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

ISBN-10: 354038426X

ISBN-13: 9783540384267

ISBN-10: 3540545093

ISBN-13: 9783540545095

Following Karmarkar's 1984 linear programming set of rules, a number of interior-point algorithms were proposed for numerous mathematical programming difficulties reminiscent of linear programming, convex quadratic programming and convex programming typically. This monograph offers a research of interior-point algorithms for the linear complementarity challenge (LCP) that's often called a mathematical version for primal-dual pairs of linear courses and convex quadratic courses. a wide relatives 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 category contains a number of very important subclasses akin to confident semi-definite matrices, P-matrices, P*-matrices brought during this monograph, and column adequate matrices. The family members comprises not just the standard strength relief algorithms but in addition course following algorithms and a damped Newton procedure for the LCP. the most themes are worldwide convergence, international linear convergence, and the polynomial-time convergence of strength relief algorithms incorporated within the family.

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We further observe Z(6)={¢eRn:(l+46)E(i+E(I<0 for every I c N ) iE! i¢I from the definitions of the index sets I÷(¢) and _r (¢). Hence Z(~) turns out the interior of the polyhedral cone {~ERn:(l+4a)~'/+~-':~¢i_<0 i~I i~l forevery [ c N } that contains the nonpositive orthant R~. See Figure 12. It should be noted that the class P,, which is the union of P,(6) (6 > 0), remains a proper subclass of CS although the intersection of Z(t:) (t: _> 0) becomes R ~_ \ {0}. Finally in this section, we remark that all the classes mentioned so far, Po, CS, P,, P,(a), PSD, P and SS, enjoy a nice property that if a matrix M belongs to one of these classes then every principal submatrix of M also belongs to the dass.

The Class of Linear Complementarity with P0-Matrices Problems In this section we investigate linear complementarity problems with Po-matrices. 1, we introduce some characteristic numbers related to Po-matrices. 3. 4 gives an example of NP-complete linear complementarity problems with Po-matrices. 1. Some Characteristic Numbers Related to P0-Matrices An n x n matrix M is called a Po-matrix when all the principal manors of M are nonnegative and a P-matrix when they are positive. We also say that M belongs to the classes Po or P, respectively.

Similar studies were also done in Bayer and Lagarias [5, 6] and Megiddo and Shub [43]. 2). 8) is a descent direction of the potential function f . Recall that the search direction (din, dy) depends on the parameter fl E [0,1] and can be regarded as a convex combination of the centering direction (dinc, dy ~) and the affine scaling direction (dina, dya), which correspond to the cases fi = 1 and fl = 0, respectively. 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++.

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A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems by Masakazu Kojima, Nimrod Megiddo, Toshihito Noma, Akiko Yoshise

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