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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.)

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  • $\begingroup$ I doubt that there is a well-accepted answer in the literature. I would perhaps suggest the name "span". $\endgroup$ – Gamow Jul 24 '15 at 5:46
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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.

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  • $\begingroup$ Thanks, this is the terminology for which I was looking. Here is a recent paper about "Structural Complexity of Multiobjective NP Search Problems" <dx.doi.org/10.1007/978-3-642-29344-3_29>. $\endgroup$ – argentpepper Jul 24 '15 at 19:59

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