The Non-linear Case
In non-linear programming, the constraints are not necessarily linear. Nonetheless, many of the same principles apply.
To ensure that the global maximum of a non-linear problem can be identified easily, the problem formulation often requires that the functions be convex and have compact lower level sets.
This is the significance of the Karush–Kuhn–Tucker conditions. They provide necessary conditions for identifying local optima of non-linear programming problems. There are additional conditions (constraint qualifications) that are necessary so that it will be possible to define the direction to an optimal solution. An optimal solution is one that is a local optimum, but possibly not a global optimum.
Read more about this topic: Duality (optimization)
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