What is the difference between primal simplex and dual simplex

We then have to choose between the cyan and the white dots, but, as the white dot has a worse dual value, we cannot go there. The cyan dot is feasible, thus it is the dual optimum. Well, it is useful for our understanding of the primal simplex. The dual simplex is very useful if our initial point is non-feasible, in which case the primal simplex is useless and we actually need to solve a subproblem called phase I, in order to find a feasible primal point.

In both cases, we actually add new constraints to the primal linear program that make our current base infeasible. We can therefore apply the dual simplex to re-optimize.

The dual simplex is actually the reason why the variants of the simplex method work better than the interior point method on several important classes of problems. Just like for the primal simplex, the dual simplex performs well in case of non-degeneracy.

However, if the primal simplex reaches a degenerated point, it may have a lot of trouble to get out of it because of the many bases that can represent the degenerated point. Just like in my article on duality, we moved the primal blue constraint to create a primal degenerated solution. As a result, the primal green dot is actually the merging of the primal orange, white and pink dots.

Yet, moving a primal constraint does not affect dual constraints. It does, however, modify the dual objective function, thus the level curves. Since the orange, white and pink primal base correspond to the same point, their have the same objective value, which implies that dual orange, white and pink dots are on the same dual level curve.

The degenerated primal point could actually be associated with the dual polyhedron defined by the convex hull of the three dual dots. This concept of dual polyhedron is very important for column generation. In our case, things remain simple because there is no much degeneracy and the dimension is very low. Thus, the path we used earlier is still valid and is actually the one taken. However, in case of more degeneracy and higher dimension, the dual polyhedron could be enormous, and the dual simplex would be travelling all over the dual polyhedron.

It may take a very long time for us to leave this dual polyhedron. In cases of higher dimension and higher degeneracy the number of such pivots could be exponential. The simplex methods are amazing methods that exploit the structure of linear programs.

In order to solve even larger problems, a better management of the bases must be done. However, they may be slow in cases of degeneracy which actually are very common in many classical modelings. Other technics are being developed to deal with degeneracy including the dual variable stabilization read this article of mine or the improved primal simplex.

Overall, the main application of linear programs concerns integer linear programming , which can model a very large range of problems. Many additional technics need to be added though, including the branch and bound , the cutting plane method , or, more recently, the integer simplex. Much research still needs to be done to improve integer linear programming.

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Save my name, email, and website in this browser for the next time I comment. An important difference between the dual simplex method and the primal-dual method is that the primal-dual algorithm does not require a dual feasible solution to be basic. This algorithm is useful specially for solving minimum fuzzy cost flow problem in which finding an initial dual feasible solution turns out to be a trivial task.

Primal simplex for LP gives the same result as dual simplex for its dual LP? But if both methods need standard form min objective function then how do I know if it's primal or dual? In both cases I need an identity matrix, so artificial variables would be the same, isn't that right? I'll look into the algorithms that computers use to determine whether it is better to do Simplex or Dual Simplex--I'll keep you updated. Show 1 more comment. DalyaG DalyaG 1 1 silver badge 3 3 bronze badges.

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Create a free Team What is Teams? Learn more. When should I use dual Simplex over primal Simplex? Ask Question. Asked 2 years, 5 months ago. Active 2 years, 5 months ago.

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