Complexity analysis and numerical implementation of a full-Newton step interior-point algorithm for LCCO

In this paper, we present a primal-dual interior point algorithm for linearly constrained convex optimization (LCCO). The algorithm uses only full-Newton step to update iterates with an appropriate proximity measure for controlling feasible iterations near the central path during the solution proces...

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Published in:	Numerical algorithms Vol. 70; no. 2; pp. 393 - 405
Main Authors:	Achache, Mohamed, Goutali, Moufida
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.10.2015 Springer Nature B.V
Subjects:	Algebra Algorithms Complexity Computer Science Convexity Numeric Computing Numerical Analysis Optimization Original Paper Polynomials Theory of Computation Interior point methods 90C51 Linearly constrained convex optimization Short-step primal-dual algorithms 90C25 Complexity of algorithms
ISSN:	1017-1398, 1572-9265
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Abstract	In this paper, we present a primal-dual interior point algorithm for linearly constrained convex optimization (LCCO). The algorithm uses only full-Newton step to update iterates with an appropriate proximity measure for controlling feasible iterations near the central path during the solution process. The favorable polynomial complexity bound for the algorithm with short-step method is obtained, namely O ( n log n 𝜖 ) which is as good as the linear and convex quadratic optimization analogue. Numerical results are reported to show the efficiency of the algorithm.
AbstractList	In this paper, we present a primal-dual interior point algorithm for linearly constrained convex optimization (LCCO). The algorithm uses only full-Newton step to update iterates with an appropriate proximity measure for controlling feasible iterations near the central path during the solution process. The favorable polynomial complexity bound for the algorithm with short-step method is obtained, namely O(nlognðoe-) which is as good as the linear and convex quadratic optimization analogue. Numerical results are reported to show the efficiency of the algorithm. In this paper, we present a primal-dual interior point algorithm for linearly constrained convex optimization (LCCO). The algorithm uses only full-Newton step to update iterates with an appropriate proximity measure for controlling feasible iterations near the central path during the solution process. The favorable polynomial complexity bound for the algorithm with short-step method is obtained, namely O ( n log n 𝜖 ) which is as good as the linear and convex quadratic optimization analogue. Numerical results are reported to show the efficiency of the algorithm.
Author	Achache, Mohamed Goutali, Moufida
Author_xml	– sequence: 1 givenname: Mohamed surname: Achache fullname: Achache, Mohamed email: achache_m@yahoo.fr organization: Laboratoire de Mathématiques Fondamentales et Numériques, Université Ferhat Abbas de Sétif1 – sequence: 2 givenname: Moufida surname: Goutali fullname: Goutali, Moufida organization: Département de Mathématiques, Faculté des Sciences, Université Ferhat Abbas de Sétif1
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References	Peng, Roos, Terlaky (CR6) 2000; 49 Darvay (CR3) 2003; 5 Roos, Terlaky, Vial (CR7) 1997 Achache, Goutali (CR2) 2012; LVII Darvay (CR4) 2009; LIV Zhang, Bai, Wang (CR9) 2008; 12 Ye (CR8) 1997 Achache (CR1) 2006; 25 Grana Drummond, Svaiter (CR5) 1999; 102 M Achache (9955_CR2) 2012; LVII Z Darvay (9955_CR3) 2003; 5 Z Darvay (9955_CR4) 2009; LIV Y Ye (9955_CR8) 1997 C Roos (9955_CR7) 1997 M Zhang (9955_CR9) 2008; 12 LM Grana Drummond (9955_CR5) 1999; 102 M Achache (9955_CR1) 2006; 25 J Peng (9955_CR6) 2000; 49
References_xml	– volume: 102 start-page: 223 issue: N2 year: 1999 end-page: 237 ident: CR5 article-title: On well definedness of the central path publication-title: J. Optim. Theory Appl. doi: 10.1023/A:1021768121263 – volume: 5 start-page: 51 issue: 1 year: 2003 end-page: 92 ident: CR3 article-title: New interior-point algorithms in linear optimization publication-title: J. Adv. Model. optim. – volume: LIV start-page: 121 issue: 2 year: 2009 end-page: 138 ident: CR4 article-title: A predictor-corrector algorithm for linearly constrained convex optimization publication-title: Studia Univ. Babes-Bolyai. Ser. Inform. – year: 1997 ident: CR8 publication-title: Interior point algorithms. Theory and Analysis doi: 10.1002/9781118032701 – volume: 12 start-page: 475 issue: 6 year: 2008 end-page: 780 ident: CR9 article-title: A new primal-dual path-following interior-point algorithm for linearly constrained convex optimization publication-title: J. Shanghai Univ. doi: 10.1007/s11741-008-0602-3 – volume: 25 start-page: 97 year: 2006 end-page: 110 ident: CR1 article-title: A new primal-dual path-following method for convex quadratic programming publication-title: Comput. Appl. Math. – volume: 49 start-page: 23 year: 2000 end-page: 39 ident: CR6 article-title: New complexity analysis of the primal-dual Newton method for linear optimization publication-title: Ann. Oper. Res. doi: 10.1023/A:1019280614748 – volume: LVII start-page: 48 issue: 1 year: 2012 end-page: 58 ident: CR2 article-title: A primal-dual interior point algorithm for convex quadratic programs publication-title: Studia Univ. Babes-Bolyai. Ser. Inform. – year: 1997 ident: CR7 publication-title: Theory and Algorithms for linear optimization. An interior Point Approach – volume: 25 start-page: 97 year: 2006 ident: 9955_CR1 publication-title: Comput. Appl. Math. doi: 10.1590/S0101-82052006000100005 – volume-title: Theory and Algorithms for linear optimization. An interior Point Approach year: 1997 ident: 9955_CR7 – volume: 5 start-page: 51 issue: 1 year: 2003 ident: 9955_CR3 publication-title: J. Adv. Model. optim. – volume: 49 start-page: 23 year: 2000 ident: 9955_CR6 publication-title: Ann. Oper. Res. doi: 10.1023/A:1019280614748 – volume: 12 start-page: 475 issue: 6 year: 2008 ident: 9955_CR9 publication-title: J. Shanghai Univ. doi: 10.1007/s11741-008-0602-3 – volume: LIV start-page: 121 issue: 2 year: 2009 ident: 9955_CR4 publication-title: Studia Univ. Babes-Bolyai. Ser. Inform. – volume: LVII start-page: 48 issue: 1 year: 2012 ident: 9955_CR2 publication-title: Studia Univ. Babes-Bolyai. Ser. Inform. – volume: 102 start-page: 223 issue: N2 year: 1999 ident: 9955_CR5 publication-title: J. Optim. Theory Appl. doi: 10.1023/A:1021768121263 – volume-title: Interior point algorithms. Theory and Analysis year: 1997 ident: 9955_CR8 doi: 10.1002/9781118032701
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SubjectTerms	Algebra Algorithms Complexity Computer Science Convexity Numeric Computing Numerical Analysis Optimization Original Paper Polynomials Theory of Computation
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Title	Complexity analysis and numerical implementation of a full-Newton step interior-point algorithm for LCCO
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