I have been using An Introduction to Optimization by Chong and Zȧk. In it there is a chapter about standard form and converting to it. This chapter makes sense to me. I think that I understand the idea of adding variables and whatnot. Now, I am reading the chapter on arbitrage detection in the foreign exchange market in the book Optimization Methods in Finance. The linear program is given as:
$$max \ \ y_1$$
$$s.t. \ \ y_k = \sum_{i=1}^5 a_{ik}x_{ik} - \sum_{j=1}^5 x_{kj}, \ \ k = 1,...,5$$ $$x_{ij} \geq 0$$$$y_k \geq 0$$$$y_1 \leq 1$$
I can understand the reason that this is the linear program, but I am not sure how one would go about converting it into the standard form so that it could be solved via simplex method.
My initial thoughts were that I could start by changing it to a minimization problem. This flips the inequality on the constraint. From here I got very confused. My understanding is that I need the constraint to be greater than $0$ (not $1$). Perhaps then I should deal with that in the summation above, but how? Additionally, I have always seem $x$ as a vector in these problems. However, now $x$ is a matrix and I am unsure of how I would write this in matrix form. It seems to me that both $A$ and $x$ would be $5\times5$ matrices. I am thinking then that $\sum_{i=1}^5 a_{ik}x_{ik} = $ the diagonal of $x^TA$. How do you recover that?
To recap I am unsure of how to handle the first summation and the constraint on the objective function when converting to standard form. I get the sense that I am either making a mistake or over complicating things somewhere. Any help would be greatly appreciated!