Multivariate function convexity example

Question

Check whether the following polynomial functions are convex or concave. $$\begin{aligned} g(x,y,z) &= 2x^2y^2 - y^4 +y^2-2yz+z^2+x^2\\\\ h(x,y,z,t) &= x^2-2y^2 -z^2+t^2-xy+xz-2yt+xt\end{aligned}$$

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I started with the Hessian on $g$ to see if it is positive (or negative) semidefinite: $$ \begin{pmatrix} 4y^2+2 & 8xy &0 \\8xy & 4x^2-12y^2+2 & -2 \\ 0 & -2 & 2 \end{pmatrix} $$ First minor is just $4y^2 + 2$ which is non negative $\forall{x,y,z \in \mathbb{R}}$. I got stuck on the second minor: $$ \begin{pmatrix} 4y^2+2 & 8xy \\8xy & 4x^2-12y^2+2 \\ \end{pmatrix} $$ here the determinant is: $$ (4y^2+2)(4x^2-12y^2+2)-64x^2y^2 = 16x^2y^2-48y^4+8y^2+8x^2-24y^2+4-64x^2y^2 = \\ = 8x^2+4+y^2(16x^2-48y^2-64x^2-24+8) = 8x^2+4-16y^2(3x^2+3y^2+1) $$ this however, is negative when $x=1, y=1, z \in \mathbb{R}$. This makes the minor method fail (as I have first a plus sign then a minus sign - If it was the other way, with the minus and then the plus, then I could continue to the third minor). What else could be done here? Does that mean the function is neither convex not concave?

When it comes to the second example $h(x,y,z,t) = x^2-2y^2 -z^2+t^2-xy+xz-2yt+xt$, the Hessian looks as follows: $$ \begin{pmatrix} 2 & -1 & 1 & 1 \\ -1 & -4 & 0 & -2 \\ 1 & 0 & -2 & 0 \\1 &-2 & 0 & 2 \end{pmatrix} $$ Here the minors equal to (in order from matrix $1 \times 1$ to $4 \times 4)$ = $2, -7, 22, 48$. This again makes the minor method fail. I also checked the eigenvalues with Wolfram Alpha and they seem hard to compute and also have two negative and two positive eigen values(which I presume does not conclude the result).

Again, does that mean the function is neither convex not concave?

I appreciate your help.

Answer 1

Here are some results:

• a symmetric matrix $A$ is positive semidefinite $\iff$ every principal minor is non negative (not only the leading ones!)

Hence the first matrix is not positive semidefinite at some point, as you proved.

• a symmetric matrix is negative semidefinite $\iff$ every principal minor times $(-1)^o$ is non negative ($o$= order of the minor)

Hence the first matrix is neither negative semidefinite (in the last minor, take $y=0,x=1$).

The same goes for the other matrix.

• A function is convex (concave) $\iff$ the Hessian matrix is positive (negative) semidefinite at every point

Hence the first function is neither concave, nor convex, and the same goes for the second.