Katsuki Fujisawa

Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm

The ground-state fermion second-order reduced density matrix (2-RDM) is determined variationally using itself as a basic variable. As necessary conditions of the N-representability, we used the positive semidefiniteness conditions, P, Q, and G conditions that are described in terms of the 2-RDM. The variational calculations are performed by using recently developed semidefinite programming algorithm (SDPA). The calculated energies of various closed- and open-shell atoms and molecules are excellent, overshooting only slightly the full-CI energies. There was no case where convergence was not achieved. The calculated properties also reproduce well the full-CI results.

Exploiting sparsity in primal-dual interior-point methods for semidefinite programming

The Helmberg-Rendl-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro and Nesterov-Todd search directions have been used in many primal-dual interior-point methods for semidefinite programs. This paper proposes an efficient method for computing the two directions when the semidefinite program to be solved is large scale and sparse.

Implementation and evaluation of SDPA 6.0 (Semidefinite Programming Algorithm 6.0)

SDP (SemiDefinite Programming) is one of the most attractive optimization models. It has many applications from various fields such as control theory, combinatorial and robust optimization, and quantum chemistry. The SDPA (SemiDefinite Programming Algorithm) is a software package for solving general SDPs based on primal-dual interior-point methods with the HRVW/KSH/M search direction. It is written in C++ with the help of LAPACK for numerical linear algebra for dense matrix computation. The purpose of this paper is to present a brief description of the latest version of the SDPA and its high performance for large scale problems through numerical experiments and comparisons with some other major software packages for general SDPs.

Exploiting sparsity in semidefinite programming via matrix completion II: Implementation and numerical results

Abstract. In Part I of this series of articles, we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs (SDPs) when solving them by primal-dual interior-point methods. This framework is based on some results about positive semidefinite matrix completion, and it can be embodied in two different ways. One is by a conversion of a given sparse SDP having a large scale positive semidefinite matrix variable into an SDP having multiple but smaller positive semidefinite matrix variables. The other is by incorporating a positive definite matrix completion itself in a primal-dual interior-point method. The current article presents the details of their implementations. We introduce new techniques to deal with the sparsity through a clique tree in the former method and through new computational formulae in the latter one. Numerical results over different classes of SDPs show that these methods can be very efficient for some problems.

Semi-definite programming for topology optimization of trusses under multiple eigenvalue constraints

Abstract Topology optimization problem of trusses for specified eigenvalue of vibration is formulated as Semi-Definite Programming (SDP), and an algorithm is presented based on the Semi-Definite Programming Algorithm (SDPA) which utilizes extensively the sparseness of the matrices. Since the sensitivity coefficients of the eigenvalues with respect to the design variables are not needed, the SDPA is especially useful for the case where the optimal design has multiple fundamental eigenvalues. Global and local modes are defined and a procedure is presented for generating optimal topology from the practical point of view. It is shown in the examples, that SDPA has advantage over existing methods in view of computational efficiency and accuracy of the solutions, and an optimal topology with five-fold fundamental eigenvalue is found without any difficulty.

PHoM – a Polyhedral Homotopy Continuation Method for Polynomial Systems

PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedral-linear homotopy functions, based on the polyhedral homotopy theory, from input data for a given system of polynomial equations f(x)=0. The second module CMPSc traces the solution curves of the homotopy equations to compute all isolated solutions of f(x)=0. The third module Verify checks whether all isolated solutions of f(x)=0 have been approximated correctly. We describe numerical methods used in each module and the usage of the package. Numerical results to demonstrate the performance of PHoM include some large polynomial systems that have not been solved previously.

SDPARA: semiDefinite programming algorithm paRAllel version

The SDPA (SemidDefinite Programming Algorithm) is known as efficient computer software based on the primal-dual interior-point method for solving SDPs (SemiDefinite Programs). In many applications, however, some SDPs become larger and larger, too large for the SDPA to solve on a single processor. In execution of the SDPA applied to large scale SDPs, the computation of the so-called Schur complement matrix and its Cholesky factorization consume most of the computational time. The SDPARA (SemiDefinite Programming Algorithm paRAllel version) is a parallel version of the SDPA on multiple processors and distributed memory, which replaces these two parts by their parallel implementation using MPI and ScaLAPACK. Through numerical results, we show that the SDPARA on a PC cluster consisting of 64 processors attains high scalability for large scale SDPs without losing the stability of the SDPA.

Performance characteristics of Graph500 on large-scale distributed environment

Graph500 is a new benchmark for supercomputers based on large-scale graph analysis, which is becoming an important form of analysis in many real-world applications. Graph algorithms run well on supercomputers with shared memory. For the Linpack-based supercomputer rankings, TOP500 reports that heterogeneous and distributed-memory super-computers with large numbers of GPGPUs are becoming dominant. However, the performance characteristics of large-scale graph analysis benchmarks such as Graph500 on distributed-memory supercomputers have so far received little study. This is the first report of a performance evaluation and analysis for Graph500 on a commodity-processor-based distributed-memory supercomputer. We found that the reference implementation “replicated-csr” based on distributed level-synchronized breadth-first search solves a large free graph problem with 231 vertices and 235 edges (approximately 2.15 billon vertices and 34.3 billion edges) in 3.09 seconds with 128 nodes and 3,072 cores. This equates to 11 giga-edges traversed per second. We describe the algorithms and implementations of the reference implementations of Graph500, and analyze the performance characteristics with varying graph sizes and numbers of computer nodes and different implementations. Our results will also contribute to the development of optimized algorithms for the coming exascale machines.

Numerical Evaluation of SDPA (Semidefinite Programming Algorithm)

SDPA (SemiDefmite Programming Algorithm) is a C++ implementation of a Mehrotra-type primal-dual predictor-corrector interior-point method for solving the standard form semidefinite program and its dual. We report numerical results of large scale problems to evaluate its performance, and investigate how major time-consuming parts of SDPA vary with the problem size, the number of constraints and the sparsity of data matrices.

Latest developments in the SDPA family for solving large-scale SDPs

The main purpose of this chapter is to introduce the latest developments in SDPA and its family. SDPA is designed to solve large-scale SemiDefinite Programs (SDPs) faster and over the course of 15 years of development, it has been expanded into a high-performance-oriented software package. We hope that this introduction to the latest developments of the SDPA Family will be beneficial to readers who wish to understand the inside of state-of-art software packages for solving SDPs.