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I need to maximize several objective functions $ f_i(x)$, that I have arranged in a vector. $ f(x) = [ f_1(x) , f_2(x) \cdots, f_K(x) ]^T $. Essentially, my question is whether I can represent the same multi-objective function using conjunctive normal forms, as shown below $$ \max_{x \in \Omega} f(x) \equiv \max_{x \in \Omega} \bigwedge^K_{i=1} f_i(x) $$ Thank you!

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  • $\begingroup$ What does the conjunction of real-valued functions even mean? $\endgroup$ – Rodrigo de Azevedo Jun 10 '19 at 7:54
  • $\begingroup$ I am looking for a notation that allows me to maximize each of the elements in $f$. I do not know if the notation is standard or even understandable. I think it would make more sense the exchange the position of $\max$ and $\bigwedge$. I would appreciate any suggestion. $\endgroup$ – Dunkel Jun 10 '19 at 8:01
  • $\begingroup$ I don't even understand what you're trying to do. However, I know you have a type error. If $f_i$ takes values in $\mathbb R$, one cannot use it as a conjunct. $\endgroup$ – Rodrigo de Azevedo Jun 10 '19 at 8:07
  • $\begingroup$ @Rodrigo de Azevedo, I see. It is meant to be used with logical variables. Is there any alternative notation for this? Going to your question, $x$ has commonn variables $y$ (to all the functions) and private variables $z_i$ that only affect $f_i(x)$. For a fixed $y$, the problem can be split and each $f_i(x)$ be independently solved. I want a perhaps more neat notation that allows me to make this point clear. $\endgroup$ – Dunkel Jun 10 '19 at 8:38
  • $\begingroup$ You should include all information in your question, including domains and co-domains and your thoughts on the problem. Having telegraphically short questions and long comment sections benefits no one. $\endgroup$ – Rodrigo de Azevedo Jun 10 '19 at 8:55

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