jll v6.10.2+0
  [e99dba38] Qt6Wayland_jll v6.10.2+1
  [f50d1b31] Rmath_jll v0.5.2+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.24.0+0
  [ffd25f8a] XZ_jll v5.8.3+0
  [f67eecfb] Xorg_libICE_jll v1.1.2+0
  [c834827a] Xorg_libSM_jll v1.2.6+0
  [4f6342f7] Xorg_libX11_jll v1.8.13+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [935fb764] Xorg_libXcursor_jll v1.2.4+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
  [a51aa0fd] Xorg_libXi_jll v1.8.4+0
  [d1454406] Xorg_libXinerama_jll v1.1.7+0
  [ec84b674] Xorg_libXrandr_jll v1.5.6+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [a65dc6b1] Xorg_libpciaccess_jll v0.19.0+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.2.0+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
  [12413925] Xorg_xcb_util_image_jll v0.4.1+0
  [2def613f] Xorg_xcb_util_jll v0.4.1+0
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
  [35661453] Xorg_xkbcomp_jll v1.4.7+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.47.0+2
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [35ca27e7] eudev_jll v3.2.14+0
⌅ [214eeab7] fzf_jll v0.61.1+0
  [a4ae2306] libaom_jll v3.14.1+0
  [0ac62f75] libass_jll v0.17.5+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [8e53e030] libdrm_jll v2.4.134+0
  [2db6ffa8] libevdev_jll v1.13.4+0
  [f638f0a6] libfdk_aac_jll v2.0.4+0
  [36db933b] libinput_jll v1.28.1+0
  [b53b4c65] libpng_jll v1.6.58+0
  [9a156e7d] libva_jll v2.23.0+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
  [1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.13.0+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.11.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.11.0
  [de0858da] Printf v1.11.0
  [9abbd945] Profile v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [1a1011a3] SharedArrays v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.11.0
  [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 v1.1.1+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [c8ffd9c3] MbedTLS_jll v2.28.6+0
  [14a3606d] MozillaCACerts_jll v2023.12.12
  [4536629a] OpenBLAS_jll v0.3.27+1
  [05823500] OpenLibm_jll v0.8.5+0
  [efcefdf7] PCRE2_jll v10.42.0+1
  [bea87d4a] SuiteSparse_jll v7.7.0+0
  [83775a58] Zlib_jll v1.2.13+1
  [8e850b90] libblastrampoline_jll v5.11.0+0
  [8e850ede] nghttp2_jll v1.59.0+0
  [3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`