On Mehrotra-Type Predictor-Corrector Algorithms

In this paper we discuss the polynomiality of a feasible version of Mehrotra's predictor-corrector algorithm whose variants have been widely used in several interior point method (IPM)-based optimization packages. A numerical example is given that shows that the adaptive choice of centering parameter and correction terms in this algorithm may lead to small steps being taken in order to keep the iterates in a large neighborhood of the central path, which is important for proving polynomial complexity properties of this method. Motivated by this example, we introduce a safeguard in Mehrotra's algorithm that keeps the iterates in the prescribed neighborhood and allows us to obtain a positive lower bound on the step size. This safeguard strategy is also used when the affine scaling direction performs poorly. We prove that the safeguarded algorithm will terminate after at most ${\cal{O}}(n^2{\rm log} (x^0)^Ts^0/\epsilon)$ iterations. By modestly modifying the corrector direction, we reduce the iteration complexity to ${\cal{O}}(n {\rm log} (x^0)^Ts^0/\epsilon).$ To ensure fast asymptotic convergence of the algorithm, we changed Mehrotra's updating scheme of the centering parameter slightly while keeping the safeguard. The new algorithms have the same order of iteration complexity as the safeguarded algorithms but enjoy superlinear convergence as well. Numerical results using the McIPM and LIPSOL software packages are reported.

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

[2]  Mauricio G. C. Resende,et al.  A Polynomial-Time Primal-Dual Affine Scaling Algorithm for Linear and Convex Quadratic Programming and Its Power Series Extension , 1990, Math. Oper. Res..

[3]  F. Jarre,et al.  Extending mehrotra's corrector for linear programs , 1997 .

[4]  Knud D. Andersen,et al.  The Mosek Interior Point Optimizer for Linear Programming: An Implementation of the Homogeneous Algorithm , 2000 .

[5]  Yinyu Ye,et al.  On quadratic and $$O\left( {\sqrt {nL} } \right)$$ convergence of a predictor—corrector algorithm for LCP , 1993, Math. Program..

[6]  Stephen J. Wright,et al.  PCx: an interior-point code for linear programming , 1999 .

[7]  Yin Zhang,et al.  The Mehrotra Predictor-Corrector Interior-Point Method As a Perturbed Composite Newton Method , 1996, SIAM J. Optim..

[8]  Sanjay Mehrotra,et al.  On the Implementation of a Primal-Dual Interior Point Method , 1992, SIAM J. Optim..

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

[10]  李幼升,et al.  Ph , 1989 .

[11]  Yin Zhang,et al.  On polynomiality of the Mehrotra-type predictor—corrector interior-point algorithms , 1995, Math. Program..

[12]  Jacek Gondzio,et al.  Further development of multiple centrality correctors for interior point methods , 2008, Comput. Optim. Appl..

[13]  Kurt M. Anstreicher,et al.  A New Infinity-Norm Path Following Algorithm for Linear Programming , 1995, SIAM J. Optim..

[14]  G. Sonnevend.  An "analytical centre" for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming , 1986 .

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

[16]  Clovis C. Gonzaga,et al.  Complexity of Predictor-Corrector Algorithms for LCP Based on a Large Neighborhood of the Central Path , 1999, SIAM J. Optim..

[17]  Jean-Philippe Vial,et al.  Theory and algorithms for linear optimization - an interior point approach , 1998, Wiley-Interscience series in discrete mathematics and optimization.

[18]  Sanjay Mehrotra.  On finding a vertex solution using interior point methods , 1991 .

[19]  Yin Zhang,et al.  Solving large-scale linear programs by interior-point methods under the Matlab ∗ Environment † , 1998 .

[20]  Jeffrey C. Lagarias,et al.  Karmarkar's linear programming algorithm and Newton's method , 1991, Math. Program..

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

[22]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[23]  Jacek Gondzio,et al.  Multiple centrality corrections in a primal-dual method for linear programming , 1996, Comput. Optim. Appl..

[24]  Shinji Mizuno,et al.  On Adaptive-Step Primal-Dual Interior-Point Algorithms for Linear Programming , 1993, Math. Oper. Res..

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

[26]  C. Cartis.  Some disadvantages of a Mehrotra-type primal-dual corrector interior point algorithm for linear programming , 2009 .

[27]  Sanjay Mehrotra,et al.  Convergence Conditions and Krylov Subspace--Based Corrections for Primal-Dual Interior-Point Method , 2005, SIAM J. Optim..

[28]  Jiming Peng,et al.  A Predictor-Corrector Algorithm for Linear Optimization Based on a Specific Self-Regular Proximity Function , 2005, SIAM J. Optim..

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

[30]  F. Potra,et al.  Predictor-corrector method for linear complementarity problems with polynomial complexity and superlinear convergence , 1995 .