I would like to get some help about the next problem: Is function $f(\theta)=\sum_{i=1}^{k}\sum_{j \neq i}||\theta_i||_2^2 ||\theta_j||_2^2$ a quasi-convex function? Where $\theta_i \in R^n$, $||\cdot||^2_2$ is the square of 2-norm, $k \geq 2$.
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1What have you tried and where are you stuck? – Ѕᴀᴀᴅ Mar 21 '19 at 11:23
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A sum of convex functions is convex. This fact should simplify the task. – nicomezi Mar 21 '19 at 11:43
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Because of a sum of convex functions is convex, I want to prove the convexity of function $f(\theta)$ by means of decomposition. However, $||\theta_i||^2_2||\theta_j||^2_2$ is not a convex function, so I can't prove the convexity of function $f(\theta)$, but function $f(\theta)$ is probably a quasi-convex function, but I don't know how to prove it. – Dajiang Lei Mar 21 '19 at 12:24