Unified Analysis of Kernel-Based Interior-Point Methods for P*(Kappa)-Linear Complementarity Problems

We present an interior-point method for the $P_*(\kappa)$-linear complementarity problem (LCP) that is based on barrier functions which are defined by a large class of univariate functions called eligible kernel functions. This class is fairly general and includes the classical logarithmic function and the self-regular functions, as well as many non-self-regular functions as special cases. We provide a unified analysis of the method and give a general scheme on how to calculate the iteration bounds for the entire class. We also calculate the iteration bounds of both long-step and short-step versions of the method for several specific eligible kernel functions. For some of them we match the best known iteration bounds for the long-step method, while for the short-step method the iteration bounds are of the same order of magnitude. As far as we know, this is the first paper that provides a unified approach and comprehensive treatment of interior-point methods for $P_*(\kappa)$-LCPs based on the entire class of eligible kernel functions. (The title of this article has been corrected.)

[1]  James Renegar,et al.  A mathematical view of interior-point methods in convex optimization , 2001, MPS-SIAM series on optimization.

[2]  Kees Roos,et al.  Primal-Dual Interior-Point Algorithms for Semidefinite Optimization Based on a Simple Kernel Function , 2005, J. Math. Model. Algorithms.

[3]  Mihai Anitescu,et al.  An infeasible-interior-point predictor-corrector algorithm for theP*-geometric LCP , 1997 .

[4]  G. Lesaja,et al.  A Class of Large-Update and Small-Update Primal-Dual Interior-Point Algorithms for Linear Optimization , 2008 .

[5]  Florian A Potra,et al.  A superlinearly convergent predictor-corrector method for degenerate LCP in a wide neighborhood of the central path with $$O(\sqrt{n}L)$$-iteration complexity , 2007, Math. Program..

[6]  M. Anitescu,et al.  Equivalence Between Different Formulations of the Linear Complementarity Problem , 1997 .

[7]  H. Väliaho,et al.  P∗-matrices are just sufficient , 1996 .

[8]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[9]  Jiming Peng,et al.  Self-regular functions and new search directions for linear and semidefinite optimization , 2002, Math. Program..

[10]  Jianming Miao.  A quadratically convergent $$O((\kappa + 1)\sqrt n L)$$ -iteration algorithm for theP*(κ)-matrix linear complementarity problem , 1995, Math. Program..

[11]  Xingsi Li,et al.  An infeasible-start path-following method for monotone LCPs , 2003 .

[12]  Shuzhong Zhang,et al.  An O(\sqrtn L) Iteration Primal-dual Path-following Method, Based on Wide Neighborhoods and Large Updates, for Monotone LCP , 2005, SIAM J. Optim..

[13]  Florian A. Potra,et al.  A Large-Step Infeasible-Interior-Point Method for the P*-Matrix LCP , 1997, SIAM J. Optim..

[14]  Keyvan Amini,et al.  Exploring complexity of large update interior-point methods for P*(kappa) linear complementarity problem based on Kernel function , 2009, Appl. Math. Comput..

[15]  Stephen J. Wright,et al.  A Superlinear Infeasible-Interior-Point Affine Scaling Algorithm for LCP , 1996, SIAM J. Optim..

[16]  Yinyu Ye,et al.  On homogeneous and self-dual algorithms for LCP , 1997, Math. Program..

[17]  Xing Liu,et al.  Corrector-Predictor Methods for Sufficient Linear Complementarity Problems in a Wide Neighborhood of the Central Path , 2006, SIAM J. Optim..

[18]  Kees Roos,et al.  A Comparative Study of Kernel Functions for Primal-Dual Interior-Point Algorithms in Linear Optimization , 2004, SIAM J. Optim..

[19]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[20]  Xing Liu,et al.  Predictor–corrector methods for sufficient linear complementarity problems in a wide neighborhood of the central path , 2005, Optim. Methods Softw..

[21]  Nimrod Megiddo,et al.  A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems , 1991, Lecture Notes in Computer Science.

[22]  Goran Lesaja,et al.  A New Class of Polynomial Interior-Point Algorithms for P *(κ)-Linear Complementarity Problems , 2008 .

[23]  Jiming Peng,et al.  Self-regularity - a new paradigm for primal-dual interior-point algorithms , 2002, Princeton series in applied mathematics.

[24]  N. Megiddo.  Pathways to the optimal set in linear programming , 1989 .

[25]  Cornelis Roos,et al.  Primal–dual interior-point algorithms for second-order cone optimization based on kernel functions , 2009 .

[26]  Jiming Peng,et al.  New Complexity Analysis of Primal-Dual Newton Methods for P *(κ) Linear Complementarity Problems , 2000 .

[27]  Stephen J. Wright,et al.  A superquadratic infeasible-interior-point method for linear complementarity problems , 1994, Math. Program..

[28]  Y. H. Lee,et al.  Complexity of large-update interior point algorithm for P*(kappa) linear complementarity problems , 2007, Comput. Math. Appl..