I am pretty sure this doesn't work, unless I miss something: your algorithm will not start moving in most cases.
Assume your variable x
is taken in R^n
.
A polyhedron of the form Ax <= b
is contained in a 'maximal' affine subspace P
of dimension p <= n
, and usually p
is much smaller than n
(you will have a lot of equality constraints, which can be implicit or explicit: you cannot assume that the expression of P
is simple to obtain from A
and b
).
Now assume you can find an initial point x_0
(which is far from being obvious, btw) ; there is very little chance that the direction of the gradient c
is a feasible direction. You would need to consider the projection of the direction c
on P
, and this is very difficult to do in practice (how would you compute such projection?).
Then, what you want in your step (3) is not the normal direction of the hyperplane you reached, but again its projection on P
(visualize the polyedron as a 2d polyedron embedded in a 3d space can help).
There is a very good reason why barrier functions are used in the interior point methods: it is very difficult to describe in practice the geometry of the high-dimension convex sets from the constraints (even simple ones like polyedrons), and things that "seems obvious" when you draw a picture on a piece of paper will not usually work when the dimension of the polyedron increases.
One last point is that your algorithm would not give the exact solution, whereas the simplex does in theory (and I read somewhere it can be done in practice by working with exact rational numbers).