With multi-objective optimization, there are many ways you can combine/compare the objective variables in your analysis. The problem is that there isn't a "right" way to do it. It depends entirely on what the problem actually is and what the variables represent. Your best bet is likely to maximize something like $k+a*t$ where $a$ is an arbitrary positive value. Once you have an answer, see if you like the resultant $k$ and $t$, and modify $a$ as necessary until you find a solution you are happy with.
As for the actual optimization, not having an analytical expression for the functions isn't the end of the world, but not having any information is going to make it rough. If you can assume some level of smoothness/continuity, even if it is only piece-wise, you can use a root finding algorithm on a derivative approximation to find local maxima (there are many more sophisticated methods than this, but I am unfamiliar with them. Other people here could likely point you in the right direction). If you can establish convexity, you could extend it to global optimality.
True black box multi-objective optimization isn't exactly an easy problem, but a few assumptions and an iterative process with a objective reduction should get you an answer that is acceptable (assuming one exists).