A new library of structured semidefinite programming instances

Klerk, Etienne de; Sotirov, Renata

doi:10.1080/10556780902896608

Cited by 9 publications

(10 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The GMP can employ arbitrary large precision and we used approximately 77 significant decimal digits if not specified, and the QD have approximately 32 (double-double) or 64 (quad-double) significant decimal digits, whereas commonly used double precision of ANSI/IEEE Standard 754-1985 [10] has only approximately 16 significant decimal digits. The original necessity is from quantum chemistry problems [11] and since the first release of the SDPA-GMP in 2008/4/10 and SDPA-QD and -DD in 2009/3/22, different researchers have applied to some numerically difficult SDPs like; Polynomial Optimization Problems [12], new bounds for Kissing number [13] and miscellaneous problems [14]. We also developed important libraries MPACK (MBLAS/MLAPACK) [15], multipleprecision version of the well-known and widely-used BLAS (Basic Linear Algebra Subprograms) [16] and LAPACK (Linear Algebra PACKage) libraries [17].…”

Section: Introductionmentioning

confidence: 99%

A numerical evaluation of highly accurate multiple-precision arithmetic version of semidefinite programming solver: SDPA-GMP, -QD and -DD.

Nakata

2010

2010 IEEE International Symposium on Computer-Aided Control System Design

View full text Add to dashboard Cite

Semidefinite programming (SDP) is an important branch of optimization and has wide range of applications: engineering, industry, chemistry, mathematics, etc. However, obtaining very accurate optimum for a semidefinite programming is difficult in general, especially for ill-posed ones. In this paper, we evaluated numerically highly accurate SDP solvers; SDPA-GMP, -QD and -DD, which employ multiple-precision arithmetic and mainly developed by our group. We applied to some problems from SDPLIB benchmark which contains some ill-posed problems as well. The SDPA-GMP, and -QD solved problems very accurately, whereas SDPA failed or obtained inaccurate optima. The SDPA-DD may be used for compensation for accuracy and speed. We also investigated the convergence behaviors which agreed well what theories indicated.

show abstract

Section: Introductionmentioning

confidence: 99%

A numerical evaluation of highly accurate multiple-precision arithmetic version of semidefinite programming solver: SDPA-GMP, -QD and -DD.

Nakata

2010

2010 IEEE International Symposium on Computer-Aided Control System Design

View full text Add to dashboard Cite

show abstract

“…In recent years, there have been developments on exploring the structure of the data matrices in semi-definite programming problems, see, for example, [9][10][11][12]). This is an important issue, especially when the data matrices are of high dimensions and the application of these results to the problems considered in this paper should be the subject of future research.…”

Section: Discussionmentioning

confidence: 99%

Strong practical stability and stabilization of uncertain discrete linear repetitive processes

Dąbkowski

Gałkowski

Bachelier

et al. 2011

Numerical Linear Algebra App

View full text Add to dashboard Cite

Repetitive processes are a distinct class of 2D systems of both theoretical and practical interest. The stability theory for these processes originally consisted of two distinct concepts termed asymptotic stability and stability along the pass, respectively, where the former is a necessary condition for the latter. Recently applications have arisen where asymptotic stability is too weak, and stability along the pass is too strong for meaningful progress to be made. This, in turn, has led to the concept of strong practical stability for such cases, where previous work has formulated this property and obtained necessary and sufficient conditions for its existence together with Linear Matrix Inequality based tests, which then extend to allow robust control law design. This paper develops considerably simpler, and hence computationally more efficient, stability tests that also extend to allow control law design.

show abstract

“…We note the libraries [9,33] have additional instances on which our method was not effective (S opt = S f ull ); we do not report results for these instances.…”

Section: Remarkmentioning

confidence: 99%

Dimension reduction for semidefinite programs via Jordan algebras

Permenter

Parrilo

2019

Math. Program.

View full text Add to dashboard Cite

We propose a new method for simplifying semidefinite programs (SDP) inspired by symmetry reduction. Specifically, we show if an orthogonal projection map satisfies certain invariance conditions, restricting to its range yields an equivalent primal-dual pair over a lower-dimensional symmetric cone-namely, the cone-of-squares of a Jordan subalgebra of symmetric matrices. We present a simple algorithm for minimizing the rank of this projection and hence the dimension of this subalgebra. We also show that minimizing rank optimizes the direct-sum decomposition of the algebra into simple ideals, yielding an optimal "blockdiagonalization" of the SDP. Finally, we give combinatorial versions of our algorithm that execute at reduced computational cost and illustrate effectiveness of an implementation on examples. Through the theory of Jordan algebras, the proposed method easily extends to linear and second-order-cone programming and, more generally, symmetric cone optimization.Address(es) of author(s) should be given

show abstract

A new library of structured semidefinite programming instances

Cited by 9 publications

References 35 publications

A numerical evaluation of highly accurate multiple-precision arithmetic version of semidefinite programming solver: SDPA-GMP, -QD and -DD.

A numerical evaluation of highly accurate multiple-precision arithmetic version of semidefinite programming solver: SDPA-GMP, -QD and -DD.

Strong practical stability and stabilization of uncertain discrete linear repetitive processes

Dimension reduction for semidefinite programs via Jordan algebras

Contact Info

Product

Resources

About