Linear programming with inequality constraints treated lexicographically
If you want to have only initial constraints to be strictly equal and rest of them should be strictly less

First, add regular constraints:
$$
A_i x \le b_i
$$
Next, add checks for equality as in https://blog.adamfurmanek.pl/2015/09/12/ilp-part-4/ :
$$
e_i = A_i x \stackrel{?}{=} b_i
$$
Finally, we want only some initial of these variable to be true, and for that we can use material implication as in https://blog.adamfurmanek.pl/2015/08/22/ilp-part-1/ :
$$
e_{i+1} \implies e_i = 1
$$
or we can simply add the following constraints:
$$
e_{i+1} \le e_i
$$

Disclaimer: I am the author of the blog posts but I don't want to advertise here, only include them as a reference.

If you want some initial of the constraints to be strictly equal, the next one should be strictly less, and rest of them can be not satisfied

Do not add any regular constraints.

Create equality constraints as before:
$$
e_i = A_i x \stackrel{?}{=} b_i
$$

Calculate prefix conjunction of equality constraints:
$$
E_i = e_1 \wedge e_2 \wedge \ldots \wedge e_{i}
$$

Variables $E_i$ indicates whether first $i$ constraints are strictly equal.

Next, calculate less than constraints:
$$
l_i = A_i x \stackrel{?}{&lt;} b_i
$$

Finally, make sure that after $i$ initial strictly equal constraints there is a strictly less constraint or another strictly equal constraint satisfied:
$$
E_i \implies \left( l_{i+1} \vee E_{i+1} \right) = 1
$$