Consider a 3 x 3 matrix A for this example
$$A=\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{pmatrix}$$
Let's assume,in general, that the $$\lambda_i, i\in \text{[0,1,2,3..,n]}$$ are the Eigen-values, if the matrix is of size {n x n}
I want to maximize the sum of all these eigenvalues $$\max\text[\sum_{i=1}^n\lambda_i]$$
Following conditions are to be met:
1. Only values which are 0 can be changed, values which are 1 remain fixed
2. Only specific discrete changes are allowed, that is
$$a_{ij}=[-1,0,1], \text{where i=rows and j=column}$$
There are works which discuss optimization problem related to second smallest eigen values https://www.sciencedirect.com/science/article/pii/S000510981830308X
But this work goes beyond my ability to understand. Besides there is another question is StackExchange linked here that talks about the similar query but it doesn't talk about constraints and also it is talking about the linear sum of matrices.
So how do I address this problem? Any help, please.
P.S Please, if you think my question is wrong and not reasonable then kindly explain to me the reason and I will remove it. Please don't downvote otherwise I won't be able to ask questions in the future as this is first one.
this is more of a comment, which I am not yet allowed to make, so here it is:
You might want to consider Boyd's lecture. In the 4th lecture, he is talking about Minimizing spectral radius of nonnegative matrix, which is a geometric program:
$$ \text{maximize } λ$$ $$ \text{subject to } \sum_{j=1}^n A(x)_{ij}v_j/(λv_i) \leq 1, \forall i = 1, . . . , n$$
which I hope is the same problem or at least similar and it gives you ideas.