The multi-objective optimization problem in my case is defined below:

• Objective 1: Minimize $f_1(X_1,X_2)=C_1X_1+C_2X_2+C_3X_1^2+C_4X_2^2+C_5X_1^2X_2^2$

• Objective 2: Minimize $f_2(X_1,X_2)=D_1X_1+D_2X_2+D_3X_1^2+D_4X_2^2+D_5X_1^2X_2^2$

• Objective 3: Minimize $f_3(X_2)=-E_1X_2$

• Constraint 1: $ 0 \le X_1 \le A_1$

• Constraint 2: $ 0 \le X_2 \le A_2$

• Constraint 3: $ X_1 + X_2 \le A_3$

Here, $C_1,C_2,C_3,C_4,C_5,D_1,D_2,D_3,D_4,D_5,E_1,A_1,A_2$ and $A_3$ are positive constant numbers. $X_1$ and $X_2$ are decision variables.

I have used a real-coding method for chromosome coding and used NSGA-II for optimization. I used NSGA-II since it is the most commonly adopted method for arriving at a Pareto front. However, to use NSGA-II, is it always required to prove the NP-Hardness of a problem?

To calculate the computational complexity of a NSGA-II problem, we need to calculate both the complexity for each generation and the number of generations^[1].

Let's say that the complexity is $\mathcal O(G\cdot F\cdot P^2)=\mathcal O(100\cdot3\cdot100^2)$. Here, $G$ is number of generations, $F$ is the number of objective functions, and $P$ is the population size.

How do we justify the use of NSGA-II in this case?

_Reference

_{[1] Curry, D. M., Dagli, C. H. (2014). Computational Complexity Measures for Many-objective Optimization Problems. Procedia Computer Science. 36:185-191.}