Semidefinite and Second-Order Cone Programming
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Second-order cone programming
Semidefinite Programming
Interior point method
Conic optimization
Second-order cone programming
Semidefinite Programming
Interior point method
Conic optimization
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Semidefinite Programming
Linear matrix inequality
Conic optimization
Semidefinite embedding
Second-order cone programming
Interior point method
Matrix (chemical analysis)
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In this paper we continue the development of a theoretical foundation for efficient primal-dual interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled (see Yu. E. Nesterov and M. J. Todd, Math. Oper. Res., 22 (1997), pp. 1--42). The class of problems under consideration includes linear programming, semidefinite programming, and convex quadratically constrained, quadratic programming problems. For such problems we introduce a new definition of affine-scaling and centering directions. We present efficiency estimates for several symmetric primal-dual methods that can loosely be classified as path-following methods. Because of the special properties of these cones and barriers, two of our algorithms can take steps that typically go a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier.
Interior point method
Second-order cone programming
Conic optimization
Quadratic growth
Semidefinite Programming
Conic section
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In this paper we continue the development of a theoretical foundation for efficient primal-dual interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled (see Yu. E. Nesterov and M.J. Todd, Math. Oper. Res., 22 (1997), pp. 1--42). The class of problems under consideration includes linear programming, semidefinite programming, and convex quadratically constrained, quadratic programming problems. For such problems we introduce a new definition of affine-scaling and centering directions. We present efficiency estimates for several symmetric primal-dual methods that can loosely be classified as path-following methods. Because of the special properties of these cones and barriers, two of our algorithms can take steps that typically go a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier.
Interior point method
Second-order cone programming
Quadratic growth
Semidefinite Programming
Conic optimization
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We propose a homogeneous primal-dual interior-point method to solve sum-of-squares optimization problems by combining non-symmetric conic optimization techniques and polynomial interpolation. The approach optimizes directly over the sum-of-squares cone and its dual, circumventing the semidefinite programming (SDP) reformulation which requires a large number of auxiliary variables. As a result, it has substantially lower theoretical time and space complexity than the conventional SDP-based approach. Although our approach avoids the semidefinite programming reformulation, an optimal solution to the semidefinite program can be recovered with little additional effort. Computational results confirm that for problems involving high-degree polynomials, the proposed method is several orders of magnitude faster than semidefinite programming.
Semidefinite Programming
Conic optimization
Semidefinite embedding
Interior point method
Second-order cone programming
Conic section
Least-squares function approximation
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In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many applications in engineering and combinatorial optimization. Although semidefinite programs are much more general than linear programs, they are not much harder to solve. Most interior-point methods for linear programming have been generalized to semidefinite programs. As in linear programming, these methods have polynomial worst-case complexity and perform very well in practice. This paper gives a survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution.
Semidefinite Programming
Semidefinite embedding
Second-order cone programming
Interior point method
Conic optimization
Linear matrix inequality
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In this article, we generalize primal-dual interior-point method, which was studied by Bai et al. [3 Y. Q. Bai , M. El Ghami , and C. Roos ( 2003 ). A new efficient large-update primal-dual interior-point method based on a finite barrier . SIAM J. Optim. 13 ( 3 ): 766 – 782 .[Crossref] , [Google Scholar]] for linear optimization, to convex quadratic optimization over symmetric cone. The symmetrization of the search directions used in this article is based on the Nesterov and Todd scaling scheme. By employing Euclidean Jordan algebras, we derive the iteration bounds that match the currently best known iteration bounds for large- and small-update methods, namely, and , respectively, which are as good as the ones for the linear optimization analogue. Moreover, this unifies the analysis for linear optimization, convex quadratic optimization, second-order cone optimization, semidefinite optimization, convex quadratic semidefinite optimization, and symmetric optimization.
Interior point method
Conic optimization
Second-order cone programming
Cone (formal languages)
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Semidefinite Programming
Conic optimization
Second-order cone programming
Semidefinite embedding
Convex cone
Duality (order theory)
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Semidefinite Programming
Second-order cone programming
Conic optimization
Quadratic growth
Cone (formal languages)
Semidefinite embedding
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Citations (2)
We propose a homogeneous primal-dual interior-point method to solve sum-of-squares optimization problems by combining non-symmetric conic optimization techniques and polynomial interpolation. The approach optimizes directly over the sum-of-squares cone and its dual, circumventing the semidefinite programming (SDP) reformulation which requires a large number of auxiliary variables. As a result, it has substantially lower theoretical time and space complexity than the conventional SDP-based approach. Although our approach avoids the semidefinite programming reformulation, an optimal solution to the semidefinite program can be recovered with little additional effort. Computational results confirm that for problems involving high-degree polynomials, the proposed method is several orders of magnitude faster than semidefinite programming.
Semidefinite Programming
Conic optimization
Semidefinite embedding
Interior point method
Second-order cone programming
Conic section
Least-squares function approximation
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Citations (3)