# How to get the minimum and maximum of one convex function?

Condition: $h,f\in \mathbb{C}^{N\times1}, \text{where}f =\hat{f} + e \text{ and } e^H e \leq 1,\ \ \ Q=h^Hff^Hh$. The Lagrangian function of $Q$ is $\mathcal{L} = h^H(\hat{f} + e)(\hat{f} + e)^Hh + \lambda (e^H e - 1)$ which is convex, where the $\lambda \geq0$. In my opinion, we can only get the minimum of $Q$ using the Karush-Kuhn-Tucker (KKT) conditions.

But I found that one author got the minimun and maximun of the function, why? The paper is here, the process is omitted. I can compute the minimum, but the maximum is got only when $\lambda < 0$, which violates the limits $\lambda \geq0$. Maybe there is anything I did not understand.

Could anybody tell me what's wrong with my thoughts. Thanks!!

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I cannot see that the author maximises $Q$ anywhere. From what I can see, an upper bound for $T_2$ is found (note that the optimisation in (16) is constrained by the requirement (8)). – Mårten W Oct 1 '13 at 15:20
Thanks for your reply! Actually both $T_1$ and $T_2$ are constrained by (8). I think that $T_1$ and $T_2$ have the same forms in essence. The author said he got the result with similar technique,i.e., KKT conditions. But I can not get it using KKT. – begforopt Oct 1 '13 at 15:31