Let $X$ be the set of $n$ by $n$ matrices with elements $0\leq x_{ij}\leq 1$ for all $(i,j)$ and such that $\sum_{j=1}^n x_{ij} \leq \bar{x}$ for all $i$ and where $\bar{x}$ is some number such that $0<\bar{x}<1$.
Define the Leontief inverse as $L(x)=(I-x)^{-1}$, note that given the restrictions on $X$ the matrix $(I-x)$ is invertible for all $x\in X$. I am interested in the following maximization problem $$ \max_{x\in X} \alpha^TL(x)\,\beta $$ where $\alpha$ and $\beta$ are column vectors.
I can take the first-order conditions of this problem but I don't know if they are sufficient. So a solution to the first-order conditions is no guarantee of optimality.
My questions are:
- Is there a clever way to solve this problem?
- Can we show that first-order conditions are sufficient?
- Is there a reference that would tackle these types of problems?
Many thanks!