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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 $x_i \geq 0 \, \forall i$.

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  • $\begingroup$ Are you using the $2$-norm? $\endgroup$ – Rodrigo de Azevedo Apr 2 '17 at 12:38
  • $\begingroup$ @RodrigodeAzevedo yes $\endgroup$ – michaelsnowden Apr 2 '17 at 16:51
  • $\begingroup$ Your goal is to find a point that satisfies the constraints? $\endgroup$ – Rodrigo de Azevedo Apr 2 '17 at 16:55
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    $\begingroup$ @RodrigodeAzevedo Yeah, but I don't know what I was thinking when I made this question because I could just take any nonnegative vector $x$, then let $x_0=x/||Ax||$. $\endgroup$ – michaelsnowden Apr 2 '17 at 17:00

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