Abstract. In this article, we consider solvers for large-scale trust-region subproblems when the quadratic model is defined by a limited-memory symmetric rank-one (L-SR1) quasi-Newton matrix. We propose a solver that exploits the compact representation of L-SR1 matrices. Our approach makes use of both an orthonormal basis for the eigenspace of the L-SR1 matrix and the ShermanMorrison-Woodbury formula to compute global solutions to trust-region subproblems. To compute the optimal Lagrange multiplier for the trust-region constraint, we use Newton's method with a judicious initial guess that does not require safeguarding. A crucial property of this solver is that it is able to compute high-accuracy solutions even in the so-called hard case. Additionally, the optimal solution is determined directly by formula, not iteratively. Numerical experiments demonstrate the effectiveness of this solver.

The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory ("Argonne"). Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan. http://energy.gov/downloads/doe-public-accessplan COAP manuscript No.

We consider a family of dense initializations for limited-memory quasi-Newton methods. The proposed initialization exploits an eigendecomposition-based separation of the full space into two complementary subspaces, assigning a different initialization parameter to each subspace. This family of dense initializations is proposed in the context of a limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) trust-region method that makes use of a shape-changing norm to define each subproblem. As with L-BFGS methods that traditionally use diagonal initialization, the dense initialization and the sequence of generated quasi-Newton matrices are never explicitly formed. Numerical experiments on the CUTEst test set suggest that this initialization together with the shape-changing trust-region method outperforms other L-BFGS methods for solving general nonconvex unconstrained optimization problems. While this dense initialization is proposed in the context of a special trust-region method, it has broad applications for more general quasi-Newton trust-region and line search methods. In fact, this initialization is suitable for use with any quasi-Newton update that admits a compact representation and, in particular, any member of the Broyden class of updates.

We propose two limited-memory BFGS (L-BFGS) trust-region methods for large-scale optimization with linear equality constraints. The methods are intended for problems where the number of equality constraints is small. By exploiting the structure of the quasi-Newton compact representation, both proposed methods solve the trustregion subproblems nearly exactly, even for large problems. We derive theoretical global convergence results of the proposed algorithms, and compare their numerical effectiveness and performance on a variety of large-scale problems.

Background: Grouping samples with low prevalence of positives into pools and testing these pools can achieve considerable savings in testing resources compared with individual testing in the context of COVID-19. We review published pooling matrices, which encode the assignment of samples into pools and describe decoding algorithms, which decode individual samples from pools. Based on the findings we propose new one-round pooling designs with high compression that can efficiently be decoded by combinatorial algorithms. This expands the admissible parameter space for the construction of pooling matrices compared to current methods. Results: By arranging samples in a grid and using polynomials to construct pools, we develop direct formulas for an Algorithm (Polynomial Pools (PP)) to generate assignments of samples into pools. Designs from PP guarantee to correctly decode all samples with up to a specified number of positive samples. PP includes recent combinatorial methods for COVID-19, and enables new constructions that can result in more effective designs. Conclusion: For low prevalences of COVID-19, group tests can save resources when compared to individual testing. Constructions from the recent literature on combinatorial methods have gaps with respect to the designs that are available. We develop a method (PP), which generalizes previous constructions and enables new designs that can be advantageous in various situations.