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48
Monotonicity of primaldual interiorpoint algorithms for semidefinite programming problems
, 1998
"... We present primaldual interiorpoint algorithms with polynomial iteration bounds to find approximate solutions of semidefinite programming problems. Our algorithms achieve the current best iteration bounds and, in every iteration of our algorithms, primal and dual objective values are strictly imp ..."
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Cited by 181 (34 self)
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We present primaldual interiorpoint algorithms with polynomial iteration bounds to find approximate solutions of semidefinite programming problems. Our algorithms achieve the current best iteration bounds and, in every iteration of our algorithms, primal and dual objective values are strictly improved.
On the NesterovTodd direction in semidefinite programming
 SIAM Journal on Optimization
, 1996
"... Nesterov and Todd discuss several pathfollowing and potentialreduction interiorpoint methods for certain convex programming problems. In the special case of semidefinite programming, we discuss how to compute the corresponding directions efficiently, how to view them as Newton directions, and how ..."
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Cited by 108 (22 self)
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Nesterov and Todd discuss several pathfollowing and potentialreduction interiorpoint methods for certain convex programming problems. In the special case of semidefinite programming, we discuss how to compute the corresponding directions efficiently, how to view them as Newton directions, and how to take Mehrotra predictorcorrector steps in this framework. We also provide some computational results suggesting that our algorithm is more robust than alternative methods.
Superlinear Convergence Of A Symmetric PrimalDual Path Following Algorithm For Semidefinite Programming
 SIAM Journal on Optimization
, 1996
"... This paper establishes the superlinear convergence of a symmetric primaldual path following algorithm for semidefinite programming under the assumptions that the semidefinite program has a strictly complementary primaldual optimal solution and that the size of the central path neighborhood tends t ..."
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Cited by 50 (5 self)
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This paper establishes the superlinear convergence of a symmetric primaldual path following algorithm for semidefinite programming under the assumptions that the semidefinite program 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 MizunoTodd Ye predictorcorrector method for linear programming which is known to be quadratically 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 superlinearly with order 2 1+2 \Gamma2r . The proof relies on a careful analysis of the central path for semidefinite programming. It is shown that under the strict complementarity assumption, the primaldual central path converges to the analytic center of the primald...
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 dete ..."
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Cited by 47 (9 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...
A superlinearly convergent predictorcorrector method for degenerate LCP in a wide neighborhood of the central path with O (√n L)iteration complexity
, 2006
"... ..."
Initialization in Semidefinite Programming Via a SelfDual SkewSymmetric Embedding
, 1996
"... The formulation of interior point algorithms for semidefinite programming has become an active research area, following the success of the methods for large scale linear programming. Many interior point methods for linear programming have now been extended to the more general semidefinite case, bu ..."
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Cited by 32 (10 self)
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The formulation of interior point algorithms for semidefinite programming has become an active research area, following the success of the methods for large scale linear programming. Many interior point methods for linear programming have now been extended to the more general semidefinite case, but the initialization problem remained unsolved. In this paper we show that the initialization strategy of embedding the problem in a selfdual skewsymmetric problem can also be extended to the semidefinite case. This way the initialization problem of semidefinite problems is solved. This method also provides solution for the initialization of quadratic programs and it is applicable to more general convex problems with conic formulation. Key words: Semidefinite programming, complementarity, skewsymmetric embedding, initialization, selfdual problems, central path. iii 1 Introduction The extension of interior point algorithms from linear programming (LP) to semidefinite programmi...
Polynomial Convergence of a New Family of PrimalDual Algorithms for Semidefinite Programming
, 1996
"... This paper establishes the polynomial convergence of a new class of (feasible) primaldual interiorpoint path following algorithms for semidefinite programming (SDP) whose search directions are obtained by applying Newton method to the symmetric central path equation (P T XP ) 1=2 (P \Gamma1 ..."
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Cited by 24 (8 self)
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This paper establishes the polynomial convergence of a new class of (feasible) primaldual interiorpoint path following algorithms for semidefinite programming (SDP) whose search directions are obtained by applying Newton method to the symmetric central path equation (P T XP ) 1=2 (P \Gamma1 SP \GammaT )(P T XP ) 1=2 \Gamma I = 0; where P is a nonsingular matrix. Specifically, we show that the shortstep path following algorithm based on the Frobenius norm neighborhood and the semilongstep path following algorithm based on the operator 2norm neighborhood have O( p nL) and O(nL) iterationcomplexity bounds, respectively. When P = I, this yields the first polynomially convergent semilongstep algorithm based on a pure Newton direction. Restricting the scaling matrix P at each iteration to a certain subset of nonsingular matrices, we are able to establish an O(n 3=2 L) iterationcomplexity for the longstep path following method. The resulting subclass of search direct...
Duality And SelfDuality For Conic Convex Programming
, 1996
"... This paper considers the problem of minimizing a linear function over the intersection of an affine space with a closed convex cone. In the first half of the paper, we give a detailed study of duality properties of this problem and present examples to illustrate these properties. In particular, we i ..."
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Cited by 20 (7 self)
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This paper considers the problem of minimizing a linear function over the intersection of an affine space with a closed convex cone. In the first half of the paper, we give a detailed study of duality properties of this problem and present examples to illustrate these properties. In particular, we introduce the notions of weak/strong feasibility or infeasibility for a general primaldual pair of conic convex programs, and then establish various relations between these notions and the duality properties of the problem. In the second half of the paper, we propose a selfdual embedding with the following properties: Any weakly centered sequence converging to a complementary pair either induces a sequence converging to a certificate of strong infeasibility, or induces a sequence of primaldual pairs for which the amount of constraint violation converges to zero, and the corresponding objective values are in the limit not worse than the optimal objective value(s). In case of strong duality, ...
Implementation of PrimalDual Methods for Semidefinite Programming Based on Monteiro and Tsuchiya Newton Directions and their Variants
 TECHNICAL REPORT, SCHOOL INDUSTRIAL AND SYSTEMS ENGINEERING, GEORGIA TECH., ATLANTA, GA 30332
, 1997
"... Monteiro and Tsuchiya [23] have proposed two primaldual Newton directions for semidefinite programming, referred to as the MT directions, and established polynomial convergence of path following methods based on them. This paper reports some computational results on the performance of interiorpoin ..."
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Cited by 20 (3 self)
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Monteiro and Tsuchiya [23] have proposed two primaldual Newton directions for semidefinite programming, referred to as the MT directions, and established polynomial convergence of path following methods based on them. This paper reports some computational results on the performance of interiorpoint predictorcorrector methods based on the MT directions and a variant of these directions, called the SChMT direction. We discuss how to compute these directions efficiently and derive their corresponding computational complexities. A main feature of our analysis is that computational formulae for these directions are derived from a unified point of view which entirely avoids the use of Kronecker product. Using this unified approach, we also present schemes to compute the AlizadehHaeberlyOverton (AHO) direction, the NesterovTodd direction and the HRVW/KSH/M direction with computational complexities (for dense problems) better than previously reported in the literature. Our computational...
A Unified Analysis for a Class of LongStep PrimalDual PathFollowing InteriorPoint Algorithms for Semidefinite Programming
 MATH. PROGRAMMING
, 1998
"... We present a unified analysis for a class of longstep primaldual pathfollowing algorithms for semidefinite programming whose search directions are obtained through linearization of the symmetrized equation of the central path HP (XS) j [P XSP \Gamma1 + (PXSP \Gamma1 ) T ]=2 = ¯I, introduce ..."
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Cited by 19 (0 self)
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We present a unified analysis for a class of longstep primaldual pathfollowing algorithms for semidefinite programming whose search directions are obtained through linearization of the symmetrized equation of the central path HP (XS) j [P XSP \Gamma1 + (PXSP \Gamma1 ) T ]=2 = ¯I, introduced by Zhang. At an iterate (X; S), we choose a scaling matrix P from the class of nonsingular matrices P such that PXSP \Gamma1 is symmetric. This class of matrices includes the three wellknown choices, namely: P = S 1=2 and P = X \Gamma1=2 proposed by Monteiro, and the matrix P corresponding to the NesterovTodd direction. We show that within the class of algorithms studied in this paper, the one based on the NesterovTodd direction has the lowest possible iterationcomplexity bound that can provably be derived from our analysis. More specifically, its iterationcomplexity bound is of the same order as that of the corresponding longstep primaldual pathfollowing algorithm for linear...