Download A Unified Approach to Interior Point Algorithms for Linear by Masakazu Kojima, Nimrod Megiddo, Toshihito Noma, Akiko PDF

By Masakazu Kojima, Nimrod Megiddo, Toshihito Noma, Akiko Yoshise

Following Karmarkar's 1984 linear programming set of rules, a variety of interior-point algorithms were proposed for varied mathematical programming difficulties akin to linear programming, convex quadratic programming and convex programming quite often. This monograph provides a examine of interior-point algorithms for the linear complementarity challenge (LCP) that is often called a mathematical version for primal-dual pairs of linear courses and convex quadratic courses. a wide kin of strength aid algorithms is gifted in a unified means for the category of LCPs the place the underlying matrix has nonnegative crucial minors (P0-matrix). This type comprises numerous vital subclasses similar to optimistic semi-definite matrices, P-matrices, P*-matrices brought during this monograph, and column enough matrices. The kin comprises not just the standard power aid algorithms but in addition direction following algorithms and a damped Newton process for the LCP. the most subject matters are international convergence, international linear convergence, and the polynomial-time convergence of power aid algorithms integrated within the family.

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8 ) ) . (z, y), respectively. n. , p(v) = ( . +. 20) of equations. ~+, and consider the system of differential equations = --Vp(v) = --2 ( ilvll2 v-xe . 24) 52 The continuous steepest descent method traces a trajectory, a solution curve of the system of differential equations above, from a given initial point v(0) = v ° E R~+. By the relation u = V2e, we have/~ = 2V/~. 25) Choose the parameters/3 e (0,1) and u > 0 such that/3 = n/(n + v). 20). 5 that the choice of the parameters fl and v above gives rise to a large reduction in the potential function f.

12. ~n/~-l)(1-~). n-~' n (ii) - log ~ - (n - 1) log ~ < - ~ log s, < - ( n - 1) log ~ - log{n - (n - 1)~1. n - 1 ~=1 Proof: Let S={sER '~:eTs=n )~(s) = lls - ell = and minsi=~}, IEN sl -- 1) ~ for every s e R++. 44 The function )~(s) has both its maximum and minimum over the compact set S. We will evaluate the maximum and minimum values of the function ~(s) to derive the inequality (i). Let s* E R n and a, E R ~ be such that s" = s. (n-(n-1)~', = ~, ~ , . . , 1' n - l ' ~)T, "'" n - l ' ~ " We will show below that s* is a maximizer of ~(s) over S and that s, is a minimizer of )~(s) over S.

10). 20). 14. Let V denote, as usual the gradient operator. 23) vf~"(x'u)r du =--7-' dy ~ dx" dY~ dy e Vf(n~, y ) r ( d:~ for every (~, y) e S++. 10). eT(Ydz + Xdy) ~y _ 1 T / zTY = -(, #), _ n = -2-T-Ze-- X - I Y - l e zy = (Yd~ + X d y ) - xg) (by __ (_ ) ~w 2 -- This completes the proof, ~r (by ( 4 . 8 ) ) . (z, y), respectively. n. , p(v) = ( . +. 20) of equations. ~+, and consider the system of differential equations = --Vp(v) = --2 ( ilvll2 v-xe . 24) 52 The continuous steepest descent method traces a trajectory, a solution curve of the system of differential equations above, from a given initial point v(0) = v ° E R~+.

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