Iterative Solution of Nonlinear Equations in Several Variables

By J. M. Ortega, Werner C. Rheinboldt. 2014

1,446 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Numerical Analysis, Numerical Methods & Scientific Computing

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Citations in the corpus

Fast and Furious Convergence: Stochastic Second Order Methods under Interpolation (2019, methodology): “We use the notions of Q and R-convergence rates (Ortega and Rheinboldt, 1970; Nocedal and Wright, 2006) reviewed in Appendix G.”
Optimization Methods for Large-Scale Machine Learning (2016, background): “aining error satises R n(w k) Rn O(ˆk); (3.9) where R ndenotes the minimal value of R n. The rate of convergence exhibited here is refereed to as R-linear convergence in the optimization literature [117] and geometric convergence in the machine learning research community; we shall simply refer to i”
Group Knowledge Transfer: Collaborative Training of Large CNNs on the Edge (2020, background): “(2) and (5)), we propose a variant of Alternating Minimization (AM) [14, 15, 16, 17, 18, 19] to solve the reformulated optimization problem as follows: where )) , and z s = f s ( W s ; argmin where )) , and Where the ∗ superscript notation in above equations presents related random variable W is…”
DeltaGrad: Rapid retraining of machine learning models (2020, background): “There exists a constant c 0 such that for all w 1 and w 2 , An assumption specifc to Quasi-Newton methods is the strong independence of the weight updates: the smallest singular value of the normalized weight updates is bounded away from zero (Ortega & Rheinboldt, 1970; Conn et al., 1991).”
Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods (2013, methodology): “For the convergence analysis of Algorithm 1, we shall of course use the elementary but important descent lemma (see for example [50] 3.”
Global Convergence of ADMM in Nonconvex Nonsmooth Optimization (2015, background): “o an iteration that is similar to coordinate descent. 1.3 Related literature Our work is related to block coordinate-descent (BCD) algorithms for unconstrained optimization. BCD can be traced back to [43] for solving linear systems and also to [25,63,2,53], where the objective function fis assumed t”
Group Knowledge Transfer: Federated Learning of Large CNNs at the Edge (2020, methodology): “n transfer knowledge from many compact CNNs trained at the edge to a large CNN trained at a cloud server. The essence of FedGKT is that it reformulates FL as an alternating minimization (AM) approach [13, 14, 15, 16, 17, 18], which optimizes two random variables (the edge model and the server model)”
MintNet: Building Invertible Neural Networks with Masked Convolutions (2019, background): “Then the Ostrowski Theorem ( cf ., Theorem 10.1.3. in [23]) shows that the sequence { x 1 , x 2 , · · · , x t } obtained by x t = g ( x t − 1 ; α, z ) converges locally to x ∗ as t → ∞ .”
Predictive Control for Linear and Hybrid Systems (2017, background): “In the following, we give a brief overview on practical inexact line search methods; see [224] for a thorough treatment of the subject.”
Enabling New Flexibility in the SUNDIALS Suite of Nonlinear and Differential/Algebraic Equation Solvers (2020, methodology): “For the root-finding case, KINSOL provides inexact and modified Newton methods [25, 58] which may optionally utilize a line search strategy[27].”
On critical points of quadratic low-rank matrix optimization problems (2020, methodology): “For ALS, the asymptotic rate is computed from a block triangular decomposition of the Euclidean Hessian; see [22].”