William Karush obtained necessary conditions for inequality-constrained optimization in his1939 master's thesis. The result was independently rediscovered and published by Harold W. Kuhn and Albert W. Tucker, producing the name Karush-Kuhn-Tucker conditions.[ref]
https://www.math.u-szeged.hu/~hajnal/courses/MSc_Optimalizalas/2014/Cottle%20-%20William%20Karush%20and%20the%20KKT%20theorem.pdf. Richard W. Cottle, William Karush and the KKT Theorem, Documenta Mathematica extra volume2012, printed pp255–257
Their distinguishing idea is that an inequality constraint can be active or inactive: a nonnegative Lagrange multiplier pairs with its constraint through complementary slackness. A constraint qualification is needed for the usual necessary-condition theorem; a stationary Lagrangian equation alone is insufficient.
William Karush is best known for his contributions to mathematical optimization and for formulating the Karush-Kuhn-Tucker (KKT) conditions in the context of constrained optimization problems. The KKT conditions are a set of necessary conditions for a solution in nonlinear programming to be optimal, especially when dealing with inequality constraints. These conditions are fundamental in various fields, including economics, engineering, and operations research, as they provide a method for solving optimization problems.
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