# How to plot feasible objective space of a Generic Multiobjective Optimization problem?

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

$min{\space}F(\bf{X})=[f_1(x),...,f_n(x)]$
$h_k(x)=0{\space\space\space} k=1,...,n_e$
$g_i(x)\leq0{\space\space\space} i=1,...,n$
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:

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.

• Can you give the definition of "feasible design space of objective functions" for those of us not intimately familiar with multiobjective optimization terminology? – Rahul Feb 9 '15 at 3:46
• Thank you for you comment @Rahul . The definitions are added to the questions and the question is edited as well. – Electricman Feb 9 '15 at 7:09