The Analytic Center Cutting Plane Method with Semidefinite Cuts

Oskoorouchi, Mohammad R.; Goffin, Jean-Louis

doi:10.1137/s1052623400374148

Cited by 16 publications

(26 citation statements)

References 42 publications

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“…Cutting plane methods can also be employed to replace a large conic constraint with several smaller constraints; for example, a semidefinite constraint can be replaced by linear and/or second order cone constraints. For more information on cutting plane methods for semidefinite programming and second order cone programming, see [1,2,3,6,10,11,12,14,15,16]. We've previously described a selective orthonormalization procedure for polyhedral relaxations of the convex feasibility problem [7].…”

Section: Conic Cutmentioning

confidence: 99%

“…The variables of this problem are the primal directions d x i and it is necessary to include constraints based on Dikin ellipsoids in order to ensure that the restart point has an appropriate potential function value. For details, see [10,11,12,2].…”

Section: Proofmentioning

confidence: 99%

“…The proofs of global convergence in the literature have examined upper and lower bounds on the dual potential function value (see, for example, [2,4,5,10,11,12]). The lower bound increases more quickly than the upper bound, and the algorithm must terminate before the two bounds meet.…”

Section: Global Convergencementioning

confidence: 99%

“…Nesterov [8] provided such a bound for linear programming, and this was used in the work of Goffin et al [4,5]. This result was generalized by Oskoorouchi and Goffin to the case of a single semidefinite cut [10] and a single SOCP cut [11], and then to the case of multiple SOCP cuts by Oskoorouchi and Mitchell [12]. Basescu and Mitchell [2] extended the result further to general conic cuts.…”

Section: Proofmentioning

confidence: 99%

“…The complexity of the algorithms of Oskoorouchi and Goffin for semidefinite programming [10] and second order cone programming [11] and the algorithm of Basescu and Mitchell [2] depend on a condition number. If the selective Gram-Schmidt orthonormalization procedure is used to modify the cuts, a fixed positive lower bound (namely a simple function of ω) can be placed on these condition numbers, so the dependence on the condition number can be removed.…”

Section: Theoremmentioning

confidence: 99%

See 4 more Smart Citations

Selective Gram–Schmidt orthonormalization for conic cutting surface algorithms

Mitchell

Basescu

2007

Math Meth Oper Res

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It is not straightforward to find a new feasible solution when several conic constraints are added to a conic optimization problem. Examples of conic constraints include semidefinite constraints and second order cone constraints. In this paper, a method to slightly modify the constraints is proposed. Because of this modification, a simple procedure to generate strictly feasible points in both the primal and dual spaces can be defined. A second benefit of the modification is an improvement in the complexity analysis of conic cutting surface algorithms. Complexity results for conic cutting surface algorithms proved to date have depended on a condition number of the added constraints. The proposed modification of the constraints leads to a stronger result, with the convergence of the resulting algorithm not dependent on the condition number.

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Section: Conic Cutmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

Section: Global Convergencementioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

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Selective Gram–Schmidt orthonormalization for conic cutting surface algorithms

Mitchell

Basescu

2007

Math Meth Oper Res

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Interior Point and Semidefinite Approaches in Combinatorial Optimization

Krishnan¹,

Terlaky²

Graph Theory and Combinatorial Optimization

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Conic programming, especially semidefinite programming (SDP), has been regarded as linear programming for the 21st century. This tremendous excitement was spurred in part by a variety of applications of SDP in integer programming (IP) and combinatorial optimization, and the development of efficient primal-dual interior-point methods (IPMs) and various first order approaches for the solution of large scale SDPs. This survey presents an up to date account of semidefinite and interior point approaches in solving NP-hard combinatorial optimization problems to optimality, and also in developing approximation algorithms for some of them. The interior point approaches discussed in the survey have been applied directly to non-convex formulations of IPs; they appear in a cutting plane framework to solving IPs, and finally as a subroutine to solving SDP relaxations of IPs. The surveyed approaches include non-convex potential reduction methods, interior point cutting plane methods, primal-dual IPMs and first-order algorithms for solving SDPs, branch and cut approaches based on SDP relaxations of IPs, approximation algorithms based on SDP formulations, and finally methods employing successive convex approximations of the underlying combinatorial optimization problem.

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Recent Progress in Interior-Point Methods: Cutting-Plane Algorithms and Warm Starts

Engau¹

2011

International Series in Operations Research &Amp; Management Science

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The Analytic Center Cutting Plane Method with Semidefinite Cuts

Cited by 16 publications

References 42 publications

Selective Gram–Schmidt orthonormalization for conic cutting surface algorithms

Selective Gram–Schmidt orthonormalization for conic cutting surface algorithms

Interior Point and Semidefinite Approaches in Combinatorial Optimization

Recent Progress in Interior-Point Methods: Cutting-Plane Algorithms and Warm Starts

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