Conclusion
It was the duality theorem for linear programming-that is, a purely theoretical result-that sparked the interest of Kuhn and Tucker. It was the duality theory they wanted to extend to the general (quadratic) nonlinear case. It is in this respect that I find the development of the duality theorem in linear programming so crucial for the emergence of nonlinear programming.
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Kjeldsen, T.H. The emergence of nonlinear programming: interactions between practical mathematics and mathematics proper. The Mathematical Intelligencer 22, 50–54 (2000). https://doi.org/10.1007/BF03025283
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DOI: https://doi.org/10.1007/BF03025283