Large-step interior-point algorithm for linear optimization based on a new wide neighbourhood

The interior-point algorithms can be classified in multiple ways. One of these takes into consideration the length of the step. In this way, we can speak about large-step and short-step methods, that work in different neighbourhoods of the central path. The large-step algorithms work in a wide neighbourhood, while the short-step ones determine the new iterates that are in a smaller neighbourhood. In spite of the fact that the large-step algorithms are more efficient in practice, the theoretical complexity of the short-step ones is generally better. Ai and Zhang introduced a large-step interior-point method for linear complementarity problems using a wide neighbourhood of the central path, which has the same complexity as the best short-step methods. We present a new wide neighbourhood of the central path. We prove that the obtained large-step primal–dual interior-point method for linear programming has the same complexity as the best short-step algorithms.

[1]  Jiming Peng,et al.  Primal-Dual Interior-Point Methods for Second-Order Conic Optimization Based on Self-Regular Proximities , 2002, SIAM J. Optim..

[2]  István Maros,et al.  Parallel search paths for the simplex algorithm , 2017, Central Eur. J. Oper. Res..

[3]  Hongwei Liu,et al.  An $${O(\sqrt{n}L)}$$ iteration primal-dual second-order corrector algorithm for linear programming , 2011, Optim. Lett..

[4]  Hongwei Liu,et al.  A New Strategy in the Complexity Analysis of an Infeasible-Interior-Point Method for Symmetric Cone Programming , 2015, J. Optim. Theory Appl..

[5]  Florian A. Potra,et al.  Predictor-corrector algorithm for solvingP*(κ)-matrix LCP from arbitrary positive starting points , 1996, Math. Program..

[6]  Behrouz Kheirfam.  A predictor-corrector interior-point algorithm for P∗(κ)$P_{*}(\kappa )$-horizontal linear complementarity problem , 2013, Numerical Algorithms.

[7]  Stephen J. Wright.  Primal-Dual Interior-Point Methods , 1997, Other Titles in Applied Mathematics.

[8]  Florian A. Potra,et al.  Interior Point Methods for Sufficient Horizontal LCP in a Wide Neighborhood of the Central Path with Best Known Iteration Complexity , 2014, SIAM J. Optim..

[9]  Hossein Mansouri,et al.  A Class of Path-Following Interior-Point Methods for $$P_*(\kappa )$$P∗(κ)-Horizontal Linear Complementarity Problems , 2015 .

[10]  T. Terlaky,et al.  EP Theorem for Dual Linear Complementarity Problems , 2009 .

[11]  Yinyu Ye,et al.  Interior point algorithms: theory and analysis , 1997 .

[12]  Kees Roos,et al.  Unified Analysis of Kernel-Based Interior-Point Methods for P*(Kappa)-Linear Complementarity Problems , 2010, SIAM J. Optim..

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

[14]  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..

[15]  Yang Li,et al.  A New Class of Large Neighborhood Path-Following Interior Point Algorithms for Semidefinite Optimization with O(√n log (Tr(X0S0)/ε)) Iteration Complexity , 2010, SIAM J. Optim..

[16]  Liang Fang,et al.  A new O(sqrt(n)L)-iteration predictor-corrector algorithm with wide neighborhood for semidefinite programming , 2014, J. Comput. Appl. Math..

[17]  Tibor Illés,et al.  A Mizuno-Todd-Ye type predictor-corrector algorithm for sufficient linear complementarity problems , 2007, Eur. J. Oper. Res..

[18]  Hossein Mansouri,et al.  An infeasible interior-point algorithm with full-Newton steps for $$P_*(\kappa )$$P∗(κ) horizontal linear complementarity problems based on a kernel function , 2016 .

[19]  Florian A. Potra,et al.  The Mizuno-Todd-Ye algorithm in a larger neighborhood of the central path , 2002, Eur. J. Oper. Res..

[20]  Zsolt Darvay.  New Interior Point Algorithms in Linear Programming , 2003 .

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

[22]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[23]  Hongwei Liu,et al.  A New Wide Neighborhood Primal–Dual Infeasible-Interior-Point Method for Symmetric Cone Programming , 2013, J. Optim. Theory Appl..

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

[25]  Tibor Illés,et al.  Polynomial affine-scaling algorithms for P*(k) linear complementary problems , 1997 .

[26]  Tibor Illés,et al.  A polynomial path-following interior point algorithm for general linear complementarity problems , 2010, J. Glob. Optim..

[27]  Florian A. Potra,et al.  A superlinearly convergent predictor-corrector method for degenerate LCP in a wide neighborhood of the central path with -iteration complexity , 2004, Math. Program..

[28]  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..

[29]  Hossein Mansouri,et al.  An $$O\left(\sqrt{n}L\right)$$OnL wide neighborhood interior-point algorithm for semidefinite optimization , 2017 .

[30]  Tibor Illés,et al.  Polynomial Interior Point Algorithms for General Linear Complementarity Problems , 2010, Algorithmic Oper. Res..