Polynomial Convergence of Infeasible-Interior-Point Methods over Symmetric Cones

Rangarajan, Bharath Kumar

doi:10.1137/040606557

Cited by 86 publications

(35 citation statements)

References 14 publications

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“…Therefore, (26) yields x = 0, and it follows from (25) that x = 0. Since A has full row rank, (21) implies y = 0. Thus the linear system of equations (19) has only zero solution, and hence G (z) is nonsingular.…”

Section: Algorithm 41 (A Smoothing Newton-type Methods For Socp)mentioning

confidence: 99%

A smoothing Newton-type method for second-order cone programming problems based on a new smoothing Fischer-Burmeister function

Fang¹,

Feng²

2011

Comput. Appl. Math.

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Abstract.A new smoothing function of the well known Fischer-Burmeister function is given.Based on this new function, a smoothing Newton-type method is proposed for solving secondorder cone programming. At each iteration, the proposed algorithm solves only one system of linear equations and performs only one line search. This algorithm can start from an arbitrary point and it is Q-quadratically convergent under a mild assumption. Numerical results demonstrate the effectiveness of the algorithm.Mathematical subject classification: 90C25, 90C30, 90C51, 65K05, 65Y20.

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Section: Algorithm 41 (A Smoothing Newton-type Methods For Socp)mentioning

confidence: 99%

A smoothing Newton-type method for second-order cone programming problems based on a new smoothing Fischer-Burmeister function

Fang¹,

Feng²

2011

Comput. Appl. Math.

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“…Luo and Xiu [10] first established a theoretical framework of path-following interior-point algorithms for the Cartesian P * (κ)-SCLCP and proved the global convergence and the iteration complexities of the proposed algorithms. In addition to Faybusovich's results [3,4], Rangarajan [11] proposed the first infeasible interior-point method (IIPM) for SCLCP. Yoshise [18] was the first to analyze IPMs for nonlinear complementarity problems over symmetric cones.…”

Section: Introductionmentioning

confidence: 99%

An interior-point method for the Cartesian P*(k)-linear complementarity problem over symmetric cones

Kheirfam¹

2014

ORiON

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A novel primal-dual path-following interior-point algorithm for the Cartesian P * (κ)-linear complementarity problem over symmetric cones is presented. The algorithm is based on a reformulation of the central path for finding the search directions. For a full Nesterov-Todd step feasible interior-point algorithm based on the new search directions, the complexity bound of the algorithm with small-update approach is the best-available bound.

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“…This is most notable for symmetric cone programming [1,3,8,14]) where Jordan algebraic structures of symmetric cones were used. The primal-dual algorithms and their analyses in this paper can be applied to symmetric cone programming as a special case of homogeneous cones programming.…”

Section: Introductionmentioning

confidence: 99%

A T-Algebraic Approach to Primal-Dual Interior-Point Algorithms

Chua¹

2009

SIAM J. Optim.

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Abstract. Three primal-dual interior-point algorithms for homogeneous cone programming are presented. They are a short-step algorithm, a large-update algorithm, and a predictor-corrector algorithm. These algorithms are described and analyzed based on a characterization of homogeneous cone via T -algebra. The analysis show that the algorithms have polynomial iteration complexity.Key words and phrases. Homogeneous cone programming; T-algebra; Primal-dual interior-point algorithm. IntroductionPrimal-dual interior-point algorithms-first designed for linear programming (see, e.g., [19]), and subsequently extended to semidefinite programming (see, e.g., [18, Part II]) and symmetric cone programming (see, e.g., [13])-are the most widely used interior-point algorithms. At the same time, they are able to achieve the best iteration complexity bound known to date. These strongly motivates further development of primal-dual interior-point algorithms for wider classes of optimization problems.Various studies to extend primal-dual algorithms beyond symmetric cone programming involves the v-space approach; i.e., the use of scaling points. Tunçel [15] showed that primal-dual symmetric algorithms that use the v-space approach can only be designed for symmetric cone programming. By dropping primal-dual symmetry, Tunçel [16] designed primal-dual algorithms based on the v-space approach. However, polynomial iteration complexity bounds were established only for symmetric cone programming. This paper focuses on primal-dual interior-point algorithms for homogeneous cone programming.Homogeneous cone programming, which will be formally defined in the next section, is a class of convex programming problems. It includes, as a proper sub-class, symmetric cone programming. While there is a finite number of non-isomorphic symmetric cones of each dimension, this number is uncountable for homogeneous cones when the dimension is at least eleven [17]. Thus the collection of homogeneous cones is significantly larger than the subclass of symmetric cones.On the other hand, the author [4] proved that every homogeneous cone can be represented as the intersection of some cone of symmetric positive definite matrices with a suitable linear subspace. Thus all homogeneous cone programming problems can be formulated as semidefinite programming problems, which are special cases of symmetric cone programming problems. However it is argued in the same paper that there are advantages in designing algorithms specifically for homogeneous cone programming as the ranks of these cones-which determines the iteration complexity of interior-point The primal-dual algorithms in this paper rely on a characterization of homogeneous cones via T -algebras.Algebraic structures of convex cones had been successfully exploited in the design and analysis of interior-point algorithms for convex conic programming. This is most notable for symmetric cone programming [1,3,8,14]) where Jordan algebraic structures of symmetric cones were used. The primal-dual algorithms and their anal...

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Polynomial Convergence of Infeasible-Interior-Point Methods over Symmetric Cones

Cited by 86 publications

References 14 publications

A smoothing Newton-type method for second-order cone programming problems based on a new smoothing Fischer-Burmeister function

A smoothing Newton-type method for second-order cone programming problems based on a new smoothing Fischer-Burmeister function

An interior-point method for the Cartesian P*(k)-linear complementarity problem over symmetric cones

A T-Algebraic Approach to Primal-Dual Interior-Point Algorithms

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