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I have a concave function $f: \mathbb{R}^n \to \mathbb{R}$ with convex functions $g_i: \mathbb{R}^n \to \mathbb{R}$ and affine functions $h_j:\mathbb{R}^n \to \mathbb{R}$. I am interested in both calculating the minimum and the maximum of $f(x)$ under the constraints that $g_i(x) \leq 0$ and $h_j(x) = 0$ for all $i,j$. The KKT conditions give necessary conditions and, as is stated on Wikipedia, also give sufficient conditions for the maximization problem in this specific case. Can the KKT conditions be modified in the case of the minimization problem to also give sufficient conditions? Or is there some different method applicable in this case?

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