Do you have any reason to believe it *is* convex? In the space of nonlinear problems, convexity is the exception, not the rule. Convexity is something to be proven, not assumed.

Consider the scalar case; that is, $m=n=1$. Then the problem is
$$\min_{y,w\geq 0}(x-yw)^2=\min_{y,w\geq 0}x^2-2xyw+y^2w^2$$

The gradient and Hessian of $\phi_x(y,w)=x^2-2xyw-y^2w^2$ is
$$\nabla\phi_x(y,w)=\begin{bmatrix} 2yw^2 - 2xw \\ 2y^2w - 2xy \end{bmatrix}$$
$$\nabla^2\phi_x(y,w)=\begin{bmatrix} 2w^2 & 4yw - 2x \\ 4yw - 2x & 2y^2 \end{bmatrix}$$
The Hessian is not positive semidefinite for all $x,y,w\geq 0$. For example,
$$\nabla^2\phi_1(2,1)=\begin{bmatrix} 2 & 6 \\ 6 & 8 \end{bmatrix}, \quad
\lambda_{\min}(\nabla^2\phi_1(2,1))=-1.7082$$