Title: Δύο σημαντικοί σταθμοί στην ιστορία του γραμμικού προγραμματισμού : Simplex and Karmarkar’ s Method
Abstract:Linear programming plays an important role in our lives. Due to revolutionary
developments both in computer technology and algorithms for linear optimization, pro-
blems that could not be solved yea...Linear programming plays an important role in our lives. Due to revolutionary
developments both in computer technology and algorithms for linear optimization, pro-
blems that could not be solved years ago, due to a required computational time of one
year, say, can now be solved within some minutes. Until 1984, the method of choice for
solving linear optimization problems was the Simplex Method. George Dantzig’s develop-
ment of the Simplex method is one of the most popular and most important methods of
finding the solution to the linear programming problems. Since the initial formulation in
1947, this method has been constantly improved. The Simplex method proceeds from one
vertex of the feasible region defined by the constraints of problem to a neighboring ver-
tex. Klee and Minty showed that, in the worst case, the Simplex method has exponential
complexity in the size of the problem. The question that then arises is whether there is
another algorithm for linear programming that has polynomial complexity. This question
was answered by Karmarkar in 1984. He produced a polynomial-time algorithm called the
projective algorithm for linear programming that ran in much better time. Karmarkar’s
algorithm uses interior points of the feasible region to approximate the optimal solution.
Consequently, this master thesis was aim to study Simplex method and the interior
point method (Karmarkar Method), the principle idea behind the methods and the theory
that is used in the development of the methods. In the end, the Karmarkar’s algorithm is
compared to the simplex method by showing a solution to a linear programming problem
obtained by the both two method.Read More
Publication Year: 2020
Publication Date: 2020-02-18
Language: en
Type: dissertation
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