There's a course that I do remotely and I have a homework question which I have no idea how to answer. I did look up a lot in google and did not find any good examples - only loads of information and theory.
The question is as follows:
Use the epsilon constraint method to produce a representation of the Pareto optimal
solutions for problem:
min (x1^2,x2^2)
s.t.
x1+x2-x3>=1
x1,x2,x3>=0
I'm pretty sure that it is kinda easy but really have no idea how to start. Would love to get some explanation.
Thanks in advance
The idea behind $\varepsilon$-constraint is to move all but one objective function in the constraint set. Hence, if we keep the second objective of the original problem and move the first one to the constraint set, we end up with the problem
$$ \min x_2^2 \quad \text{s.t.} \quad x_1^2 \leq \varepsilon,\; x_1+x_2-x_3 \geq 1,\; x \geq 0_3. $$
It is quite easy to see that all nondominated points (Pareto points) of the original problem are located somewhere inside $[0,1]^2$. To obtain a representation of the Pareto front using the $\varepsilon$-constraint method, one needs to solve the scalarized problem above for different choices of $\varepsilon$. For example for $\varepsilon \in \{0,0.1,0.2,\ldots,1\}$.