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I read that it is possible to convert a multi-objective optimization problem into single objective by using weighted sum method. I wanted to know if it is a good idea to convert a two objective optimization problem into single objective by multiplication. for example:

Goal: minimize $f_1(x)$, maximize $f_2(x)$. So is it a good idea to calculate $f_2(x)/f_1(x)$, and use it as an optimal value.

PS: Any related link to academic literature would be helpful as well. Thanks.

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  • $\begingroup$ First thing that comes to my mind is that multiplying or dividing the objective functions you may end up with non-linear objective. Is that relevant to you to have a linear objective function? $\endgroup$
    – Libra
    Oct 2, 2013 at 6:11
  • $\begingroup$ No. I don't mind having a linear or a non-linear objective function . $\endgroup$
    – A. K.
    Oct 2, 2013 at 11:57
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    $\begingroup$ You may have a look at this: theory.stanford.edu/~megiddo/pdf/rational.pdf $\endgroup$
    – Libra
    Oct 2, 2013 at 12:49
  • $\begingroup$ @Libra thanks for sharing the link. In case the link expires the the paper is: "COMBINATORIAL OPTIMIZATION WITH RATIONAL OBJECTIVE FUNCTIONS" By NIMROD MEGIDDO from Tel Aviv University $\endgroup$
    – A. K.
    Nov 16, 2022 at 17:34

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In a multi-objective optimization, the objectives to be optimized are conflict. For the convenience of the description, supposing all the objectives are to be minimized, because the maximizing problem can be transformed to the minimizing problems by multiplying $-1$.

The "conflict" means that this is no single solution can simultaneously satisfy all objectives, but a set of solutions. These solutions form the Pareto-front in the objective space, and also these solutions are called Pareto-optimal solutions, and form the Pareto Set in the decision space. In addition, these solutions should evenly distribute on the Pareto-front.

In solving multi-objective optimization, our goal is to obtain these Pareto-optima solutions. Weight-sum method can not result in the optimal solutions that evenly distribute on the Pareto-front, therefore, this method cannot be used in this regard.

Typically, there are few good algorithms that convert a multi-objective optimization problem to several single-objective optimization problems. such as MOEA/D. In addition, NSGA-II performs very well in solving multi-objective optimization.

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