Optimization based solvers

[ErikQQY]

Part of #336

Some TODOs:

• Multi-points BVP are encountering some DI issues, but two-point BVP are fine, need to dig deeper to see what's going wrong.
• Need some patches in SciMLBase.jl and DiffEqBase.jl for better interface.
• More docstrings and documentations

[github-actions]

Benchmark Results

Click to check benchmark results

	master	5cc940a...	master / 5cc940a...
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK2()	0.578 ± 0.022 s	0.612 ± 0.021 s	0.945 ± 0.048
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK3()	13.7 ± 0.55 ms	13.8 ± 0.41 ms	0.995 ± 0.05
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK4()	3.15 ± 0.25 ms	3.12 ± 0.13 ms	1.01 ± 0.091
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK5()	9.42 ± 1.3 ms	9.28 ± 1.3 ms	1.01 ± 0.2
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK6()	1.62 ± 0.29 ms	1.62 ± 0.29 ms	1 ± 0.25
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = false)	1.84 ± 0.66 ms	1.83 ± 0.67 ms	1 ± 0.51
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = true)	3.11 ± 0.99 ms	3.08 ± 1 ms	1.01 ± 0.46
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = false)	0.0662 ± 0.012 s	0.0658 ± 0.0057 s	1.01 ± 0.2
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = true)	0.0796 ± 0.014 s	0.0893 ± 0.023 s	0.891 ± 0.27
Simple Pendulum/IIP/Shooting(Tsit5())	0.236 ± 0.077 ms	0.236 ± 0.076 ms	0.998 ± 0.46
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK2()	0.734 ± 0.0083 s	0.76 ± 0.017 s	0.966 ± 0.025
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK3()	16.8 ± 4.4 ms	16.5 ± 4.8 ms	1.02 ± 0.4
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK4()	3.72 ± 0.41 ms	3.61 ± 0.2 ms	1.03 ± 0.13
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK5()	10.9 ± 1.8 ms	10.9 ± 1 ms	0.997 ± 0.19
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK6()	1.9 ± 0.21 ms	1.9 ± 0.22 ms	0.999 ± 0.16
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = false)	3.54 ± 3.1 ms	3.57 ± 2.9 ms	0.991 ± 1.2
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = true)	6.06 ± 5.3 ms	6.09 ± 5.2 ms	0.996 ± 1.2
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = false)	0.121 ± 0.0065 s	0.119 ± 0.0039 s	1.02 ± 0.064
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = true)	0.142 ± 0.0038 s	0.152 ± 0.0067 s	0.932 ± 0.048
Simple Pendulum/OOP/Shooting(Tsit5())	0.632 ± 0.062 ms	0.639 ± 0.059 ms	0.989 ± 0.13
time_to_load	5.3 ± 0.034 s	5.4 ± 0.11 s	0.982 ± 0.021

### Benchmark Plots A plot of the benchmark results has been uploaded as an artifact to the workflow run for this PR. Go to "Actions"->"Benchmark a pull request"->[the most recent run]->"Artifacts" (at the bottom).