A self-concordant exponential kernel function for primal–dual interior-point algorithm

Self-concordant (SC) function is essential to analyzing primal–dual interior-point methods with polynomial-time complexity bound. In this paper, we introduce an exponential kernel function and prove that it is an SC function. Furthermore, we investigate its properties and find it is neither an eligible kernel function nor an SC barrier function. Nevertheless, based on this SC exponential kernel function, we present primal–dual interior-point algorithm for solving linear optimization problems. We analyse the algorithm and derive the complexity bound for large-update methods, which coincides with that of the algorithm based on the logarithmic function. Finally, the comparative numerical results are reported.

[1]  Osman Güler,et al.  Barrier Functions in Interior Point Methods , 1996, Math. Oper. Res..

[2]  A. Nemirovski,et al.  Interior-point methods for optimization , 2008, Acta Numerica.

[3]  Raphael A. Hauser,et al.  Square-Root Fields and the "V-Space" Approach to Primal-Dual Interior-Point Methods for Self-Scaled , 1999 .

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

[5]  C. Roos,et al.  Local Self-Concordance of Barrier Functions Based on Kernel Functions , 2012 .

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

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

[9]  Michael J. Todd,et al.  Primal-Dual Interior-Point Methods for Self-Scaled Cones , 1998, SIAM J. Optim..

[10]  Akiko Yoshise,et al.  A primal barrier function Phase I algorithm for nonsymmetric conic optimization problems , 2012 .

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

[12]  Robert Chares.  Cones and interior-point algorithms for structured convex optimization involving powers andexponentials , 2009 .

[13]  Y. Nesterov.  Towards Nonsymmetric Conic Optimization , 2006 .

[14]  Yinyu Ye,et al.  A homogeneous interior-point algorithm for nonsymmetric convex conic optimization , 2014, Mathematical Programming.

[15]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[16]  Yurii Nesterov,et al.  Introductory Lectures on Convex Optimization - A Basic Course , 2014, Applied Optimization.

[17]  Yurii Nesterov,et al.  Towards non-symmetric conic optimization , 2012, Optim. Methods Softw..

[18]  François Glineur,et al.  Topics in Convex Optimization: Interior-Point Methods, Conic Duality and Approximations , 2001 .

[19]  Michael J. Todd,et al.  Self-Scaled Barriers and Interior-Point Methods for Convex Programming , 1997, Math. Oper. Res..

[20]  Y. Nesterov.  Constructing Self-Concordant Barriers for Convex Cones , 2006 .