Also, the reference I alluded to is the following: ``` @book{book, author = {Kotz, Samuel and Kozubowski, Tomasz and Podgorski, Krzysztof}, year = {2001}, month = {01}, pages = {48}, title = {The Laplace Distribution and Generalizations}, isbn = {0-8176-4166-1}, doi = {10.1007/978-1-4612-0173-1_5} } ```

Oh, well, basically we have $\lambda = b$.

@Dieter Kadelka You are perfectly right, in fact I initially learned about these in a class named "Stochastic Processes", so in makes so much sense, but since I'm translating everything in my head as I write, I simply didn't make the connection right away.

If by linear programming you mean something else than Newton's method, I am very much interested. As I need to ultimately code the result, anything easier and/or more efficient than Newton's method will be received with open arms. Would you mind elaborating a bit? Thank you sir.

I tried with a $10\times{}10$ matrix using Excel's solver and it took about half an hour. That's much too slow for my purpose. I doubt the order of the $\boldsymbol{P}$ matrix will ever exceed $n = 10$ states though. In fact, I won't allow it too.

Whether I minimize the trace or the sum the squares of all elements on the diagonal, my current issue (under the Newton's method approach) is that my "main" unknown is a matrix ($\boldsymbol{P}$, whereas the Lagrange multipliers are scalars, and I'm confused about how to wrap those 3 under a common structure so that I can recursively update it and converge toward a solution. Should I add rows and columns to the unknown $\boldsymbol{P}$ matrix so that it incorporates $\lambda_1$ and $\lambda_2$ on the main diagonal?

I totally agree with the EDIT section, this is something I have observed in practice using Excel's solver. What remains encouraging is that through this scheme, $P_{11}$ will nonetheless be less than $\pi_1$, given that $\pi_1 \geq 0.5$, ergo the diagonal has still been minimized in some way.