Get A Unified Approach to Interior Point Algorithms for Linear PDF

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, quite a few interior-point algorithms were proposed for varied mathematical programming difficulties equivalent to linear programming, convex quadratic programming and convex programming typically. This monograph provides 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 relations of power relief algorithms is gifted in a unified method for the category of LCPs the place the underlying matrix has nonnegative relevant minors (P0-matrix). This category contains quite a few vital subclasses similar to confident semi-definite matrices, P-matrices, P*-matrices brought during this monograph, and column adequate matrices. The kin includes not just the standard power relief algorithms but in addition direction following algorithms and a damped Newton approach for the LCP. the most issues are worldwide convergence, international linear convergence, and the polynomial-time convergence of power relief algorithms incorporated within the family.

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In order to overcome the difficulty, we reduce the LCP to an artificial linear complementarity problem with an apparent interior feasible point from which we can start the UIP method. 1) where M ' is an n' x n' matrix, q' E R "~ and N' = {1,2,... ,n'}. We also use the symbols S~. and S~+ for the set of all the feasible solutions and all the interior feasible solutions to the LCP'~ respectively: S~ = {(z', y') > 0 : y' = M ' z ' + q'}, s~+ = { ( ~ , ' , ¢ ) > o : ¢ = M ' ~ ' + q ' } . The artificial problem LCP' must have a readily available initial point (¢,1, yn) E S'+ + for the UIP method.

3). 4) of ~/. We assume that all the entries of M and q are integral. 5) i=1 L = E ~ [log~(Imql + 1)] + /=1 j = l Flog2(tqd+ 1)] + 2 Ng~(~ + 1)] + n(n + i). 6) i=1 Here [z] denotes the smallest integer not less than z E R. 3. Obviously L >_ L. ~ Every minor of the matrix ( - M zI + 1) . 7); hence Pma~ < 2r'/n 2. Te. 4). 10) lies in the interior S~+ of the feasible region of the LCP' and serves as an initial point of the UIP method applied to the LCP'.

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. One can easily verify this property. 3. 1). , v~) be positive diagonal matrices. Introducing new variable 31 (1 Ho Figure 12: The relations ~mong the sets R~, Z(~;), H0 and H__. 1), we obtain the LC'--P: Find ~n (~, ~t) E R 2" such that 9 = / ~ r $ + t~, (5¢,~t) >_ 0 and ~ifll = 0 (i E N).

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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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