Abstract
In this paper we are generalizing the efficient kernel function with trigonometric barrier term given by (M. Bouafia, D. Benterki and A. Yassine, J. Optim. Theory Appl. 170 (2016) 528–545). Using an elegant and simple analysis and under some easy to check conditions, we explore the best complexity result for the large update primal-dual interior point methods for linear optimization. This complexity estimate improves results obtained in (X. Li and M. Zhang, Oper. Res. Lett. 43 (2015) 471–475; M.R. Peyghami and S.F. Hafshejani, Numer. Algo. 67 (2014) 33–48; M. Bouafia, D. Benterki and A. Yassine, J. Optim. Theory Appl. 170 (2016) 528–545). Our comparative numerical experiments on some test problems consolidate and confirm our theoretical results according to which the new kernel function has promising applications compared to the kernel function given by (M. Bouafia and A. Yassine, Optim. Eng. 21 (2020) 651–672). Moreover, the comparative numerical study that we have established favors our new kernel function better than other best trigonometric kernel functions (M. Bouafia, D. Benterki and A. Yassine, J. Optim. Theory Appl. 170 (2016) 528–545; M. Bouafia and A. Yassine, Optim. Eng. 21 (2020) 651–672).
Highlights
Polynomial time Interior Point Methods IPMs for solving linear programming were first proposed by Karmarkar [7]
Our analysis shows that the worst case iteration complexity of large-update IPMs for sOol(v︀√inngloLgOn plorgobnl)e︀.ms based on the new kernel function meets the so far best known iteration complexity, i.e., ε
Q = 2, we obtain the best known complexity bound for large-update methods, namely O (︀ n log nε )︀
Summary
Polynomial time Interior Point Methods IPMs for solving linear programming were first proposed by Karmarkar [7]. Zhang [8], presented another trigonometric barrier function, which has These results improve the complexity bound obtained by El Ghami et al. In 2016, Bouafia et al [4], we proposed the first with trigonometric barrier terms for interior point methods in LO. In 2020, Bouafia and Yassine [3], investigated a new efficient twice parametric kernel function that combines the parametric classic function with the parametric kernel function trigonometric barrier term given by Bouafia et al [4] to develop primal-dual interior-point algorithms for solving linear programming problems. Using some mild and standard conditions, the worst case iteration complexity bound of the large-update primal dual IPMs based on the new proposed kernel function is driven. Throughout the paper, ‖‖ denotes the 2-norm of a vector
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