## An Asymptotical O(...)-iteration Path-following Linear Programming Algorithm That Uses Wide Neighborhoods (1994)

### BibTeX

@MISC{Hung94anasymptotical,

author = {Pi-Fang Hung and Yinyu Ye},

title = {An Asymptotical O(...)-iteration Path-following Linear Programming Algorithm That Uses Wide Neighborhoods },

year = {1994}

}

### OpenURL

### Abstract

Path-following linear programming (LP) algorithms generate a sequence of points within certain neighborhoods of a central-path C, which prevent iterates from prematurely getting too close to the boundary of the feasible region. Depending on their norm used, these neighborhoods include N2(fi), N1(fi) and N \Gamma 1(fi), where fi 2 (0; 1), and C ae N2(fi) ae N1(fi) ae N \Gamma 1(fi) for each fi 2 (0; 1): A paradox is that among all existing (infeasible or feasible) path-following algorithms, the theoretical iteration complexity, O(pnL), of small-neighborhood (N2) algorithms is significantly better than the complexity, O(nL), of wide-neighborhood (N \Gamma 1) algorithms, while in practice wide-neighborhood algorithms outperform small-neighborhood ones by a big margin. Here, n is the number of LP variables and L is the LP data length. In this paper, we present an O(n n+1 2n L)-iteration (infeasible) primal-dual high-order algorithm that uses wide neighborhoods. Note that this iteration bound is asymptotical O(pnL), i.e., the best bound for small-neighborhood algorithms, as n increases.