I have two optimization problems, both of whose inputs are from the set $I$ and whose solutions are from the set $S$, one a minimization with objective function $m_{\min}$ and one a maximization with objective function $m_{\max}$. I am studying what I would like to call the join or direct sum of these two problems, the problem of maximizing $m_{\max}(x, w) - m_{\min}(x, w)$, though I don't know the correct name for such a problem. In my particular case, each problem on its own has a trivial algorithm that produces optimal solutions, but together, the problem is intractable. Has the structural complexity or approximability of optimization problems of this form been studied in a general way?(The fact that one is a maximization problem and one is a minimization problem is not crucial here, they could both be the same type of optimization.)
What do you call the join of two optimization problems?
cc.complexity theory;optimization
I think this falls in the category of multi-objective function optimization problems. To be specific, it's a linear scalarization of an optimization problem with two objective functions.I don't know this field very well, but I expect their complexity has been studied.