Difference between simplex method and revised simplex method
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We use one of the key features of LP: Strong duality. As Paul said, the simplex starts with a feasible but not optimal solution. Moreover, we start with an extreme vertex of the feasible region. At each step we move to a extreme vertex “adjacent” to the current one in order to improve the objective value. We do so until it is possible and we prove optimality by constructing a dual feasible solution.
Now, let us look at dual simplex. We start with a superoptimal solution which of course has to be infeasible. At each step we want to make it closer to a feasible region. In a way, analogous to simplex we start with a vertex that is at the intersection of some constraints. At each step we move closer to the feasible region by making less constraints violated at the expense of reducing objective value. We repeat until we move to the feasible region and use strong duality to prove optimality.
Now, let us look at dual simplex. We start with a superoptimal solution which of course has to be infeasible. At each step we want to make it closer to a feasible region. In a way, analogous to simplex we start with a vertex that is at the intersection of some constraints. At each step we move closer to the feasible region by making less constraints violated at the expense of reducing objective value. We repeat until we move to the feasible region and use strong duality to prove optimality.
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