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The phrase "are the optimal solution" is correct and usable in written English.
It can be used when discussing the best or most effective answer to a problem or situation.
Example: "After analyzing all the options, we concluded that these strategies are the optimal solution for increasing productivity."
Alternatives: "represent the best solution" or "are the most effective answer."
Exact(7)
Collaborative efforts, among academia, industry, and regulatory agencies, are the optimal solution for ensuring a solid future for this promising manufacturing technology.
For instance, z∗ and K∗ are the optimal solution of (5).
Moreover, ({boldsymbol {p}^, t_{s}^) are the optimal solution of maximizing L AF (p,t s,Z 1) for given (boldsymbol {Z}_{1}^).
Many sustainable and adaptable design solutions, such as recycling/up-cycling, have been explored for the given problem, but researchers concluded that neither are the optimal solution [2].
Therefore, by the weak duality, we see that (v(P =v(D)) and (bar {u}=(bar{u}_{1},bar{u}_{2})in U) and (y^{ast}in Y^{ast}) are the optimal solution of (D).
Thus, by the weak duality, we see that (v tilde{P})=v(D)) and (bar {u}in U), (y^{ast}in Y^{ast}) are the optimal solution of (D).
Similar(53)
Suppose (boldsymbol {X}_{k}^,forall k), are the optimal solutions to the SDP relaxation of (23).
Then (46) and (47) are the optimal solutions of the problem (14).
α iy, i = 1, ⋯, l, y ≠ yi are the optimal solutions of the problem (8) ~(10).
In the long run, education, prevention, and a therapeutic vaccine are the optimal solutions for reducing the stigma associated with both EVD and HIV.
These are the optimal solutions in both cases and suggest a high level of flexibility in the way that chimpanzees use their spatial memory.
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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