Erwin Kalvelagen

Just reduction in nz count? Or are there also other things going on.

Somewhat difficult to read (tex does not render), but it looks ok. Now you have a unit starting basis. Phase I: Plugin the correct objective, start iterating. When finished make sure all artificials are non-basic. Then drop the artificials. and start Phase II with original objective. Your book should explain this in great (even mind numbing) detail (most likely with spelled out examples).

It will solve the augmented problem using standard simplex (small wrinkle: we need to pivot out the artificials before terminating). Then remove the articifials and start Phase II. You book should have a chapter on this.

No Gauss elimination needed yet. To find an initial basic feasible solution most textbooks will augment the problem using these artificials and then start a "phase 1" simplex method. Again in practice simplex methods work very differently.

In practice software will do this for you, so you don't have to worry about that. But assuming you want to solve this using a textbook "tableau" method, you need to look up in your book a technique called "artificial variables". (Note that practical revised simplex methods don't use artificials it is merely used to keep students busy).

Sure.

Very good.

Sorry I don't of no tool for that. As a practitioner I am not often worried about producing a dual formulation of a model. The easiest way to check is solve both primal and dual models. The primal model gives x plus duals y. The dual model gives y + duals x.

The concept of a slack should be explained in a very early chapter.

Chosen values and actual values are not terms I am familiar with. Slack is difference between the lhs (i.e. a'x for a given x) and the rhs (i.e. b). I.e. Ax + s = b.

For normal simplex method this gap concept is not important. It is more of a theoretical thing. For some interior points that work on the primal and the dual it is a measure that is really used. It has for certain implications in the economic interpretation of optimal solutions (think of sensitivity analysis).

I believe gap=0 does not imply optimality. Things can be infeasible or unbounded.

I am not following you here. I can recommend Vanderbei's Linear Programming book. It has many more details than can be covered using stackexchange.

Checking if the solution is zero? That has no meaning in this context. I think you mean checking if a solution is optimal. There must be a paragraph or even a whole chapter in your book about optimality conditions for an LP. Yes, optimality implies no gap.

You have to be precise with "solution". It can mean feasible solution or optimal solution depending on the context. Optimal means gap=0.