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For a constrained optimization problem, in general the KKT conditions are a necessary but not sufficient condition for a point to be the local maxima/minima of the objective function.

Is it always true that if the point is not a local maxima/minima, it must be a saddle point of the objective function?

For simplicity, we can assume that all functions involved are differentiable at least once. But I don't want to make any assumptions about second derivative or convexity.

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  • $\begingroup$ Math@SE is a better choice of venue for this question. $\endgroup$
    – whuber
    Nov 30, 2011 at 23:00

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No, it isn't always true. You need convexity of both the objective function $f(x)$ and the inequality constraint functions $g(x) \le 0$ at the point $\bar{x}$ where the KKT conditions are satisfied, and the equality constraints $h(x)=0$ need to be affine (both constraint conditions only for those constraints that are active) to ensure that it is.

Nonlinear Programming: Theory and Algorithms by Bazaraa, Sherali, and Shetty has a section on this in the 2nd Edition.

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