Finding Maxima and Minima of Ordinary Differential Equation Solutions

Setup

In this tutorial we will show how to use Optim.jl to find the maxima and minima of solutions. Let's take a look at the double pendulum:

#Constants and setup
using OrdinaryDiffEq
initial = [0.01, 0.01, 0.01, 0.01]
tspan = (0.,100.)

#Define the problem
function double_pendulum_hamiltonian(udot,u,p,t)
    α  = u[1]
    lα = u[2]
    β  = u[3]
    lβ = u[4]
    udot .=
    [2(lα-(1+cos(β))lβ)/(3-cos(2β)),
    -2sin(α) - sin(α+β),
    2(-(1+cos(β))lα + (3+2cos(β))lβ)/(3-cos(2β)),
    -sin(α+β) - 2sin(β)*(((lα-lβ)lβ)/(3-cos(2β))) + 2sin(2β)*((lα^2 - 2(1+cos(β))lα*lβ + (3+2cos(β))lβ^2)/(3-cos(2β))^2)]
end

#Pass to solvers
poincare = ODEProblem(double_pendulum_hamiltonian, initial, tspan)
ODEProblem with uType Vector{Float64} and tType Float64. In-place: true
timespan: (0.0, 100.0)
u0: 4-element Vector{Float64}:
 0.01
 0.01
 0.01
 0.01
sol = solve(poincare, Tsit5())
retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 193-element Vector{Float64}:
   0.0
   0.08332584852065579
   0.24175300587841853
   0.4389533535703127
   0.6797301355043014
   0.9647629621490508
   1.3179425637594349
   1.7031226016307728
   2.0678503967116617
   2.4717899847517866
   ⋮
  95.8457309586563
  96.3577910122243
  96.92913461915474
  97.44679415429573
  97.96248479179103
  98.51183391850897
  99.0608253308051
  99.58284388126884
 100.0
u: 193-element Vector{Vector{Float64}}:
 [0.01, 0.01, 0.01, 0.01]
 [0.009170687380405334, 0.006669000455384281, 0.012420525490765841, 0.00826
6408515192909]
 [0.007673275265972504, 0.00037461737897660443, 0.016442590227730397, 0.004
636827483318277]
 [0.006125974419239289, -0.007305450189721187, 0.019967371084231897, -0.000
3364979830896869]
 [0.004966110662711131, -0.01630851653373806, 0.021440659476204722, -0.0067
05037098400474]
 [0.0047955683310194714, -0.026238103489235838, 0.01882432520883759, -0.013
913364556753736]
 [0.0060546798253553686, -0.03712455187908053, 0.010055702788069564, -0.021
038127478647375]
 [0.007900784412908646, -0.04667606960847394, -0.002673581831574513, -0.025
18303627203377]
 [0.008276510489473166, -0.05278433365633976, -0.012731546444725367, -0.025
25804037623962]
 [0.00552349681674124, -0.05525250414492613, -0.016843881882621835, -0.0218
98963191274153]
 ⋮
 [-0.014886751154788403, 0.04233275827248491, 0.0136282832580092, 0.0180290
82291419467]
 [-0.008190258536393156, 0.054422679804409874, 0.009448013826704854, 0.0177
4006800908217]
 [0.004124711787695587, 0.05674878820505975, -0.00515418739191979, 0.017596
983103942972]
 [0.013079718118471138, 0.048077043077395416, -0.01377066122508919, 0.01828
6648610391296]
 [0.015316040241448815, 0.03163095955755212, -0.008956991644884404, 0.01711
8404049844594]
 [0.011115490017375213, 0.00992901822063005, 0.007297481421219374, 0.010353
371812537674]
 [0.005713878919291721, -0.011787427051187821, 0.02050806401368854, -0.0023
10458905852316]
 [0.004211439726126673, -0.029911199361470703, 0.018750446422905413, -0.015
650712294907165]
 [0.005741239607321043, -0.04165385985159563, 0.007413270184094278, -0.0233
4897852528026]

In time, the solution looks like:

using Plots; gr()
plot(sol, vars=[(0,3),(0,4)], leg=false, plotdensity=10000)

while it has the well-known phase-space plot:

plot(sol, vars=(3,4), leg=false)

Local Optimization

Let's fine out what some of the local maxima and minima are. Optim.jl can be used to minimize functions, and the solution type has a continuous interpolation which can be used. Let's look for the local optima for the 4th variable around t=20. Thus our optimization function is:

f = (t) -> sol(t,idxs=4)
#1 (generic function with 1 method)

first(t) is the same as t[1] which transforms the array of size 1 into a number. idxs=4 is the same as sol(first(t))[4] but does the calculation without a temporary array and thus is faster. To find a local minima, we can simply call Optim on this function. Let's find a local minimum:

using Optim
opt = optimize(f,18.0,22.0)
Results of Optimization Algorithm
 * Algorithm: Brent's Method
 * Search Interval: [18.000000, 22.000000]
 * Minimizer: 1.863213e+01
 * Minimum: -2.793164e-02
 * Iterations: 11
 * Convergence: max(|x - x_upper|, |x - x_lower|) <= 2*(1.5e-08*|x|+2.2e-16
): true
 * Objective Function Calls: 12

From this printout we see that the minimum is at t=18.63 and the value is -2.79e-2. We can get these in code-form via:

println(opt.minimizer)
println(opt.minimum)
18.632127451866573
-0.02793163565154488

To get the maximum, we just minimize the negative of the function:

f = (t) -> -sol(first(t),idxs=4)
opt2 = optimize(f,0.0,22.0)
Results of Optimization Algorithm
 * Algorithm: Brent's Method
 * Search Interval: [0.000000, 22.000000]
 * Minimizer: 1.399975e+01
 * Minimum: -2.269411e-02
 * Iterations: 13
 * Convergence: max(|x - x_upper|, |x - x_lower|) <= 2*(1.5e-08*|x|+2.2e-16
): true
 * Objective Function Calls: 14

Let's add the maxima and minima to the plots:

plot(sol, vars=(0,4), plotdensity=10000)
scatter!([opt.minimizer],[opt.minimum],label="Local Min")
scatter!([opt2.minimizer],[-opt2.minimum],label="Local Max")

Brent's method will locally minimize over the full interval. If we instead want a local maxima nearest to a point, we can use BFGS(). In this case, we need to optimize a vector [t], and thus dereference it to a number using first(t).

f = (t) -> -sol(first(t),idxs=4)
opt = optimize(f,[20.0],BFGS())
* Status: success

 * Candidate solution
    Final objective value:     -2.588588e-02

 * Found with
    Algorithm:     BFGS

 * Convergence measures
    |x - x'|               = 1.11e-04 ≰ 0.0e+00
    |x - x'|/|x'|          = 4.78e-06 ≰ 0.0e+00
    |f(x) - f(x')|         = 1.68e-10 ≰ 0.0e+00
    |f(x) - f(x')|/|f(x')| = 6.49e-09 ≰ 0.0e+00
    |g(x)|                 = 8.44e-12 ≤ 1.0e-08

 * Work counters
    Seconds run:   0  (vs limit Inf)
    Iterations:    4
    f(x) calls:    16
    ∇f(x) calls:   16

Global Optimization

If we instead want to find global maxima and minima, we need to look somewhere else. For this there are many choices. A pure Julia option is BlackBoxOptim.jl, but I will use NLopt.jl. Following the NLopt.jl tutorial but replacing their function with out own:

import NLopt, ForwardDiff

count = 0 # keep track of # function evaluations

function g(t::Vector, grad::Vector)
  if length(grad) > 0
    #use ForwardDiff for the gradients
    grad[1] = ForwardDiff.derivative((t)->sol(first(t),idxs=4),t)
  end
  sol(first(t),idxs=4)
end
opt = NLopt.Opt(:GN_ORIG_DIRECT_L, 1)
NLopt.lower_bounds!(opt, [0.0])
NLopt.upper_bounds!(opt, [40.0])
NLopt.xtol_rel!(opt,1e-8)
NLopt.min_objective!(opt, g)
(minf,minx,ret) = NLopt.optimize(opt,[20.0])
println(minf," ",minx," ",ret)
NLopt.max_objective!(opt, g)
(maxf,maxx,ret) = NLopt.optimize(opt,[20.0])
println(maxf," ",maxx," ",ret)
Error: ArgumentError: Package ForwardDiff not found in current path:
- Run `import Pkg; Pkg.add("ForwardDiff")` to install the ForwardDiff packa
ge.
plot(sol, vars=(0,4), plotdensity=10000)
scatter!([minx],[minf],label="Global Min")
scatter!([maxx],[maxf],label="Global Max")
Error: UndefVarError: minx not defined

Appendix

These tutorials are a part of the SciMLTutorials.jl repository, found at: https://github.com/SciML/SciMLTutorials.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 tutorial, do the following commands:

using SciMLTutorials
SciMLTutorials.weave_file("tutorials/ode_extras","03-ode_minmax.jmd")

Computer Information:

Julia Version 1.6.2
Commit 1b93d53fc4 (2021-07-14 15:36 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-11.0.1 (ORCJIT, znver2)
Environment:
  JULIA_DEPOT_PATH = /root/.cache/julia-buildkite-plugin/depots/a6029d3a-f78b-41ea-bc97-28aa57c6c6ea
  JULIA_NUM_THREADS = 16

Package Information:

      Status `/var/lib/buildkite-agent/builds/5-amdci4-julia-csail-mit-edu/julialang/scimltutorials-dot-jl/tutorials/ode_extras/Project.toml`
  [f3b72e0c] DiffEqDevTools v2.27.2
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  [14d82f49] Xorg_libpthread_stubs_jll v0.1.0+3
  [c7cfdc94] Xorg_libxcb_jll v1.13.0+3
  [cc61e674] Xorg_libxkbfile_jll v1.1.0+4
  [12413925] Xorg_xcb_util_image_jll v0.4.0+1
  [2def613f] Xorg_xcb_util_jll v0.4.0+1
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
  [35661453] Xorg_xkbcomp_jll v1.4.2+4
  [33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
  [c5fb5394] Xorg_xtrans_jll v1.4.0+3
  [8f1865be] ZeroMQ_jll v4.3.2+6
  [3161d3a3] Zstd_jll v1.5.0+0
  [0ac62f75] libass_jll v0.14.0+4
  [f638f0a6] libfdk_aac_jll v0.1.6+4
  [b53b4c65] libpng_jll v1.6.38+0
  [a9144af2] libsodium_jll v1.0.20+0
  [f27f6e37] libvorbis_jll v1.3.6+6
  [1270edf5] x264_jll v2020.7.14+2
  [dfaa095f] x265_jll v3.0.0+3
  [d8fb68d0] xkbcommon_jll v0.9.1+5
  [0dad84c5] ArgTools
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8bb1440f] DelimitedFiles
  [8ba89e20] Distributed
  [f43a241f] Downloads
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [b27032c2] LibCURL
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions
  [44cfe95a] Pkg
  [de0858da] Printf
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML
  [a4e569a6] Tar
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll
  [deac9b47] LibCURL_jll
  [29816b5a] LibSSH2_jll
  [c8ffd9c3] MbedTLS_jll
  [14a3606d] MozillaCACerts_jll
  [4536629a] OpenBLAS_jll
  [bea87d4a] SuiteSparse_jll
  [83775a58] Zlib_jll
  [8e850ede] nghttp2_jll
  [3f19e933] p7zip_jll