Complexity Analysis of an Interior Point Algorithm for the Semidefinite Optimization Based on a Kernel Function with a Double Barrier Term
In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization（SDO） problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barri...
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| Published in: | Acta mathematica Sinica. English series Vol. 31; no. 3; pp. 543 - 556 |
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| Format: | Journal Article |
| Language: | English |
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Heidelberg
Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society
01.03.2015
Springer Nature B.V |
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| ISSN: | 1439-8516, 1439-7617 |
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| Abstract | In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization（SDO） problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q1 〉 q2 〉 1, the algorithm has O（（q1 ＋ 1） nq1＋1/2（q1-q2）logn/ε）and O（（q1 ＋ 1）2（q1-q2）^3q1-2q2＋1√n logn/c） complexity results for large- and small-update methods, respectively. |
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| AbstractList | (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization (SDO) problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q sub(1) > q sub(2) > 1, the algorithm has ... and ... complexity results for large- and small-update methods, respectively. In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization (SDO) problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q 1 > q 2 > 1, the algorithm has and complexity results for large- and small-update methods, respectively. In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization（SDO） problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q1 〉 q2 〉 1, the algorithm has O（（q1 ＋ 1） nq1＋1/2（q1-q2）logn/ε）and O（（q1 ＋ 1）2（q1-q2）^3q1-2q2＋1√n logn/c） complexity results for large- and small-update methods, respectively. |
| Author | Mohamed ACHACHE |
| AuthorAffiliation | Laboratoire de Mathematiques Fondamentales et Numeriques,Faculte des Sciences, Universite Ferhat Abbas Sdtif 1, Algerie |
| Author_xml | – sequence: 1 givenname: Mohamed surname: Achache fullname: Achache, Mohamed email: achache_m@yahoo.fr organization: Laboratoire de Mathématiques Fondamentales et Numériques, Faculté des Sciences, Université Ferhat Abbas Sétif 1 |
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| Cites_doi | 10.1007/s101070200296 10.1137/S1052623403423114 10.1017/CBO9780511840371 10.1137/0805002 10.1016/S0377-2217(02)00275-8 10.1016/j.na.2009.05.078 10.1137/1038003 10.1080/1055678021000090024 10.1080/10556789908805745 10.1016/j.cam.2010.12.012 10.1590/S0101-82052006000100005 10.1016/j.na.2009.05.064 |
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| Copyright | Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2015 |
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| DOI | 10.1007/s10114-015-1314-4 |
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| Keywords | kernel functions 90C51 65K05 complexity of algorithms 90C25 90C22 Semidefinite optimization primal-dual interior point methods large and small-update algorithms |
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| Notes | 11-2039/O1 Semidefinite optimization, kernel functions, primal-dual interior point methods, large andsmall-update algorithms, complexity of algorithms In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization（SDO） problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q1 〉 q2 〉 1, the algorithm has O（（q1 ＋ 1） nq1＋1/2（q1-q2）logn/ε）and O（（q1 ＋ 1）2（q1-q2）^3q1-2q2＋1√n logn/c） complexity results for large- and small-update methods, respectively. SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-1 ObjectType-Feature-2 content type line 23 |
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| References | Choi, Lee (CR6) 2009; 71 Horn, Johnson (CR17) 1991 Liu, Li (CR9) 2011; 235 El Ghami, Steihaug, Roos (CR7) 2006 Cho (CR10) 2009; 13 Bai, Roos, El Ghami (CR5) 2002; 17 Vanderberghe, Boyd (CR16) 1996; 38 Roos, Terlaky, Vial (CR14) 1997 Peng, Roos, Terlaky (CR13) 2002 Peng, Roos, Terlaky (CR11) 2002; 143 El Ghami (CR8) 2005 Achache (CR1) 2006; 25 Bai, El Ghami, Roos (CR3) 2004; 15 Bai, Wang, Roos (CR4) 2009; 70 Peng, Roos, Terlaky (CR12) 2002; 93 Todd (CR15) 1999; 11 Alizadeh (CR2) 2006; 5 M Ghami El (1314_CR7) 2006 J Peng (1314_CR12) 2002; 93 L Liu (1314_CR9) 2011; 235 M Achache (1314_CR1) 2006; 25 J Peng (1314_CR11) 2002; 143 Y Q Bai (1314_CR3) 2004; 15 G M Cho (1314_CR10) 2009; 13 Y Q Bai (1314_CR4) 2009; 70 J Peng (1314_CR13) 2002 L Vanderberghe (1314_CR16) 1996; 38 R A Horn (1314_CR17) 1991 C Roos (1314_CR14) 1997 Y Q Bai (1314_CR5) 2002; 17 F Alizadeh (1314_CR2) 2006; 5 B K Choi (1314_CR6) 2009; 71 M J Todd (1314_CR15) 1999; 11 M Ghami El (1314_CR8) 2005 |
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| SubjectTerms | Algorithms Barriers Complexity Eigenvalues Functions (mathematics) Kernel functions Mathematical analysis Mathematical programming Mathematics Mathematics and Statistics Optimization Polynomials Semidefinite programming Studies Texts 优化 内核函数 内点算法 复杂性分析 多项式复杂性 屏障 期限 路径跟踪 |
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| Title | Complexity Analysis of an Interior Point Algorithm for the Semidefinite Optimization Based on a Kernel Function with a Double Barrier Term |
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