Abstract
In this chapter, we define an interior-point method for linear programming that is called a path-following method. Recall that for the simplex method we required a two-phase solution procedure. The path-following method is a one-phase method. This means that the method can begin from a point that is neither primal nor dual feasible and it will proceed from there directly to the optimal solution.
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The path-following algorithm introduced in this chapter has its origins in a paper by Kojima et al. (1989). Their paper assumed an initial feasible solution and therefore was a true interior-point method. The method given in this chapter does not assume the initial solution is feasible-it is a one-phase algorithm. The simple yet beautiful idea of modifying the Kojima-Mizuno-Yoshise primal-dual algorithm to make it into a one-phase algorithm is due to Lustig (1990).
Of the thousands of papers on interior-point methods that have appeared in the last decade, the majority have included convergence proofs for some version of an interior-point method. Here, we only mention a few of the important papers. The first polynomial-time algorithm for linear programming was discovered by Khachian (1979). Khachian’s algorithm is fundamentally different from any algorithm presented in this book. Paradoxically, it proved in practice to be inferior to the simplex method. N.K. Karmarkar’s pathbreaking paper (Karmarkar 1984) contained a detailed convergence analysis. His claims, based on preliminary testing, that his algorithm is uniformly substantially faster than the simplex method sparked a revolution in linear programming. Unfortunately, his claims proved to be exaggerated, but nonetheless interior-point methods have been shown to be competitive with the simplex method and usually superior on very large problems. The convergence proof for a primal-dual interior-point method was given by Kojima et al. (1989). Shortly thereafter, Monteiro & Adler (1989) improved on the convergence analysis. Two recent survey papers, Todd (1995) and Anstreicher (1996), give nice overviews of the current state of the art. Also, a soon-to-be-published book by Wright (1996) should prove to be a valuable reference to the reader wishing more information on convergence properties of these algorithms.
The homotopy method outlined in Exercise 17.5 is described in Nazareth (1986) and Nazareth (1996). Higher-order path-following methods are described (differently) in Carpenter et al. (1993).
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© 2001 Robert J. Vanderbei
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Vanderbei, R.J. (2001). A Path-Following Method. In: Linear Programming. International Series in Operations Research & Management Science, vol 37. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-5662-3_17
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DOI: https://doi.org/10.1007/978-1-4757-5662-3_17
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