An improving procedure of the interior projective method for linear programming

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Abstract

We propose in this study, a practical modification of Karmarkar’s projective algorithm for linear programming problems. This modification leads to a considerable reduction of the cost and the number of iterations.

This claim is confirmed by many interesting numerical experimentations.

Introduction

We consider the linear programming problem(P)minctx=z,Ax=b,x0,where ARm×n with rg(A)=m<n,c,xRn and bRm.

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) as(F)Ax=b˜,x0,whereA=ActR(m+1)×nandb˜=bzRm+1.

On the other hand, we can consider the problem (F) as an optimization problem of the following linear programming problem (via artificial variable):(AP)minλ,Ax+λq=b˜,(x,λ)0,where q=b˜-Aa,a is a fixed arbitrary point in R+n.

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 as(PL)minc˜tx˜,Bx˜=b˜,x˜0,where

  • x˜=(x,λ)tRn+1,

  • c˜=

Examples of fixed sizes

Example 01

A=1-10111,b=01andc=110t.The optimal value is z = 0.

The exact optimal solution is x=[001]t.

Comparison chart

MethodxzIterTime
Ye–Lustig algorithm(000.9999)t0.0000040:0:0:31
Modified algorithm(001.0000)t0.0000010:0:0:16

Example 02

A=231230-21,b=20andc=4120t.The optimal value is z = 0.67.

The exact optimal solution is x=[00.6700]t.

Comparison chart

MethodxzIterTime
Ye–Lustig algorithm(00.666600)t0.6666070:0:0:47
Modified algorithm(0.00040.66350.00220.0031)t0.6700030:0:0:31

Example 03

A=1-11121-121112,b=345andc=3213t.The

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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