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A simplified homogeneous and selfdual linear programming algorithm and its implementation
 Annals of Operations Research
, 1996
"... 1 Introduction Consider the linear programming (LP) problem in the standard form: (LP) minimize cT x ..."
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Cited by 56 (5 self)
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1 Introduction Consider the linear programming (LP) problem in the standard form: (LP) minimize cT x
An Asymptotical O(...)iteration Pathfollowing Linear Programming Algorithm That Uses Wide Neighborhoods
, 1994
"... Pathfollowing linear programming (LP) algorithms generate a sequence of points within certain neighborhoods of a centralpath 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) ..."
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Pathfollowing linear programming (LP) algorithms generate a sequence of points within certain neighborhoods of a centralpath 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) pathfollowing algorithms, the theoretical iteration complexity, O(pnL), of smallneighborhood (N2) algorithms is significantly better than the complexity, O(nL), of wideneighborhood (N \Gamma 1) algorithms, while in practice wideneighborhood algorithms outperform smallneighborhood 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) primaldual highorder algorithm that uses wide neighborhoods. Note that this iteration bound is asymptotical O(pnL), i.e., the best bound for smallneighborhood algorithms, as n increases.
A WEIGHTED ANALYTIC CENTER FOR LINEAR MATRIX INEQUALITIES
, 2001
"... Let R be the convex subset of IR n defined by q simultaneous linear matrix inequalities (LMI) A (j) 0 + ∑n (j) i=1 xiA i ≻ 0, j = 1, 2,..., q. Given a strictly positive vector ω = (ω1, ω2, · · · , ωq), the weighted analytic center xac(ω) is the minimizer argmin (φω(x)) of the strictly convex fu ..."
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Let R be the convex subset of IR n defined by q simultaneous linear matrix inequalities (LMI) A (j) 0 + ∑n (j) i=1 xiA i ≻ 0, j = 1, 2,..., q. Given a strictly positive vector ω = (ω1, ω2, · · · , ωq), the weighted analytic center xac(ω) is the minimizer argmin (φω(x)) of the strictly convex function φω(x) = ∑q j=1 ωj log det[A (j)(x)] −1 over R. We give a necessary and sufficient condition for a point of R to be a weighted analytic center. We study the argmin function in this instance and show that it is a continuously differentiable open function. In the special case of linear constraints, all interior points are weighted analytic centers. We show that the region W = {xac(ω)  ω> 0} ⊆ R of weighted analytic centers for LMI’s is not convex and does not generally equal R. These results imply that the techniques in linear programming of following paths of analytic centers may require special consideration when extended to semidefinite programming. We show that the region W and its boundary are described by real algebraic varieties, and provide slices of a nontrivial real algebraic variety to show that W isn’t convex. Stiemke’s Theorem of the alternative provides a practical test of whether a point is in W. Weighted analytic centers are used to improve the location of standing points for the Stand and Hit method of identifying necessary LMI constraints in semidefinite programming.