how to get lexicographic efficient extremes by solving a scalarized (weighted sum) problem in bi-objective optimization problem?

Question

I have a bi-objective problem: $minimize (Z_1(X) and Z_2(X))$

$Z(X)=w_1*Z_1(X)+w_2*Z_2(X)$ is the weighted sum objective function that is minimized. $Z_1(X)$ and $Z_2(X)$ are two objective functions.

I want to know if there is a method or formulation to determine $w_1$ and $w_2$ such that solving the weighted sum optimization problem results in $lexmin(Z_1(X),Z_2(X))$ and $lexmin(Z_2(X),Z_1(X))$ as lexicographic optimum solutions to the bi-objective problem.

Answer 1

You cannot obtain a lexicographically ordered solution with just one optimization. You first have to minimize $Z_1(X)$. If the optimal value is $c$, you then minimize $Z_2(X)$ such that $Z_1(X) \leq c$.