Equivalences of optimization problems

Defining what it means for two optimization problemsoptimization problems to be "equivalent" is hard, and is not usually done rigorously.

The difficulty in defining this equivalence arises from optimization theory almost always being applied computationally. If we simply defined two problems being equivalent as the existence of arbitrary such functions as in the definition above, then all problems that have an optimal solution would be equivalent; we can simply let and map everything to an optimal solution.