1 High-Performance Implementation of Positive Matrix Completion for SDPs Makoto Yamashita (Tokyo Institute of Technology) Kazuhide Nakata (Tokyo Institute of Technology) 2013/10/6 INFORMS Annual Meeting 2013 The research was financially supported by the Sasakawa Sientific Research Grant from The Japan Science Society. 2013/10/6-9 Minneapolis Convention Center, Minneapolis, USA

Sparsity in SDPs 2013/10/6 INFORMS Annual Meeting Notation Only blue elements are involved in inner-product. However, we also have SDP condition.

Structural Sparsity in Spin-Glass SDPs Each node interacts with only its 6 neighbors Only the blue elements are involved in inner-product ⇒ Exploit this structural sparsity SDP condition ⇒ Positive Matrix Completion 2013/10/6 INFORMS Annual Meeting

Idea of Matrix-Completion type Interior-Point Method 2013/10/6 INFORMS Annual Meeting Without Blacks Complement Blacks

Outline of this talk Introduction of Matrix-Completion IPM Speed-up by new factorization formula Multiple threads computation Numerical results This talk corresponds to the new version of SDPA-C (SDPA with the Completion) Available at /10/6 INFORMS Annual Meeting

Standard form of SDP Primal-Dual form. 2013/10/6 INFORMS Annual Meeting

Framework of IPM 2013/10/6 INFORMS Annual Meeting

Keywords in Matrix-Completion How many elements are necessary? ⇒ Aggregate Sparsity Pattern How to convert into smaller matrices? ⇒ Chordal Graph & Maximal Cliques How to complete ? ⇒ The form of 2013/10/6 INFORMS Annual Meeting

Aggregate Sparsity Pattern Non-zero pattern in the dual side 2013/10/6 INFORMS Annual Meeting Example graph

Chordal Graph Chodal, if every cycle longer than 3 has a chord The variable matrix is decomposed by the maximal cliques. Maximal Cliques (Clique, if there is an edge between any pair of the verticies.) 2013/10/6 INFORMS Annual Meeting length4 length Chordal

Decomopostion of 2013/10/6 INFORMS Annual Meeting Blue: aggregate, Red: Chordal Grone et al The entire matrix can be positive definite.

An example of Matrix Completion 2013/10/6 INFORMS Annual Meeting Positive Definite Non-singular & Transpose Non-singular

A remarkable property of the matrix completion The matrix is fully-dense, but its inverse is sparse /10/6 INFORMS Annual Meeting

The factorization of the variable matrix 2013/10/6 INFORMS Annual Meeting Point::

Schur complement matrix with the sparse matrices 2013/10/6 INFORMS Annual Meeting

Outline of this talk Introduction of Matrix-Completion IPM Speed-up by new factorization formula Mutliple threads computation Numerical results 2013/10/6 INFORMS Annual Meeting

Review of Matrix Factorization.. Point 2013/10/6 INFORMS Annual Meeting

2013/10/6 INFORMS Annual Meeting

2013/10/6 INFORMS Annual Meeting

Speed-up by the new factorization 2013/10/6 INFORMS Annual Meeting SDPA-C6.2.0SDPA-C7.3.8Speed-up Schur Complement sec sec2.00x Total sec sec1.88x The computation time is shrunk, but there is still room to improve. Parallel computation by multiple threads Max-clique SDP

Multiple threaded Schur complement matrix 2013/10/6 INFORMS Annual Meeting Each column is independent from others. The thread that becomes idle computes the next column

The effect of multiple threads 2013/10/6 INFORMS Annual Meeting SDPA-C 6.2.0(1) SDPA-C 7.3.8(1) SDPA-C 7.3.8(2) SDPA-C 7.3.8(4) Schur sec sec sec 1.78 x sec 2.86 x Total sec sec sec 1.78 x sec 2.82 x Max-clique The number in （ ） is threads 5.31 times speed-up

New version of SDPA-C 2013/10/6 INFORMS Annual Meeting SDPA-C 6.2.1SDPA 7.3.8SDPA-C Interior-Point MethodMatrix CompletionStandardMatrix Compeltion Sparse CholeskyOur own codeMUMPSCHOLMOD &MUMPS Multiple Threads× △ ○ Callable Library× ○○ Matlab Interface× ○○

Test Environments and Test Problems CPU Xeon X5365(3.0GHz), Memory 48GB, Red Hat Linux Test Problem 1 SDP relaxation of Max-Clique Problem on lattice (p,q) Test Problem 2 Spin-glass computation in quantum chemistry 2013/10/6 INFORMS Annual Meeting

Max clique SDPs (p=400,q=10) 2013/10/6 INFORMS Annual Meeting SDPA-C SDPA-C 7.3.8(4) SDPA 7.3.8(4) SeDuMi 1.3 Schur ΔX Total New SDPA-C is the fastest. #Clique 438, Average Size 29.89, Max Size 59

Spin-glass SDPs 2013/10/6 INFORMS Annual Meeting pn#CliquesAve SizeMax SizeSDPA7.3.8SDPA-C The unit of computation time is second. The computation cost grows up mildly in SDPA-C. The clique size is almost constant. For larger SDPs, SDPA-C is more efficient.

Conclusion and future works Speed-up of Matrix-Completion IPM by the new factorization formula  More effective for larger problems Speed-up by multiple threads. We should automatically select the standard IPM or Matrix-Completion IPM 2013/10/6 INFORMS Annual Meeting