I understand the theoretical foundation of the idea, that when I want to find the minimum of a linear function over a bounded polyhedron that I kind of only need to look at the vertices and if I am located at a vertice where all outgoing edges are not "better" for the value of my objective function then I am done because for a linear function on a convex set a local minimum is also a global one. So it is reasonable to just walk over all the vertices until we find a locally minimal one.
Right now I am working with the book "Linear Programming" by Chvatal and I understand the Simplex Method the way he is explaining it. This is explained in a very intuitive more algebraic way in terms of manipulating a system of linear equations and at the same time not making the basis variables negative. He calls this representation a dictionary.
The thing I am having trouble with is that I just can't connect the two ideas.How does the Simplex Algorithm relates to the idea of "walking along edges"?