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A new polynomial-time algorithm for linear programming. (English) Zbl 0557.90065
This paper discusses a new polynomial time algorithm for linear programming (LP). It is an interior point method whose worst case computational complexity is $$0(n^{3.5}L)$$ arithmetic operations on 0(L) bit numbers, where n is the number of variables and L is the number of bits in the input. The complexity bound for this algorithm is better than that for the ellipsoid algorithm by a factor of $$0(n^{2.5}).$$
The author shows that every LP can be transformed into the form: min cx subject to $$x\in \Omega \cap \Delta$$, where $$\Omega$$ is the subspace $$\{$$ $$x: Ax=0\}$$ and $$\Delta$$ is the simplex $$\{$$ $$x: x\geq 0$$ and $$\Sigma x_ j=1\}$$, and the minimum objective value in the problem is known to be zero. His algorithm solves the LP in this form.
Let $$a_ 0=(1/n)e$$, where e is the vector of all 1’s in $$R^ n$$. Let $$B(a_ 0,r)$$, $$B(a_ 0,R)$$ be respectively the largest sphere with center $$a_ 0$$ lying in $$\Delta$$, and the smallest sphere with center $$a_ 0$$ containing $$\Delta$$. Then $$R/r=(n-1)$$. Using this he shows that if $$a_ 0$$ is feasible, $$a_ 0-r\hat c$$, where $$\hat c$$ is the normalized vector which in the orthogonal projection of c in $$\Omega$$, is chosen to the minimum objective value by a factor of (1-1/(n-1)). This is the main result on which the algorithm is based.
The algorithm is initiated with a feasible solution $$x^ 0>0$$, and it generates a descent sequence of positive feasible points $$x^ 0,x^ 1,..$$.. In the kth step, the point $$x^ k$$ is brought into the center of the simplex by a projective transformation, a step of the form described above is taken, and the inverse projective transformation is applied, leading to the next point $$x^{k+1}$$, reducing the objective function value by a factor of 0(n). The sequence of points generated, converges to a near optimal solution in polynomial time.
Reviewer: K.G.Murty

##### MSC:
 90C05 Linear programming 68Q25 Analysis of algorithms and problem complexity
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##### References:
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