Numerical block diagonalization of matrix *-algebras with application to semidefinite programming

Abstract

Semidefinite programming (SDP) is one of the most active areas in mathematical programming, due to varied applications and the availability of interior point algorithms. In this paper we propose a new pre-processing technique for SDP instances that exhibit algebraic symmetry. We present computational results to show that the solution times of certain SDP instances may be greatly reduced via the new approach.

On the number of matrices to generate a matrix ∗-algebra over the real field

“…Then it holds that X 1 , X 2 , The case where n 4 is obtained in a similar way. [2,3] and H 2 = (Q 1 X 2 Q 1 ) [2,4] . Then it holds that X 1 , X 2 = H n .…”

“…Recently, a numerical algorithm for finding such decomposition was proposed by Murota et al [20] and Maehara and Murota [17]. A variant of this algorithm was designed by de Klerk et al [4]. Maehara and Murota [19] further developed a simpler decomposition algorithm, which allows us to control numerical errors.…”

“…We say that T ⊆ M n is a matrix * -algebra if it contains the identity matrix and it is closed under linear combination, multiplication, and transposition. Matrix * -algebras have been studied extensively in engineering fields such as semidefinite programming [2,4,17,20], signal processing [18], and structural engineering [1]. In these applications, we can reduce the computational effort by exploiting the Artin-Wedderburn type structure theorem for matrix * -algebras: a matrix * -algebra can be decomposed uniquely into irreducible components with a single orthogonal matrix.…”

Copositive optimization – Recent developments and applications

“…We also obtain with Z(n) = 1 4 n 2 n−1 2 n−2 2 n−3 2 lim n→∞ cr(K n ) Z(n) ≥ lim n→∞ cr(K n,n ) Z(n, n) ≥ 0.8594 , and again the constant on the right-hand side is the best known up to now. For very recent, significant improvements in runtime via symmetry arguments to obtain these bounds we refer to [46,52].…”

Solving the general joint block diagonalization problem via linearly independent eigenvectors of a matrix polynomial

“…Even if the solutions returned by algebraic methods are of “low quality,” they are usually good initial guesses for optimization methods. In the current literature, the algebraic methods for the GJBD problem fall into two categories: One is based on matrix ∗‐algebra (see, for example, the works of de Klerk et al, Maehara and Murota, and Murota et al for the orthogonal GJBD problem and a recent generation by Cai and Liu for the nonorthogonal GJBD problem), and the other is based on a matrix polynomial (see the work of Cai et al for the Hermitian GJBD problem). In the former category, the null space of a linear operator needs to be computed, which requires flops; thus, for problems with large n values, such an approach will be quite expensive for computation.…”