Asymptotic behavior of the central path for a special class of degenerate SDP problems

This paper studies the asymptotic behavior of the central path (X(), S(), y()) as 0 for a class of degenerate semidefinite programming (SDP) problems, namely those that do not have strictly complementary primal-dual optimal solutions and whose degenerate diagonal blocks XT () and ST () of the centra...

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Vydané v:	Mathematical programming Ročník 103; číslo 3; s. 487 - 514
Hlavní autori:	Neto, João X. da Cruz, Ferreira, Orizon P., Monteiro, Renato D. C.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Heidelberg Springer 01.07.2005 Springer Nature B.V
Predmet:	Applied sciences Euclidean space Exact sciences and technology Linear programming Mathematical programming Operational research and scientific management Operational research. Management science Semidefinite programming Studies Semidefinite programming Limiting behavior Optimal solution Asymptotic behavior Primal dual method Semi definite programming Convex quadratic programming Convex quadratically constrained programming Quadratic programming Convex programming Central path
ISSN:	0025-5610, 1436-4646
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Abstract	This paper studies the asymptotic behavior of the central path (X(), S(), y()) as 0 for a class of degenerate semidefinite programming (SDP) problems, namely those that do not have strictly complementary primal-dual optimal solutions and whose degenerate diagonal blocks XT () and ST () of the central path are assumed to satisfy max{XT (), ST ()} = O(). We establish the convergence of the central path towards a primal-dual optimal solution, which is characterized as being the unique optimal solution of a certain log-barrier problem. A characterization of the class of SDP problems which satisfy our assumptions are also provided. It is shown that the re-parametrization t> 0 (X(t4), S(t4), y(t4)) of the central path is analytic at t = 0. The limiting behavior of the derivative of the central path is also investigated and it is shown that the order of convergence of the central path towards its limit point is O(). Finally, we apply our results to the convex quadratically constrained convex programming (CQCCP) problem and characterize the class of CQCCP problems which can be formulated as SDPs satisfying the assumptions of this paper. In particular, we show that CQCCP problems with either a strictly convex objective function or at least one strictly convex constraint function lie in this class. [PUBLICATION ABSTRACT]
AbstractList	This paper studies the asymptotic behavior of the central path (X(), S(), y()) as 0 for a class of degenerate semidefinite programming (SDP) problems, namely those that do not have strictly complementary primal-dual optimal solutions and whose degenerate diagonal blocks XT () and ST () of the central path are assumed to satisfy max{XT (), ST ()} = O(). We establish the convergence of the central path towards a primal-dual optimal solution, which is characterized as being the unique optimal solution of a certain log-barrier problem. A characterization of the class of SDP problems which satisfy our assumptions are also provided. It is shown that the re-parametrization t> 0 (X(t4), S(t4), y(t4)) of the central path is analytic at t = 0. The limiting behavior of the derivative of the central path is also investigated and it is shown that the order of convergence of the central path towards its limit point is O(). Finally, we apply our results to the convex quadratically constrained convex programming (CQCCP) problem and characterize the class of CQCCP problems which can be formulated as SDPs satisfying the assumptions of this paper. In particular, we show that CQCCP problems with either a strictly convex objective function or at least one strictly convex constraint function lie in this class. [PUBLICATION ABSTRACT]
Author	Ferreira, Orizon P. Neto, João X. da Cruz Monteiro, Renato D. C.
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Cites_doi	10.1007/BF01594923 10.1016/S0167-6377(00)00065-1 10.1090/conm/114/1097873 10.1137/S1052623496299187 10.1287/moor.25.3.381.12216 10.1016/S0377-2217(02)00276-X 10.1016/S0167-6377(97)00011-4 10.1080/10556789808805693 10.1016/0167-6377(88)90047-8 10.1137/S1052623494269035 10.2140/pjm.1980.88.101 10.1287/moor.15.4.662 10.1007/s002110050407 10.1287/moor.21.4.793 10.1007/3-540-54509-3 10.1287/moor.21.3.629 10.1007/978-1-4615-4381-7_10 10.1007/BF01581702 10.1515/9781400881819 10.1080/02331930290019396 10.1007/s10107-003-0463-x 10.1007/978-3-662-06409-2 10.1287/moor.23.1.39 10.1007/s101070050025 10.1007/978-1-4613-9617-8_8 10.1287/moor.16.4.842 10.1007/b105286 10.1023/A:1018339901042 10.1137/S1052623401390793 10.1137/S105262349630009X
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Title	Asymptotic behavior of the central path for a special class of degenerate SDP problems
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