, Volume 49, Issue 1-3, pp 7-21

Polynomial affine algorithms for linear programming

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Abstract

The method of steepest descent with scaling (affine scaling) applied to the potential functionq logc′x — ∑ i=1 n logx i solves the linear programming problem in polynomial time forq ⩾ n. Ifq = n, then the algorithm terminates in no more than O(n 2 L) iterations; if q ⩾ n + \(\sqrt n \) withq = O(n) then it takes no more than O(nL) iterations. A modified algorithm using rank-1 updates for matrix inversions achieves respectively O(n 4 L) and O(n 3.5 L) arithmetic computions.