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Step 2. If ( x k, y k, λ k ) is feasible to problem (4), stop.
If ( x ¯, y ¯, λ ¯ ) is feasible to problem (4), then it is also a globally optimal solution to problem (4).
Furthermore, every optimal solution of problem (7), which is feasible to problem (4), is an optimal solution for this problem.
(7) Following Proposition 2.1, we see that for ((x",y")), there exists (lambda inOmega) such that ((x",y",lambda")) is feasible to problem (4).
As I R ⊆ T R, ( x, y, λ ) is also feasible to problem (7), thus contradicting the optimality of ( x ¯, y ¯, λ ¯ ).
Suppose that ((x,y,lambda,w)) (feasible to problem (4)) is not a Pareto optimal solution to problem (4), then there exists a feasible point ((check{x},check{y},check{lambda },check{w})) to problem (4) (also feasible to problem (9)), satisfying Cx+C'yleq Ccheck{x}+C'check{y}, (12) which contradicts (11).
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From 3.3, 3.4 and 3.5, it is clear that points of the form E ( x ¯ ) + λ d are feasible to the problem P E for each λ ∈ ( 0, δ ), where δ = min ( δ 1, δ 2, δ 3 ).
All feasible solutions to problem (P) are ε-feasible.
Problem (13) to (15) may be infeasible; however, when it is feasible, this heuristic will provide a feasible solution to problem (7) to (9) which is used as the starting point of the algorithma.
Suppose that ( x ∗, y ∗, u ∗ ) is a feasible point to problem (5).
In this subsection, we propose a simple procedure to obtain a feasible point to problem (9 15) from the dual solution found with Algorithm 1.
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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