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Let $f$ be a multivariate function $f:\mathbb{R}^n \rightarrow \mathbb{R}$.

In order to check its convexity in the $\mathbb{R}^n$ domain, we can check whether its Hessian is semidefinite positive: $$ \mathbf{z}\mathbf{H}\mathbf{z}^T \geq 0 \quad \forall \mathbf{z} \in \mathbb{R}^n $$

But I al only interested in a given region. For instance, I want to check convexity for reals between 0 and 1. How should I adapt the above equation?

My guess is that I should check it for $\mathbf{z}$ that are in that region of interest, but I don't find any reference confirming that.

For instance, I have $$ f(x, y) = (1-x)(1-y) $$ 3-D plot

whose Hessian is: $$ H = \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix} $$

And then: $$ \mathbf{z}^T \mathbf{H} \mathbf{z} = 2z_1z_2 \geq 0 \quad \forall z_1, z_2 \in [0,1] $$

Thus, I would say that this function is convex. But I'm confused because, in the plot, I rather see that it is convex in some directions and concave in others.

alberto
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  • Did you look at my solution at all? I gave you necessary and sufficient condition.

    the $z$'s are directions, and they have nothing to do with points on Domain!

     – Red shoes Jun 05 '17 at 08:55

1 Answers1

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First of all, your region must be convex. So taking into account a convex region is given. if region is open then you have same criteria .

in general, even your region is not open, (say $C$ is your region) then you must have $$\mathbf{z}\mathbf{H}\mathbf{z}^T \geq 0 \quad \forall \mathbf{z} \in \text{aff(C)-aff(C)}$$

For those who have problem with $aff$ : equivalently let $x_0$ be arbitrary point in $C$ then above condition can be equivalently written as $$\mathbf{z}\mathbf{H}\mathbf{z}^T \geq 0 \quad \forall \mathbf{z} \in \text{Span}(C-x_0)$$

Red shoes
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