TL;DR
This paper proposes an inexact interior-point relaxation method (iIPRM) for LO problems that does not require its iterates to remain strictly positive and employs a sparse preconditioner tailored to the structure of the normal matrix.
We introduce a class of additive reduced basis preconditioners designed to accelerate the iterative solution of large-scale linear systems arising from discretized parametrized PDEs. The main idea is to regularize the inherently singular reduced-order approximation by adding simple correction terms: either a scaled ide...
This work proposes a multiscale primal-dual interior-point relaxation method (IPRM), which can handle instances whose full formulations contain trillions of transport variables and develops an effective support-identification strategy based on the approximate solutions obtained by IPRM.
Collocation methods based on Birkhoff interpolation at Gaussian-type points are well-conditioned for one-dimensional initial and boundary value problems [Wang et al., {\em SIAM J. Sci. Comput.} 36 (2014)], but the underlying construction does not extend directly to multiple dimensions. We address the long-standing ill-...
A globally convergent regularized Newton method with positive definite regularization for solving nonsmooth optimization problems that replaces the identity matrix in traditional algorithms with a general positive-definite symmetric matrix to regularize the generalized Hessian.
Interior-point methods (IPMs) are among the most widely used algorithms for constrained optimization, yet their Newton-based search directions require costly second-order information and large linear-system solves. Learning to optimize offers cheaper updates learned from data, but the singular behavior of logarithmic b...
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