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 regularly. This monograph provides a learn 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 kin of power aid algorithms is gifted in a unified manner for the category of LCPs the place the underlying matrix has nonnegative central minors (P0-matrix). This category contains a number of very important subclasses similar to confident semi-definite matrices, P-matrices, P*-matrices brought during this monograph, and column enough matrices. The kin comprises not just the standard capability relief algorithms but in addition direction following algorithms and a damped Newton process for the LCP. the most issues are international convergence, worldwide linear convergence, and the polynomial-time convergence of power relief algorithms integrated within the family.

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**Extra resources for A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems**

**Example text**

K). -matrix. 4. An N P - C o m p l e t e Linear Complementarity Problem with a P0-Matrix In this section we show that the class of all the linear complementarity problems with Po-matrices is NP-complete. For this purpose we consider the knapsack problem: Given an integer vector a = ( a l , a 2 , . . , a t , ) r and an integer b, find a solution U = ( U l , U 2 , . . , U m ) T ~-. R m" of a:ru = b and u~ e {0,1} ( i = 1 , 2 , . . , m ) . 16) The knapsack problem is known to be NP-complete (see, for example, Schrijver [61]).

4. 1 holds. ~) is a one-dimensional smooth cu~e. 1) as t (> 0) tends to O. In particular, the LCP has a solution. 3. 2. The Jacobian matrix of the restriction of u to S++ with respect to z is Y + X M , where X = diag z and Y = diag y. 1). Hence the mapping u is a diffeomorphism between S++ and/P++. Thus we have shown (i). 2) of the path of centers Sc,,. We will show (iii). Let [ > 0. 1, the subset {u-Z(te) : 0 < t < t-} of the path of centers Scan is bounded. Hence there is at least one accumulation point of u -~ (re) as t --~ 0.

Subclasses of the P0-Matrices We will be concerned with some subclasses of the class P0: SS (the class of skewsymmetric matrices), PSD (the class of positive semi-definite matrices), P, P,(a), P, and CS (the class of column sufficient matrices). 1 for their definitions. Let a be any nonnegative real number. 4) where 1+(() = {i E N : ~[Al~]i > 0}, I_(() = {i E N : ~[M~]i < 0}. The index sets I+(~) and I_(~) depend on the matrix M though they do not explicitly represent the dependence. 4) can be rewritten as ~TM~ + 4a ~ (i[M~]i >_ 0 for every ~ E 1~".

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

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