An improving procedure of the interior projective method for linear programming
Introduction
We consider the linear programming problemwhere with and .
In this paper, we present a modification in the phase two of Karmarkar’s algorithm presented in [4]. This modification reduces the number of iterations and the calculated times.
In fact, we try to avoid the calculation of the projection used in the algorithm at each iteration. To do that we write the problem (P) aswhereand
On the other hand, we can consider the problem (F) as an optimization problem of the following linear programming problem (via artificial variable):where is a fixed arbitrary point in .
The problem (AP) has a trivial strictly feasible solution (a, 1), and we have the following important results: Theorem 1 x∗ is a solution of (F) if and only if (x∗, ε) is an optimal solution of (AP) with x∗ ⩾ 0 and ε sufficiently small.
Section snippets
Resolution of the problem (AP)
In this Section, we are interested in the resolution of the problem (AP) using the Ye–Lustig algorithm [1].
It is known in projective method that the cost of the calculus is dominated by the projection, for instance, see [2], [3], [4], [5], [6]. To avoid this inconvenience, we propose a slight modification in phase two of Ye–Lustig method to reduce the number and the cost of calculation at each iteration.
Note that the problem (AP) can be written aswhere
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Examples of fixed sizes
Example 01 The optimal value is z∗ = 0. The exact optimal solution is Example 02 The optimal value is z∗ = 0.67. The exact optimal solution is . Example 03 TheMethod x∗ z∗ Iter Time Ye–Lustig algorithm 0.0000 04 0:0:0:31 Modified algorithm 0.0000 01 0:0:0:16 Method x∗ z∗ Iter Time Ye–Lustig algorithm 0.6666 07 0:0:0:47 Modified algorithm 0.6700 03 0:0:0:31
Conclusion
In spite of the mathematical development in the domain of the linear programming, a lot of problems remain to be developed. For this, in our survey, we treated a theoretical and numerical survey of a modification done to the level of the phase two of the Ye–Lustig algorithm. These propositions are confirmed by interesting numerical experimentations.
Thus, our survey allowed us to improve the numeric behavior of the Ye–Lustig algorithm while reducing the cost of iteration for the linear
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