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Solving sparse linear systems is a problem that arises in many scientific applications, and sparse direct solvers are a time-consuming and key kernel for those applications and for more advanced solvers such as hybrid direct-iterative solvers. For this reason, optimizing their performance on modern architectures is critical. The preprocessing steps of sparse direct solvers|ordering and block-symbolic factorization|are two major steps that lead to a reduced amount of computation and memory and to a better task granularity to reach a good level of performance when using BLAS kernels. With the advent of GPUs, the granularity of the block computation has become more important than ever. In this paper, we present a reordering strategy that increases this block granularity. This strategy relies on block-symbolic factorization to refine the ordering produced by tools such as Metis or Scotch, but it does not impact the number of operations required to solve the problem. We integrate this algorithm in the PaStiX solver and show an important reduction of the number of off-diagonal blocks on a large spectrum of matrices. This improvement leads to an increase in efficiency of up to 20% on GPUs. [ABSTRACT FROM AUTHOR]

This work is motivated by two two-sided optimization problems that arise in atomic chemistry when one is looking for a minimal set of localized orbitals with prescribed occupation numbers, respectively, a prescribed total number of electrons. We first propose an optimal analytic solution of the first problem, and then show that an optimal solution of the second problem can be found by solving a convex quadratic programming problem with box constraints and p unknowns. We prove that the latter problem is solved by the active-set method in at most 2p iterations. These investigations reveal that the optimal solutions of both problems are projections that are in C = {Y ∈ Rn×p : YTY = Ip, YT NY = Δ} for N symmetric and Δ diagonal and that these solutions are generally nonunique. The question of how one can then select a particular solution out of the set C leads to the main problem of this paper: the optimization of an arbitrary smooth function whose minimizer describes the solution of interest over C. To solve this problem, we first find that a slight modification of C is a Riemannian manifold for which geometric objects can be derived that are required to allow optimization over this manifold. Using these geometric tools we propose an augmented Lagrangian-based algorithm that guarantees global convergence to a stationary point of our main problem. The latter is shown by investigating when the LICQ is satisfied. Finally we compare this algorithm numerically with a similar algorithm that, however, does not apply these geometric tools and that is, to our knowledge, not guaranteed to converge. Our results show that our algorithm yields a significantly better performance. [ABSTRACT FROM AUTHOR]

We introduce a flexible optimization framework for nuclear norm minimization of matrices with linear structure, including Hankel, Toeplitz, and moment structures and catalog applications from diverse fields under this framework. We discuss various first-order methods for solving the resulting optimization problem, including alternating direction methods of multipliers, proximal point algorithms, and gradient projection methods. We perform computational experiments to compare these methods on system identification problems and system realization problems. For the system identification problem, the gradient projection method (accelerated by Nesterov's extrapolation techniques) and the proximal point algorithm usually outperform other first-order methods in terms of CPU time on both real and simulated data, for small and large regularization parameters, respectively, while for the system realization problem, the alternating direction method of multipliers, as applied to a certain primal reformulation, usually outperforms other first-order methods in terms of CPU time. We also study the convergence of the proximal alternating direction methods of multipliers used in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we show that for a symmetric tensor, its best symmetric rank-1 approximation is its best rank-1 approximation. Based on this result, a positive lower bound for the best rank-1 approximation ratio of a symmetric tensor is given. Furthermore, a higher order polynomial spherical optimization problem can be reformulated as a multilinear spherical optimization problem. Then, we present a modified power algorithm for solving the homogeneous polynomial spherical optimization problem. Numerical results are presented, illustrating the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study the projection onto the intersection of an affine subspace and a convex set and provide a particular treatment for the cone of positive semidefinite matrices. Among applications of this problem is the calibration of covariance matrices. We propose a Lagrangian dualization of this least-squares problem, which leads us to a convex differentiable dual problem. We propose to solve the latter problem with a quasi-Newton algorithm. We assess this approach with numerical experiments which show that fairly large problems can be solved efficiently. [ABSTRACT FROM AUTHOR]