Mathematical Programming 46 (1990) 173-190 North-Holland 173 ASYMPTOTIC BEHAVIOUR OF KARMARKAR'S METHOD FOR LINEAR PROGRAMMING Miroslav D. ASIC The Ohio State University, Newark Campus, Newark, OH, USA Vera V. KOVACEVIC-VUJCIC and Mirjana D. RADOSAVLJEVIC-NIKOLIC Faculty of Organizational Sciences, Belgrade University, Belgrade, Yugoslavia Received 3 February 1987 Revised manuscript received 15 February 1988 The asymptotic behaviour of Karmarkar's method is studied and an estimate of the rate of the objective function value decrease is given. Two possible sources of numerical instability are discussed and a stabilizing procedure is proposed. Key words': Karmarkar's algorithm, linear programming, rate of convergence. 1. Introduction Karmarkar (1984) suggested a polynomial-time algorithm for solving linear program- n L-), which is better ming problems [12]. He has proved that its complexity is O( 35 than the complexity O(n4L 2) of Khachian's method [14]. Here n is the number of variables and L is the number of bits of the input data. The new method has created a great interest not only because of this theoretical result, but primarily because of the author's claims that the practical behaviour of the method is superior to that of the simplex method. Since 1984 many researchers have studied Karmarkar's algorithm both from the theoretical and practical point of view [1, 2, 3, 5, 6, 10, 18, 20]. In order to improve the practical behaviour of the algorithm several modifications and extensions have been proposed [4, 7, 10, 11, 15, 16, 19, 21, 22, 23, 24]. It should be mentioned that the results of Barnes [4] have been independently obtained by Dikin [8] already in 1974. In this paper we shall study asymptotic behaviour of Karmarkar's method. In Section 2 we analyze some convergence properties of the method. We show that the generated sequence of points approaches the boundary uniformly, in the sense that all components which tend to zero are asymptotically proportional to the objective function value (Theorems 2 and 3). We also estimate the rate of conver- gence, i.e. the reduction in the value of the objective function, and in nondegenerate case we obtain its limit (Theorems 3 and 4). Research supported in part by Republicka zajednica za nauku SR Srbije.