CSDP, A C library for semidefinite programming

Borchers, Brian

doi:10.1080/10556789908805765

Cited by 454 publications

(378 citation statements)

References 3 publications

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“…However, we may end up with a problem having no interior, for example if we have a k-clique for which all the edge variables X i j = −1 k−1 . If this happens, we solve the SDP problem using CSDP [11] since its infeasible interior point algorithm runs well in this situation.…”

Section: Oraclementioning

confidence: 99%

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Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

Anjos

Ghaddar²,

Hupp

et al. 2013

Facets of Combinatorial Optimization

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This paper is concerned with computing global optimal solutions for maximum k-cut problems. We improve on the SBC algorithm of Ghaddar, Anjos and Liers in order to compute such solutions in less time. We extend the design principles of the successful BiqMac solver for maximum 2-cut to the general maximum k-cut problem. As part of this extension, we investigate different ways of choosing variables for branching. We also study the impact of the separation of clique inequalities within this new framework and observe that it frequently reduces the number of subproblems considerably. Our computational results suggest that the proposed approach achieves a drastic speedup in comparison to SBC, especially when k = 3. We also made a comparison with the orbitopal fixing approach of Kaibel, Peinhardt and Pfetsch. The results suggest that while their performance is better for sparse instances and larger values of k, our proposed approach is superior for smaller k and for dense instances of medium size. Furthermore, we used CPLEX for solv- ing the ILP formulation underlying the orbitopal fixing algorithm and conclude that especially on dense instances the new algorithm outperforms CPLEX by far.

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Section: Oraclementioning

confidence: 99%

“…Therefore we omit these shrinkings, but as a consequence the SDP relaxation to be solved by the oracle may have no interior. When this happens, we solve the relaxations using CSDP [11] as mentioned earlier in Section 3.1.3.…”

Section: Shrinking and Sdp Relaxations Without Interiormentioning

confidence: 99%

Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

Anjos

Ghaddar²,

Hupp

et al. 2013

Facets of Combinatorial Optimization

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“…To compute quadratic Lyapunov functions for linear or affine dynamics, semidefinite programming (SDP) [BEFB94] tools can be used, for instance CSDP [Bor99] or SeDuMi [RPS99]. In a nutshell, the problem of finding a Lyapunov function according to some parameterized template is mapped onto a nonlinear, but still convex optimization problem that can be solved numerically [Pet99].…”

Section: Proof (Of Theorem 1)mentioning

confidence: 99%

“…In particular, there will be constraints on the existence of Lyapunov functions with a particular parameterization. Specialized software (SDP solvers [Bor99,RPS99]) can be used to carry out the automatic computation of these functions. Let…”

Section: Hierarchical Verification Of Robust Safety and Stabilitymentioning

confidence: 99%

Towards Component Based Design of Hybrid Systems: Safety and Stability

Damm

Dierks

Oehlerking

et al. 2010

Time for Verification

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Abstract. We propose a library based incremental design methodology for constructing hybrid controllers from a component library of models of hybrid controllers, such that global safety and stability properties are preserved. To this end, we propose hybrid interface specifications of components characterizing plant regions for which safety and stability properties are guaranteed, as well as exception mechanisms allowing safe and stability-preserving transfer of control whenever the plant evolves towards the boundary of controllable dynamics. We then propose a composition operator for constructing hybrid automata from a library of such pre-characterized components supported by compositional and automatable proofs of hybrid interface specifications.

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“…Mosek [AA00], SDPA [YFN + 10], CSDP [Bor99]). For more details about SDP, we refer the interested reader to [VB94].…”

Section: The Polynomial σ Is Called An Sos Certificate and Guaranteesmentioning

confidence: 99%

A Sums-of-Squares extension of policy iterations

Adjé

Garoche²,

Magron

2017

Nonlinear Analysis: Hybrid Systems

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In order to address the imprecision often introduced by widening operators in static analysis, policy iteration based on min-computations amounts to considering the characterization of reachable value set of a program as an iterative computation of policies, starting from a postfixpoint. Computing each policy and the associated invariant relies on a sequence of numerical optimizations. While the early research efforts relied on linear programming (LP) to address linear properties of linear programs, the current state of the art is still limited to the analysis of linear programs with at most quadratic invariants, relying on semidefinite programming (SDP) solvers to compute policies, and LP solvers to refine invariants.We propose here to extend the class of programs considered through the use of Sums-ofSquares (SOS) based optimization. Our approach enables the precise analysis of switched systems with polynomial updates and guards. The analysis presented has been implemented in Matlab and applied on existing programs coming from the system control literature, improving both the range of analyzable systems and the precision of previously handled ones.

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CSDP, A C library for semidefinite programming

Cited by 454 publications

References 3 publications

Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

Towards Component Based Design of Hybrid Systems: Safety and Stability

A Sums-of-Squares extension of policy iterations

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