I have a linear system of the form $A x = b$ where $A$ and $b$ are known, $A$ is "square", and $\lvert b \rvert_1 = \lvert x \rvert_1 = 1$. Unfortunately, I am working in a framework that only gives me access to a dot product, not the individual vector or operator elements. The only way I have found to solve this problem is by optimizing the quantity:
$$ d^2 = \lvert b - A x \rvert_2 = \langle b, b \rangle - 2 \langle b, A x \rangle + \langle A x, A x \rangle $$
subject to the normalization $\lvert x \rvert_1 = 1$. This tends to work, but it becomes computationally challenging to compute the last term because of the double operator. (Forming the vector $A x$ is actually very difficult, so I have to compute the inner product "all at once" to use this toolset: $\langle A x, A x \rangle = x^T A^T A x$.)
Now, if this were an $L^2$ constrained problem, I could use ${d^\prime}^2 = 1 - \langle b, A x \rangle$ as the "overlap" between states $b$ and $A x$, which makes more sense to me coming from a quantum mechanics perspective, yet, because $x$ is only $L^1$ normalized, I would still have to ensure that $b$ and $A x$ are $L^2$ normalized before optimizing this way -- which takes me back to computing $\langle A x, A x \rangle$ at every step, which is, again, too expensive. The operator $A$ does guarantee that the $L^1$ norm is preserved, but the $L^2$ norm is, in general, not.
I can't think of another approach to solve this equation using only inner products. Have I missed something obvious? Or is there literature out there about solving this problem? Google has yet to be helpful on this issue, because "$L^1$ optimization" is a common search and does not seem to be related to my question, since $A$ in that problem is apparently sparse. Please explain any answers at an advanced-undergrad-core-math level, because I am not a mathematician.
