This is sort of a followup to this question, but with nonlinear constraints instead of a nonlinear objective function. $$ \begin{align*} \text{Find } x \text{ that maximizes } & (Ax)^{\top} y \\ \text{Subject to } & ||Ax|| = 1 \\ & x_i \geq 0 \; \forall \: i \in \{1\dots n\} \\ \text{Where } & A \in \mathbb{R}^{d \, \times \, n} \\ & x \in \mathbb{R}^n \\ & y \in \mathbb{R}^d \\ & A_{i,j} \geq 0 \, \forall \, i \in \{1\dots d\}, j \in \{1\dots n\} \\ & y_i \geq 0 \, \forall \, i \in \{1\dots d\} \\ & ||y|| = 1 \\ & ||a_i|| = 1 \, \forall \text{ column vectors } a_i \text{ in } A \end{align*} $$
I am using a software package which solves convex nonlinear optimization. However, I need a starting point $x^0$ that satisfies the constraints.
So, my problem is: find a vector $x^0$ such that $||Ax||=1$ and $x_i \geq 0 \, \forall i$.
