Abstract
In this paper, we first propose a new finite exponential-trigonometric kernel function that has finite value at the boundary of the feasible region. Then by using some simple analysis tools, we show that the new kernel function has exponential convexity property. We prove that the large-update primal-dual interior-point method based on this kernel function for solving linear optimization problems has \(O\left( \sqrt{n}\log n\log \frac{n}{\epsilon }\right)\) iteration bound in the worst case when the barrier parameter is taken large enough. Moreover, the numerical results reveal that the new finite exponential-trigonometric kernel function has better results than the other kernel functions.
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Bai YQ, El Ghami M, Roos C (2003) A new efficient large-update primal-dual interior-point methods based on a finite barrier. SIAM J Optim 13(3):766–782 (electronic)
Bai YQ, El Ghami M, Roos C (2004) A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization. SIAM J Optim 15(1):101–128
Bouafia M, Benterki D, Yassine A (2016a) Complexity analysis of interior point methods for linear programming based on a parameterized kernel function. RAIRO-Oper Res 50(4–5):935–949
Bouafia M, Benterki D, Yassine A (2016b) An efficient primal-dual interior point method for linear programming problems based on a new kernel function with a trigonometric barrier term. J Optim Theory Appl 170(2):528–545
Bouafia M, Benterki D, Yassine A (2018) An efficient parameterized logarithmic kernel function for linear optimization. Optim Lett 12(5):1079–1097
Cai X, Wang G, Zhang Z (2013) Complexity analysis and numerical implementation of primal-dual interior-point methods for convex quadratic optimization based on a finite barrier. Numer Algorithims 62:289–306
Cai XZ, Wang GQ, El Ghami M, Yue YJ (2014) Complexity analysis of primal-dual interior-point methods for linear optimization based on a new parametric kernel function with a trigonometric barrier term. In: Abstract and applied analysis. Art. ID 710158
El Ghami M, Guennoun ZA, Boula S, Steihaug T (2012) Interior-point methods for linear optimization based on a kernel function with a trigonometric barrier term. J Comput Appl Math 236:3613–3623
El Ghamia M, Ivanov I, Melissen JBM, Roos C, Steihaug T (2009) A polynomial-time algorithm for linear optimization based on a new class of kernel functions. J Comput Appl Math 224:500–513
Fathi-Hafshejani S, Mansouri H, Reza Peyghami M, Chen S (2018) Primal-dual interior-point method for linear optimization based on a kernel function with trigonometric growth term. Optimization 67(10):1605–1630
Karmarkar NK (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4:373–395
Kheirfam B (2012) Primal-dual interior-point algorithm for semidefinite optimization based on a new kernel function with trigonometric barrier term. Numer Algorithms 61:659–680
Kheirfam B, Moslem M (2015) A polynomial-time algorithm for linear optimization based on a new kernel function with trigonometric barrier term. Yugosl J Oper Res 25(2):233–250
Kojima M, Mizuno S, Yoshise A (1989) A primal-dual interior point algorithm for linear programming. In: Megiddo N (ed) Progress in mathematical programming: interior point and related methods. Springer, New York, pp 29–47
Li X, Zhang M (2015) Interior-point algorithm for linear optimization based on a new trigonometric kernel function. Oper Res Lett 43(5):471–475
Megiddo N (1989) Pathways to the optimal set in linear programming. In: Megiddo N (ed) Progress in mathematical programming: interior point and related methods. Springer, New York, pp 131–158
Monteiro RDC, Adler I (1989) Interior-point path following primal-dual algorithms: part I: linear programming. Math Program 44:27–41
Nesterov YE, Nemirovskii AS (1994) Interior point polynomial algorithms in convex programming, SIAM studies in applied mathematics, vol 13. SIAM, Philadelphia
Peng J, Roos C, Terlaky T (2002) Self-regularity: a new paradigm for primal-dual interior-point algorithms. Princeton University Press, Princeton
Reza Peyghami M, Amini K (2010) A kernel function based interior-point methods for solving P*(k)-linear complementarity problem. Acta Math Sinica 26(9):1761–1778
Reza Peyghami M, Fathi Hafshejani S (2014) Complexity analysis of an interior point algorithm for linear optimization based on a new poriximity function. Numer Algorithms 67:33–48
Reza Peyghami M, Fathi Hafshejani S, Shirvani L (2014) Complexity of interior-point methods for linear optimization based on a new trigonometric kernel function. J Comput Appl Math 255:74–85
Roos C, Terlaky T, Vial J-P (2005) Theory and algorithms for linear optimization: an interior point approach. Springer, New York
Sonnevend G (1986) An analytic center for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming. In: Prakopa A, Szelezsan J, Strazicky B (eds) Lecture notes in control and information sciences, vol 84. Springer, Berlin, pp 866–876
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Fathi-Hafshejani, S., Peyghami, M.R. & Fakharzadeh Jahromi, A. An efficient primal-dual interior point method for linear programming problems based on a new kernel function with a finite exponential-trigonometric barrier term. Optim Eng 21, 107–129 (2020). https://doi.org/10.1007/s11081-019-09436-3
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DOI: https://doi.org/10.1007/s11081-019-09436-3
Keywords
- Linear optimization
- Interior-point methods
- Iteration complexity
- Finite barrier function
- Large-update methods