Abstract
In this paper we present a new primal-dual affine scaling method for linear programming. The method yields a strictly complementary optimal solution pair, and also allows a polynomial-time convergence proof. The search direction is obtained by using the original idea of Dikin, namely by minimizing the objective function (which is the duality gap in the primal-dual case), over some suitable ellipsoid. This gives rise to completely new primal-dual affine scaling directions, having no obvious relation with the search directions proposed in the literature so far. The new directions guarantee a significant decrease in the duality gap in each iteration, and at the same time they drive the iterates to the central path. In the analysis of our algorithm we use a barrier function which is the natural primal-dual generalization of Karmarkar's potential function. The iteration bound is O(nL), which is a factor O(L) better than the iteration bound of an earlier primal-dual affine scaling method (Monteiro, Adler and Resende [Monteiro, R. D. C., I. Adler, M. G. C. Resende. 1990. A polynomial-time primal-dual affine scaling algorithm for linear and convex quadratic programming and its power series extension. Math. Oper. Res. 15 191–214.]).
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