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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 $0\left({n}^{3·5}L\right)$ 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 $0\left({n}^{2·5}\right)·$

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 $\left\{$ $x:Ax=0\right\}$ and ${\Delta }$ is the simplex $\left\{$ $x:x\ge 0$ and ${\Sigma }{x}_{j}=1\right\}$, and the minimum objective value in the problem is known to be zero. His algorithm solves the LP in this form.

Let ${a}_{0}=\left(1/n\right)e$, where e is the vector of all 1’s in ${R}^{n}$. Let $B\left({a}_{0},r\right)$, $B\left({a}_{0},R\right)$ be respectively the largest sphere with center ${a}_{0}$ lying in ${\Delta }$, and the smallest sphere with center ${a}_{0}$ containing ${\Delta }$. Then $R/r=\left(n-1\right)$. Using this he shows that if ${a}_{0}$ is feasible, ${a}_{0}-r\stackrel{^}{c}$, where $\stackrel{^}{c}$ 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
##### References:
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