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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.