A New Complexity Result on Solving the Markov Decision Problem
Mathematics of Operations Research
Proximity of Weighted and Layered Least Squares Solutions
SIAM Journal on Matrix Analysis and Applications
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In this paper we present a variant of Vavasis and Ye's layered-step path-following primal-dual interior-point algorithm for linear programming. Our algorithm is a predictor--corrector-type algorithm which uses from time to time the layered least squares (LLS) direction in place of the affine scaling (AS) direction. It has the same iteration-complexity bound of Vavasis and Ye's algorithm, namely ${\cal O}(n^{3.5}\log({\bar\chi_A}+n))$, where n is the number of nonnegative variables and ${\bar\chi_A}$ is a certain condition number associated with the constraint matrix A. Vavasis and Ye's algorithm requires explicit knowledge of ${\bar\chi_A}$ (which is very hard to compute or even estimate) in order to compute the layers for the LLS direction. In contrast, our algorithm uses the AS direction at the current iterate to determine the layers for the LLS direction, and hence does not require the knowledge of ${\bar\chi_A}$. A variant with similar properties and with the same complexity has been developed by Megiddo, Mizuno, and Tsuchiya [ Math. Programming, 82 (1998), pp. 339--355]. However, their algorithm needs to compute n LLS directions on every iteration, while ours computes at most one LLS direction on any given iteration.