《《1999 Probabilistic analysis of an infeasible-interior-point algorithm for linear programming》.pdfVIP

MATHEMATICS OF OPERATIONS RESEARCH Vol. 24, No. 1, February 1999 Printed in U.S.A. PROBABILISTIC ANALYSIS OF AN INFEASIBLE-INTERIOR-POINT ALGORITHM FOR LINEAR PROGRAMMING KURT M. ANSTREICHER, JUN JI, FLORIAN A. POTRA, AND YINYU YE We consider an infeasible-interior-point algorithm, endowed with a finite termination scheme, applied to random linear programs generated according to a model of Todd. Such problems have degenerate optimal solutions, and possess no feasible starting point. We use no information regarding an optimal solution in the initialization of the algorithm. Our main result is that the expected number of iterations before termination with an exact optimal solution is O(n ln(n)). 1. Introduction. A number of recent papers have attempted to analyze the probabilistic behavior of interior point algorithms for linear programming. Ye (1994) showed that a variety of algorithms, endowed with the finite termination scheme of Ye (1992) (see also Mehrotra and Ye 1993), obtain an exact optimal solution with “high probability” (probability approaching one as n 3) in no more than O(n ln(n)) iterations. Here n is the number of variables in a standard form primal problem. Several subsequent works—Huang and Ye (1991), Anstreicher, Ji, and Ye (1992), and Ji and Potra (1992)—then obtained bounds on the expected number of iterations until termination, using various algorithms and termination methods. The analysis in each of these latter papers is based on a particular random linear programming model from Todd (1991) (Model 1 with xˆ sˆ e, see Todd 1991, p. 677), which has a known initial interior solution for the primal and dual problems, and is nondegenerate with probability one. Unfortunately, we eventually rea