`__solve` interface support

Some of the subpackages in Optimization.jl do not support interfaces like SciMLBase.__solve(prob::OptimizationProblem, alg, args...; kwargs...), while others do. For example, SciMLBase has some traits for solvers that support the cache interface:

https://github.com/SciML/SciMLBase.jl/blob/6f1a1d077182cde28e951417484b17b6b771252f/src/solve.jl#L128-L139

We usually define

function __solve(prob, alg, args...; kwargs)
    cache = __init(prob, alg)
    sol = solve!(cache) # or solve!(integrator)
end

for some specific problem solvers (ODE, SDE, or BVP), but it seems the __solve is directly defined on OptimizationCache in Optimization.jl, while we are also doing similar init and solve cache things in Optimization.jl under the hood, why not use the same interface in Optimization.jl too?

Keep track of this issue from https://github.com/SciML/BoundaryValueDiffEq.jl/pull/350 for better integration of Optimization.jl.

Start with SciMLBase/src/solve.jl and the OptimizationCache implementation, then compare the Optimization.jl subpackages that define __solve with the __init and solve! flow described here. Done means the relevant optimization solvers consistently support the __solve(prob, alg, args...; kwargs...) interface and retain compatibility with the BoundaryValueDiffEq.jl integration.