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I need some clarification on multi objective optimization. I would like to know if a problem has three objectives with completely different variables, should such a problem be solved as three independent single objective optimization problem or could the problem be solved using a multiobjective optimization. I would appreciate your feedback. Thank you.

eg. Min f1=a1*x1 +a2*x2; f2= a3*x3 +a4*x4; f3=a5*x5 + a6*x6

Honey Adams
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2 Answers2

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This problem should be solved as three separate optimization problems. Since each of the functions see "separate" variables, there is no need to use the machinery from multi-objective optimization.

Multi-objective minimization seeks to simultaneously minimize several functions. You could technically call your example a "multi-objective" problem, where your vector is $x=[x_1,x_2,x_3,x_4,x_5,x_6]$. However, since your problem can be separated into solving three separate minimization problems (each of which have no effect on the solution of the other), it does not really match the spirit of the field.

Zim
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  • I appreciate your feedback. – Honey Adams Jun 04 '20 at 21:33
  • How fo you know this? Presumably* there are constraints, and those constraints might couple the variables in the different objective functions. *It wouldn't be a very "viable" optimization problem with a linear objective and no constraints. – Mark L. Stone Jun 05 '20 at 00:32
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    @MarkL.Stone that's a good point -- if the original problem had mentioned constraints which mixed the variables together, then multi-objective theory would be a nice tool to use here. – Zim Jun 05 '20 at 02:08
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At a time, you have asked too many different questions. Whether a single objective or multiobjective optimization is required, that depends on your aim of study. Whether variables can be separated or not is completely different problem, it defines the nature of objective functions and their interrelationship.

In order to get clear grasp on the multiobjective optimizatiom, you can refer to the following research article.

https://www.researchgate.net/publication/341752284_Pareto_dominance_based_Multiobjective_Cohort_Intelligence_algorithm

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http://dx.doi.org/10.1016/j.ins.2020.05.019

From the article, you can see that when objectives are conflicting, multiobjective optimization is most suitable, otherwise you can use single objective optimization.

M.Patil
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