I'm working on a problem where I have a (vector) linear recurrence relation of the form
$$ a_{n+1} = \lambda \circ a_n+b_n $$
I need to solve the following optimization problem:
$$ \min\limits_{b_n} \sum_{n=1}^\infty \|b_n\|_1 \\ \text{subject to: } \|\sigma\circ b_n\|_1 \leq B \text{ for all } n\in\mathbb{N} \\ \rho_n^m \leq \sigma^m b_n^m \text{ for all } n,m\in\mathbb{N},1\leq m \leq M\ \\ a_N = 0 \text{ for some } N\in\mathbb{N}$$
Here, $a_0$, $\lambda$, $\sigma, \rho_n \in \mathbb{R}^M$ and $B \in \mathbb{R}$ are given parameters of the system, while $b_n \in \mathbb{R}^M$ and $N\in\mathbb{N}$ are unknown, and $a_n\in\mathbb{R}^M$ is calculated via the recurrence. $\rho_n = \rho(a_n)$ is an increasing function of $a_n$ which is zero whenever $a_n = 0$. $\| \cdot \|_1$ denotes the $L_1-$norm and $\circ$ denotes the Hadamard (entrywise) product. I don't think it matters, but $\sigma = \lambda^{\circ - 1}$ is the entrywise reciprocal of $\lambda$.
Is there any literature on how to approach a problem like this $-$ even computationally? I have done some searching but haven't come up with anything that works. I have experience in graduate level optimization but I have never come across a problem like this. I can see randomized optimization possibly getting near solutions, but I want something a bit more guaranteed to produce a true minimum. I greatly appreciate any help or references for problems even somewhat similar to this one.
The context of this problem is developing an optimal strategy for paying off a series of debts with different initial balances, interest rates, minimum payments, and a fixed monthly budget, though I'm more interested in the emergent problem than the application itself.