I am interested in a particular type of optimization problem, which seems relevant to engineering applications, in particular. Suppose that I don't need a function to be fully optimized -- I can consider the problem solved when I have found any solution $x^*\in\mathbb{R}^n$ such that $f(x^*)<T$ for some target threshold $T\in\mathbb{R}$.
Furthermore, consider that my primary concern is the cost of computing functional values during each iteration $i$ -- e.g. because I must run a manual experiment or cumbersome simulation in order to determine $f(x_i)$. In such a case, is there an efficient approach to solving this problem, other than simply relying on algorithms that attempt to minimize function evaluations?
Has anyone heard of such a thing?
You can set up a feasibility problem, which is a special case of optimization problem.
Include the target objective value as a constraint, $f(x^*) \le T$, in addition to all the other constraints. If the optimization software allows, submit that to the solver without any objective function, or if need be, include a constant objective function, such as minimize 0.