As you know, a generic Multiobjective optimization problem can be stated as follows:

$h_k(x)=0{\space\space\space} k=1,...,n_e$
$g_i(x)\leq0{\space\space\space} i=1,...,n$
where $\bf{X}=[x_1, x_2, ... ,x_j]$

definitions : Objective Space is a vector space including objective functions,i.e.$[f_1(x),...,f_n(x)]$ , of the Multiobjective Optimization problem as its dimensions. It is different from solution space, which is a vector space with decision variables,i.e.$[x_1, x_2, ... ,x_j]$, of the Multiobjective Optimization problem as the dimensions.

It is obvious that no one can plot feasible solution space when number of decision variables are more than three, i.e., $j>3$. Also, It is not possible to plot feasible objective space when number of objectives are more than three, i.e., $n>3$.
I want to pull your attention to the case that we have 5 decision variables so we cannot plot the solution space, and we have three objective functions. Having three objective functions enables us to plot feasible objective space. Objective space for a MO problem including three objective functions of $f_1(.)$ , $f_2(.)$ and $f_(3)$ is shown in the figure:
enter image description here
where $\mu_1,\mu_2,\mu_3$ are three objective functions of the Multiobjective Optimization problem.

Now my question is:
How to plot feasible objective space of a Generic Multiobjective Optimization problem?
For example, imagine the problem bellow with the given constraints and tell me how can I obtain the feasible objective space similar to the one in the figure.

$f_1(X)= norm(x)^2$
$f_2(X)= 3x_1+2x_2 - x_3/3 + 0.01(x_4 - x_5)^3$ $f_3(X)= x_1^2 + 3x_2^2 + 0.2(x_3 - x_5)^3 + log(x_4^2 + x_1^2 + x_2^2 + 1)$

Subject to:
$h_1(X) = x_1 + 2x_2 - x_3 - 0.5x_4 + x_5 - 2$
$h_2(X) = 4x_1 - 2x_2 + 0.8x_3 + 0.6x_4 + 0.5x_5^2$
$g_1(X)= norm(x)^2 - 10$

Please note that, I don't expect the solution of the given problem. Please give me some applicable insights about obtaining the graphing of feasible objective space.

  • $\begingroup$ Please tell me if this is the proper forum for asking this question, there are several math forum in stackexchange that are very similar to me $\endgroup$ – Electricman Feb 8 '15 at 17:02
  • $\begingroup$ Not my area of expertise, but visualizing the Pareto front is one what this is done. Since you have the Matlab tag, maybe see this article and this FileExchange contribution. $\endgroup$ – horchler Feb 8 '15 at 17:44
  • $\begingroup$ I know how to plot Pareto front, my code does it already. But I want to plot the objective design space. This plot should be obtained from cuting the space with the given constraints. @horchler $\endgroup$ – Electricman Feb 8 '15 at 18:42

Obtaining the feasible space of a multi-objective optimization problem refers to the landscape of this problem. Actually, this is a research topic of "approximate the landscape of a multi-objective optimization problem". Here, I could give you a commonly used method in this regard.

1: Employ any good method for solving multi-objective optimization problem, such as NSGA-II.

2: In each generation, save the non-dominated solutions to an external archive.

3: When the algorithm terminates, plot the solutions in this archive.

Hope this can help you.


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