Construct explicit second-order AD for optimization BVPs

[ChrisRackauckas-Claude]

Ignore this PR until it has been reviewed by @ChrisRackauckas.

Summary

• construct the sparse optimization AD backend with an explicit SecondOrder dense backend
• preserve the existing sparsity-detector choices and the special handling for AutoZygote, AutoSymbolics, an existing SecondOrder, and NoAD
• assert that the two Ipopt regression solves emit no warnings and return successful solutions

Root cause

MIRK optimization problems currently pass a first-order AutoSparse backend to OptimizationFunction. OptimizationBase can synthesize its second-order backend later, but the generic OptimizationCache now checks the original AD type first and correctly warns that an explicit second-order backend was not provided. That warning makes both existing @test_nowarn regression cases fail after OptimizationIpopt moved to the generic cache.

This change constructs the same second-order combinations that OptimizationBase would generate, but does so before the cache is initialized. SecondOrder is imported directly from its public owner, DifferentiationInterface.

The same downgrade also exposed a separate sparse constraint-Jacobian live-parameter failure. That belongs in OptimizationBase and is fixed by SciML/Optimization.jl#1270.

Dependencies

This is the BoundaryValueDiffEq companion to:

• Recognize explicit second-order AutoSparse AD Optimization.jl#1269, which recognizes AutoSparse(SecondOrder(...)) as an explicitly supplied second-order backend
• Use live parameters in sparse constraint Jacobians Optimization.jl#1270, which forwards live parameters through generated sparse constraint-Jacobian callbacks

The OptimizationBase compatibility floor cannot be raised until those changes have a coordinated registered release; SciML/Optimization.jl#1287 is coordinating the sequential package version.

Validation

On clean current upstream/master, the Julia 1.10 MIRK Core downgrade with OptimizationIpopt 1.3.0 and released OptimizationBase reproduces the regression in MIRK Basic Tests: 197 passed, 2 failed, 1 errored (200 total). The failures are the two captured missing_second_order_ad warnings; the error is the structured-parameter sparse constraint-Jacobian callback.

With this branch plus the exact changes from Optimization PRs #1269 and #1270:

• official BoundaryValueDiffEqMIRK, GROUP=Core: passed
  • Basic: 205/205 in 87m52.1s
  • NLLS: 72/72 in 38m06.3s
  • Ensemble: 10/10 in 5m19.1s
  • Singular BVP: 8/8 in 2m48.8s
  • VectorOfVector initials: 2/2 in 1m21.1s
  • Dynamic Optimization: 6/6 in 6m40.5s
• BoundaryValueDiffEqCore, GROUP=Core: 14/14 passed
• BoundaryValueDiffEqCore, GROUP=QA: 18/18 passed
• direct NoAD preservation assertion: passed
• Runic 1.7 over the repository: passed
• git diff --check: passed

[github-actions]

Benchmark Results (Julia v1.11)

Time benchmarks

	master	77b8fbd...	master / 77b8fbd...
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK2()	0.571 ± 0.0063 s	0.591 ± 0.0049 s	0.967 ± 0.013
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK3()	12.2 ± 0.34 ms	12.3 ± 0.23 ms	0.998 ± 0.034
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK4()	2.69 ± 0.07 ms	2.73 ± 0.097 ms	0.985 ± 0.043
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK5()	3.22 ± 0.1 ms	3.2 ± 0.1 ms	1.01 ± 0.045
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK6()	1.56 ± 0.052 ms	1.57 ± 0.062 ms	0.991 ± 0.051
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = false)	1.56 ± 0.078 ms	1.53 ± 0.074 ms	1.02 ± 0.071
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = true)	3.23 ± 0.16 ms	3.18 ± 0.16 ms	1.02 ± 0.072
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = false)	0.0404 ± 0.0036 s	0.0401 ± 0.0037 s	1.01 ± 0.13
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = true)	0.0637 ± 0.0016 s	0.0614 ± 0.0016 s	1.04 ± 0.038
Simple Pendulum/IIP/Shooting(Tsit5())	0.178 ± 0.073 ms	0.184 ± 0.072 ms	0.97 ± 0.55
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK2()	0.736 ± 0.009 s	0.747 ± 0.013 s	0.985 ± 0.021
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK3()	16.4 ± 6.6 ms	16.5 ± 6.4 ms	0.997 ± 0.56
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK4()	3.31 ± 0.22 ms	3.25 ± 0.15 ms	1.02 ± 0.082
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK5()	3.94 ± 0.25 ms	3.93 ± 0.21 ms	1 ± 0.082
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK6()	1.88 ± 0.13 ms	1.87 ± 0.12 ms	1.01 ± 0.096
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = false)	3.42 ± 0.54 ms	3.39 ± 0.71 ms	1.01 ± 0.26
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = true)	6.7 ± 6.4 ms	6.56 ± 5.6 ms	1.02 ± 1.3
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = false)	0.0945 ± 0.0051 s	0.089 ± 0.0068 s	1.06 ± 0.099
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = true)	0.141 ± 0.0045 s	0.14 ± 0.0054 s	1.01 ± 0.051
Simple Pendulum/OOP/Shooting(Tsit5())	0.622 ± 0.065 ms	0.617 ± 0.051 ms	1.01 ± 0.13
time_to_load	7.59 ± 0.12 s	7.58 ± 0.12 s	1 ± 0.022

Memory benchmarks

	master	77b8fbd...	master / 77b8fbd...
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK2()	0.389 M allocs: 0.0443 GB	0.389 M allocs: 0.0443 GB	1
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK3()	0.0436 M allocs: 4.88 MB	0.0436 M allocs: 4.88 MB	1
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK4()	15.7 k allocs: 1.65 MB	15.7 k allocs: 1.65 MB	1
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK5()	22.1 k allocs: 2.04 MB	22.1 k allocs: 2.04 MB	1
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK6()	12.8 k allocs: 1.05 MB	12.8 k allocs: 1.05 MB	1
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = false)	25.5 k allocs: 1.82 MB	25.5 k allocs: 1.82 MB	1
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = true)	0.049 M allocs: 3.39 MB	0.049 M allocs: 3.39 MB	1
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = false)	0.553 M allocs: 0.0535 GB	0.553 M allocs: 0.0535 GB	1
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = true)	0.833 M allocs: 0.0778 GB	0.833 M allocs: 0.0778 GB	1
Simple Pendulum/IIP/Shooting(Tsit5())	4.64 k allocs: 0.224 MB	4.64 k allocs: 0.224 MB	1
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK2()	0.89 M allocs: 0.984 GB	0.89 M allocs: 0.984 GB	1
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK3()	0.0932 M allocs: 24.8 MB	0.0932 M allocs: 24.8 MB	1
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK4()	0.0324 M allocs: 3.96 MB	0.0324 M allocs: 3.96 MB	1
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK5()	0.045 M allocs: 4.99 MB	0.045 M allocs: 4.99 MB	1
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK6()	25.3 k allocs: 2.17 MB	25.3 k allocs: 2.17 MB	1
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = false)	0.142 M allocs: 10.2 MB	0.142 M allocs: 10.2 MB	1
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = true)	0.266 M allocs: 18.7 MB	0.266 M allocs: 18.7 MB	1
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = false)	2.52 M allocs: 0.279 GB	2.52 M allocs: 0.279 GB	1
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = true)	3.82 M allocs: 0.404 GB	3.82 M allocs: 0.404 GB	1
Simple Pendulum/OOP/Shooting(Tsit5())	0.0373 M allocs: 1.69 MB	0.0373 M allocs: 1.69 MB	1
time_to_load	0.159 k allocs: 11.2 kB	0.159 k allocs: 11.2 kB	1