Barrier and penalty methods for low-rank semidefinite programming with application to truss topology design

Soodeh, Habibi,; Kavand, Arefeh; Kočvara, Michal; Stingl, Michael

doi:10.48550/arxiv.2105.08529

“…These methods were mostly designed for exploiting sparsity rather than low rank structure. During the last months of this work an approach closely related to ours appeared in [12]. It significantly extends ideas of [33] for a deterministic preconditioning variant.…”

Section: Introductionmentioning

confidence: 80%

“…In this orthonormal case a simpler bound on the condition number may be obtained by following the argument of Th. 5.1 in [12].…”

Section: The Eigenvalues Of (Imentioning

confidence: 97%

“…The actual randomized approach and analysis, however, mainly builds on [13] and the bounding techniques presented there. For the deterministic preconditioning variant the recent work [12] provided strong guidance for analysing the condition number.…”

Section: Low Rank Preconditioning a Gram-matrix Plus Positive Diagonalmentioning

confidence: 99%

“…Their ideas and arguments heavily influenced the condition number analysis of our approach presented in theorems 2 and 9. In contrast to [12], our algorithmic approach does not require any a priori knowledge on the rank of the optimal solution. Rather, Theorem 9 and Lemma 12 motivate an estimate on the singular value induced by certain directions associated with active interior point variables, that seems to offer a good indicator for the relevance of the corresponding subspace.…”

Section: Introductionmentioning

confidence: 99%

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The conic bundle implementation of the spectral bundle method for large scale semidefinite programming solves in each iteration a semidefinite quadratic subproblem by an interior point approach. For larger cutting model sizes the limiting operation is collecting and factorizing a Schur complement of the primal-dual KKT system. We explore possibilities to improve on this by an iterative approach that exploits structural low rank properties. Two preconditioning approaches are proposed and analyzed. Both might be of interest for rank structured positive definite systems in general. The first employs projections onto random subspaces, the second projects onto a subspace that is chosen deterministically based on structural interior point properties. For both approaches theoretic bounds are derived for the associated condition number. In the instances tested the deterministic preconditioner provides surprisingly efficient control on the actual condition number. The results suggest that for large scale instances the iterative solver is usually the better choice if precision requirements are moderate or if the size of the Schur complemented system clearly exceeds the active dimension within the subspace giving rise to the cutting model of the bundle method.

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A semidefinite programming approach for the projection onto the cone of negative semidefinite symmetric tensors with applications to solid mechanics

Padovani

¹

,

Porcelli

²

2022

Calcolo

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We propose an algorithm for computing the projection of a symmetric second-order tensor onto the cone of negative semidefinite symmetric tensors with respect to the inner product defined by an assigned positive definite symmetric fourth-order tensor $${\mathcal {C}}$$ C . The projection problem is written as a semidefinite programming problem and an algorithm based on a primal-dual path-following interior point method coupled with a Mehrotra’s predictor-corrector approach is proposed. Implementations based on well-known symmetrization schemes and on direct methods are theoretically and numerically investigated taking into account tensors $${\mathcal {C}}$$ C arising in the modelling of masonry-like materials. For these special cases, indications on the preferable symmetrization scheme that take into account the conditioning of the arising linear systems are given.

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A preconditioned iterative interior point approach to the conic bundle subproblem

Helmberg

¹

2023

Math. Program.

0

View full text Add to dashboard Cite

The conic bundle implementation of the spectral bundle method for large scale semidefinite programming solves in each iteration a semidefinite quadratic subproblem by an interior point approach. For larger cutting model sizes the limiting operation is collecting and factorizing a Schur complement of the primal-dual KKT system. We explore possibilities to improve on this by an iterative approach that exploits structural low rank properties. Two preconditioning approaches are proposed and analyzed. Both might be of interest for rank structured positive definite systems in general. The first employs projections onto random subspaces, the second projects onto a subspace that is chosen deterministically based on structural interior point properties. For both approaches theoretic bounds are derived for the associated condition number. In the instances tested the deterministic preconditioner provides surprisingly efficient control on the actual condition number. The results suggest that for large scale instances the iterative solver is usually the better choice if precision requirements are moderate or if the size of the Schur complemented system clearly exceeds the active dimension within the subspace giving rise to the cutting model of the bundle method.

show abstract

Barrier and penalty methods for low-rank semidefinite programming with application to truss topology design

Cited by 3 publications

References 28 publications

A Preconditioned Iterative Interior Point Approach to the Conic Bundle Subproblem

A Preconditioned Iterative Interior Point Approach to the Conic Bundle Subproblem

A semidefinite programming approach for the projection onto the cone of negative semidefinite symmetric tensors with applications to solid mechanics

A preconditioned iterative interior point approach to the conic bundle subproblem

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