Which combinatorial problem is similar to this problem?
In a very literal sense, D.W.'s answer is correct. In the absence of any information about the cost function, there are no alternatives better than simply trying the cost function at each possible input.
However, in real-world applications, we usually know something about the cost function—and as soon as we have a little bit of structure, we can leverage that to create more efficient search strategies.
For example, if the cost function has a relatively simple closed form in terms of the inputs, you may be able to apply an optimizing SMT solver such as the widely-used Z3 from Microsoft Research (interactive tutorial here). These solvers use complex strategies to locate and exploit hidden structure in the input problem and find solutions more efficiently than brute-force search.