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	ed84cb0...	master / ed84cb0...
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK2()	0.59 ± 0.012 s	0.576 ± 0.015 s	1.03 ± 0.034
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK3()	14 ± 0.56 ms	13.6 ± 0.57 ms	1.03 ± 0.06
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK4()	3.19 ± 0.18 ms	3.14 ± 0.17 ms	1.02 ± 0.079
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK5()	9.64 ± 1.3 ms	9.32 ± 1.3 ms	1.03 ± 0.2
Simple Pendulum/IIP/BoundaryValueDiffEqMIRK.MIRK6()	1.68 ± 0.29 ms	1.63 ± 0.29 ms	1.03 ± 0.26
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = false)	1.87 ± 0.66 ms	1.9 ± 0.62 ms	0.987 ± 0.47
Simple Pendulum/IIP/MultipleShooting(10, Tsit5; grid_coarsening = true)	3.19 ± 1 ms	3.12 ± 0.97 ms	1.02 ± 0.45
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = false)	0.0677 ± 0.0077 s	0.069 ± 0.0076 s	0.981 ± 0.16
Simple Pendulum/IIP/MultipleShooting(100, Tsit5; grid_coarsening = true)	0.0835 ± 0.013 s	0.082 ± 0.014 s	1.02 ± 0.24
Simple Pendulum/IIP/Shooting(Tsit5())	0.243 ± 0.075 ms	0.246 ± 0.075 ms	0.989 ± 0.43
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK2()	0.758 ± 0.01 s	0.755 ± 0.0092 s	1 ± 0.018
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK3()	16.6 ± 5 ms	17 ± 6.9 ms	0.98 ± 0.5
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK4()	3.63 ± 0.14 ms	3.67 ± 0.22 ms	0.99 ± 0.072
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK5()	11 ± 1 ms	11.1 ± 1.7 ms	0.992 ± 0.18
Simple Pendulum/OOP/BoundaryValueDiffEqMIRK.MIRK6()	1.91 ± 0.16 ms	1.91 ± 0.23 ms	0.997 ± 0.15
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = false)	3.87 ± 3 ms	3.71 ± 3 ms	1.04 ± 1.2
Simple Pendulum/OOP/MultipleShooting(10, Tsit5; grid_coarsening = true)	6.11 ± 5.3 ms	6.45 ± 5.3 ms	0.946 ± 1.1
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = false)	0.117 ± 0.0043 s	0.128 ± 0.0067 s	0.913 ± 0.058
Simple Pendulum/OOP/MultipleShooting(100, Tsit5; grid_coarsening = true)	0.142 ± 0.0046 s	0.158 ± 0.016 s	0.899 ± 0.093
Simple Pendulum/OOP/Shooting(Tsit5())	0.636 ± 0.062 ms	0.639 ± 0.068 ms	0.997 ± 0.14
time_to_load	5.4 ± 0.026 s	5.61 ± 0.033 s	0.964 ± 0.0073

### 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).