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We use the superscript * to represent an optimal solution of an optimization problem.
Then is an optimal solution of (P1).
So is an optimal solution of.
(a) Since a bounded optimal solution of (P2) or (P3) is an optimal solution of (P1).
Notice that is an optimal solution of (P3) implies is an optimal solution of (P4) which implies is an optimal solution of (P5).
Let z be an optimal solution of this program.
Then x* is an optimal solution of (FP).
Let the optimal solution of (A.7) be.
O2 is the optimal solution of Problem (6).
Suppose that (xi^) is an optimal solution of Problem B.
Lemma 3.1 Let x ¯ be an optimal solution of (P).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com