Leafing through a few textbooks, I've noticed that the problem of initially bracketing a minimum during a line search tends be an afterthought (at least in my undergraduate texts). Are there well-established techniques or best practices for this type of problem, or are solutions typically application dependent? Can anyone recommend some references on the topic?
Usually one doubles the initial step until the Goldstein condition is violated or (in a feasible point method) the boundary is reached. Then one has a bracket. (If no such step exists, the objective function is unbounded below.) One can also use less conservative extrapolation procedures, but these require good tuning to be robust enough in a general purpose solver.
In my experience establishing the bracket is very often application-dependent. If you had real constraints or an algebraic derivation for your bracket, you'd use it of course! Usually there's an appeal to either
- nature this physically makes no sense outside this bracket
- computability this would be too hard to compute outside of the bracket
- objective solutions outside of this region are otherwise undesirable.
I'm hoping someone else can come in with a more algorithmic approach, which is what I think you're looking for here.