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168
On Extending Some PrimalDual InteriorPoint Algorithms From Linear Programming to Semidefinite Programming
 SIAM Journal on Optimization
, 1998
"... This work concerns primaldual interiorpoint methods for semidefinite programming (SDP) that use a search direction originally proposed by HelmbergRendlVanderbeiWolkowicz [5] and KojimaShindohHara [11], and recently rediscovered by Monteiro [15] in a more explicit form. In analyzing these meth ..."
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Cited by 75 (2 self)
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This work concerns primaldual interiorpoint methods for semidefinite programming (SDP) that use a search direction originally proposed by HelmbergRendlVanderbeiWolkowicz [5] and KojimaShindohHara [11], and recently rediscovered by Monteiro [15] in a more explicit form. In analyzing these methods, a number of basic equalities and inequalities were developed in [11] and also in [15] through different means and in different forms. In this paper, we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming (LP), producing basic relationships that have compact forms almost identical to their counterparts in LP. We also introduce a new formulation of the central path and variablemetric measures of centrality. These results provide convenient tools for deriving polynomiality results for primaldual algorithms extended from LP to SDP using the aforementioned and related search directions. We present examples...
Polynomial Convergence of PrimalDual Algorithms for Semidefinite Programming Based on Monteiro and Zhang Family of Directions
 School of ISyE, Georgia Institute of Technology, Atlanta, GA 30332
, 1997
"... This paper establishes the polynomialconvergence of the class of primaldual feasible interiorpoint algorithms for semidefinite programming (SDP) based on Monteiro and Zhang family of search directions. In contrast to Monteiro and Zhang's work, no condition is imposed on the scaling matrix that ..."
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Cited by 72 (11 self)
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This paper establishes the polynomialconvergence of the class of primaldual feasible interiorpoint algorithms for semidefinite programming (SDP) based on Monteiro and Zhang family of search directions. In contrast to Monteiro and Zhang's work, no condition is imposed on the scaling matrix that determines the search direction. We show that the polynomial iterationcomplexity bounds of two wellknown algorithms for linear programming, namely the shortstep pathfollowing algorithm of Kojima et al. and Monteiro and Adler, and the predictorcorrector algorithm of Mizuno et al., carry over to the context of SDP. Since Monteiro and Zhang family of directions includes the Alizadeh, Haeberly and Overton direction, we establish for the first time the polynomial convergence of algorithms based on this search direction. Keywords: Semidefinite programming, interiorpoint methods, polynomial complexity, pathfollowing methods, primaldual methods. AMS 1991 subject classification: 65K05, 90C25, 90C...
S.: Superlinear convergence of a symmetric primaldual pathfollowing algorithm for semidefinite programming
 SIAM Journal on Optimization
, 1998
"... Abstract This paper establishes the superlinear convergence of a symmetric primaldual path following algorithm for semidenite programming under the assumptions that the semidenite pro gram has a strictly complementary primaldual optimal solution and that the size of the central path neighborhood te ..."
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Cited by 63 (5 self)
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Abstract This paper establishes the superlinear convergence of a symmetric primaldual path following algorithm for semidenite programming under the assumptions that the semidenite pro gram has a strictly complementary primaldual optimal solution and that the size of the central path neighborhood tends to zero The interior point algorithm considered here closely resembles the MizunoToddYe predictorcorrectormethod for linear programmingwhere it is known to be quadrat ically convergent It is shown that when the iterates are well centered the duality gap is reduced superlinearly after each predictor step Indeed if each predictor step is succeeded by r consecutive corrector steps then the predictor reduces the duality gap superlinearlywith order
A Superlinearly Convergent PrimalDual InfeasibleInteriorPoint Algorithm for Semidefinite Programming
 Department of Mathematics, The University of Iowa, Iowa City, IA
, 1995
"... . A primaldual infeasibleinteriorpoint pathfollowing algorithm is proposed for solving semidefinite programming (SDP) problems. If the problem has a solution, then the algorithm is globally convergent. If the starting point is feasible or close to being feasible, the algorithms finds an optimal ..."
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Cited by 61 (9 self)
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. A primaldual infeasibleinteriorpoint pathfollowing algorithm is proposed for solving semidefinite programming (SDP) problems. If the problem has a solution, then the algorithm is globally convergent. If the starting point is feasible or close to being feasible, the algorithms finds an optimal solution in at most O( p nL) iterations, where n is the size of the problem and L is the logarithm of the ratio of the initial error and the tolerance. If the starting point is large enough then the algorithm terminates in at most O(nL) steps either by finding a solution or by determining that the primaldual problem has no solution of norm less than a given number. Moreover, we propose a sufficient condition for the superlinear convergence of the algorithm. In addition, we give two special cases of SDP for which the algorithm is quadratically convergent. Key words. semidefinite programming, pathfollowing, infeasibleinteriorpoint algorithm, polynomiality, superlinear convergence. AMS ...
Local Convergence of PredictorCorrector InfeasibleInteriorPoint Algorithms for SDPs and SDLCPs
 Mathematical Programming
, 1997
"... . An example of SDPs (semidefinite programs) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the MizunoToddYe type predictorcorrector primaldual interiorpoint method for LPs (linear programs) to SDPs, and suggests that we need to force the genera ..."
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Cited by 60 (4 self)
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. An example of SDPs (semidefinite programs) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the MizunoToddYe type predictorcorrector primaldual interiorpoint method for LPs (linear programs) to SDPs, and suggests that we need to force the generated sequence to converge to a solution tangentially to the central path (or trajectory). A MizunoToddYe type predictorcorrector infeasibleinteriorpoint algorithm incorporating this additional restriction for monotone SDLCPs (semidefinite linear complementarity problems) enjoys superlinear convergence under strict complementarity and nondegeneracy conditions. Key words. Semidefinite Programming, InfeasibleInteriorPoint Method, PredictorCorrectorMethod, Superlinear Convergence, PrimalDual Nondegeneracy Abbreviated Title. InteriorPoint Algorithms for SDPs y Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2121 OhOkayama, Meguroku, Tokyo 152, Japa...
Symmetric PrimalDual Path Following Algorithms for Semidefinite Programming
, 1996
"... In this paper a symmetric primaldual transformation for positive semidefinite programming is proposed. For standard SDP problems, after this symmetric transformation the primal variables and the dual slacks become identical. In the context of linear programming, existence of such a primaldual tran ..."
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Cited by 60 (11 self)
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In this paper a symmetric primaldual transformation for positive semidefinite programming is proposed. For standard SDP problems, after this symmetric transformation the primal variables and the dual slacks become identical. In the context of linear programming, existence of such a primaldual transformation is a well known fact. Based on this symmetric primaldual transformation we derive Newton search directions for primaldual pathfollowing algorithms for semidefinite programming. In particular, we generalize: (1) the short step path following algorithm, (2) the predictorcorrector algorithm and (3) the largest step algorithm to semidefinite programming. It is shown that these algorithms require at most O( p n j log ffl j) main iterations for computing an ffloptimal solution. The symmetric primaldual transformation discussed in this paper can be interpreted as a specialization of the scalingpoint concept introduced by Nesterov and Todd [12] for selfscaled conic problems. The ...
A superlinearly convergent predictorcorrector method for degenerate LCP in a wide neighborhood of the central path with O (√n L)iteration complexity
, 2006
"... ..."
A Convergence Analysis of the Scalinginvariant Primaldual Pathfollowing Algorithms for Secondorder Cone Programming
 Optim. Methods Softw
, 1998
"... This paper is a continuation of our previous paper in which we studied a polynomial primaldual pathfollowing algorithm for SOCP using an analogue of the HRVW/KSH/M direction for SDP. We develop an improved and simplified complexity analysis which can be also applied to the algorithm using the NT di ..."
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Cited by 51 (4 self)
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This paper is a continuation of our previous paper in which we studied a polynomial primaldual pathfollowing algorithm for SOCP using an analogue of the HRVW/KSH/M direction for SDP. We develop an improved and simplified complexity analysis which can be also applied to the algorithm using the NT direction. Specifically, we show that the longstep algorithm using the NT direction has O(n log " 01 ) iterationcomplexity to reduce the duality gap by a factor of ", where n is the number of the secondorder cones. The complexity for the same algorithm using the HRVW/KSH/M direction is improved to O(n 3=2 log " 01 ) from O(n 3 log " 01 ) of the previous analysis. We also show that the short and semilongstep algorithms using the NT direction (and the HRVW/KSH/M direction) have O( p n log " 01 ) and O(n log " 01 ) iterationcomplexities, respectively. keywords: secondorder cone, interiorpoint methods, polynomial complexity, primaldual pathfollowing methods. 1 Introduction...
Exploiting sparsity in semidefinite programming via matrix completion II: implementation and numerical results
"... In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primaldual interiorpoint methods. This framework is based on some results about po ..."
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Cited by 50 (17 self)
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In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primaldual interiorpoint methods. This framework is based on some results about positive semidefinite matrix completion, and it can be embodied in two di#erent ways. One is by a conversion of a given sparse SDP having a large scale positive semidefinite matrix variable into an SDP having multiple but smaller positive semidefinite matrix variables. The other is by incorporating a positive definite matrix completion itself in a primaldual interiorpoint method. The current article presents the details of their implementations. We introduce new techniques to deal with the sparsity through a clique tree in the former method and through new computational formulae in the latter one. Numerical results over di#erent classes of SDPs show that these methods can be very e#cient for some problems. Keywords: Semidefinite programming; Primaldual interiorpoint method; Matrix completion problem; Clique tree; Numerical results. # Department of Applied Physics, The University of Tokyo, 731 Hongo, Bunkyoku, Tokyo 1138565 Japan (nakata@zzz.t.utokyo.ac.jp ). + Department of Architecture and Architectural Systems, Kyoto University, Kyoto 6068501 Japan (fujisawa@ismj.archi.kyotou.ac.jp). # Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2121 OhOkayama, Meguroku, Tokyo 1528552 Japan (mituhiro@is.titech.ac.jp). The author was supported by The Ministry of Education, Culture, Sports, Science and Technology of Japan. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2121 OhOkayama, Meguroku, Toky...
Local minima and convergence in lowrank semidefinite programming
 Mathematical Programming
, 2003
"... The lowrank semidefinite programming problem (LRSDPr) is a restriction of the semidefinite programming problem (SDP) in which a bound r is imposed on the rank of X, and it is well known that LRSDPr is equivalent to SDP if r is not too small. In this paper, we classify the local minima of LRSDPr and ..."
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Cited by 44 (2 self)
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The lowrank semidefinite programming problem (LRSDPr) is a restriction of the semidefinite programming problem (SDP) in which a bound r is imposed on the rank of X, and it is well known that LRSDPr is equivalent to SDP if r is not too small. In this paper, we classify the local minima of LRSDPr and prove the optimal convergence of a slight variant of the successful, yet experimental, algorithm of Burer and Monteiro [6], which handles LRSDPr via the nonconvex change of variables X = RR T. In addition, for particular problem classes, we describe a practical technique for obtaining lower bounds on the optimal solution value during the execution of the algorithm. Computational results are presented on a set of combinatorial optimization relaxations, including some of the largest quadratic assignment SDPs solved to date.