Consider the task of minimizing $c^Tx$ subject to the constraint that $Ax \leq b$.
I had a couple of questions in relation to the simplex algorithm (applied to this problem):
How does one initialize the algorithm (i.e., identify a basic feasible solution/vertex).
Given a current vertex of $\{x:Ax \leq b\}$, how does one identify an "adjacent" vertex that possibly improves the objective.
It appears that most books/texts on the simplex algorithm discuss just the standard or canonical forms of LP. My question is how can one solve the above LP directly without having to change to the standard/canonical form (which in fact is the dual of the standard LP).
You can convert your problem to any of the other forms. The easiest thing is probably to convert it to augmented form (or slack form, if you will), since the Simplex algorithm is often described for this.
Minimizing $c^Tx$ becomes maximizing $-c^Tx$.
The constraint $Ax\leq b$ becomes $Ax + s = b, s\geq 0$, where $s$ is a new vector of slack variables.