Matrix representations of multivalued projections and least squares problems

Arias, María Laura; Contino, Maximiliano; Maestripieri, Alejandra; Marcantognini, Stefania

doi:10.1016/j.jmaa.2023.127631

Cited by 1 publication

(1 citation statement)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently, many scholars have proposed many algorithms to tackle least-squares problems, like relaxed greedy randomized coordinate descent methods [25], the parallel approximate pseudo-inverse matrix technique in conjunction with the explicit preconditioned conjugate gradient least squares scheme [26], the new randomized Gauss-Seidel method [27], the QR-Cholesky method, the implicit QR method, the bidiagonal implicit QR method [28], the randomized block coordinate descent algorithm [29], an iterative pre-conditioning technique [30], splitting-based randomized iterative methods [31], the projection method [32], the index-search-method-based inner-outer iterative algorithm [33], the greedy double subspace coordinate descent method [34], a distributed algorithm [35], and multivalued projections [36].…”

Section: Introductionmentioning

confidence: 99%

Solving Least-Squares Problems via a Double-Optimal Algorithm and a Variant of the Karush–Kuhn–Tucker Equation for Over-Determined Systems

Liu,

Kuo,

Chang

2024

Algorithms

View full text Add to dashboard Cite

A double optimal solution (DOS) of a least-squares problem Ax=b,A∈Rq×n with q≠n is derived in an m+1-dimensional varying affine Krylov subspace (VAKS); two minimization techniques exactly determine the m+1 expansion coefficients of the solution x in the VAKS. The minimal-norm solution can be obtained automatically regardless of whether the linear system is consistent or inconsistent. A new double optimal algorithm (DOA) is created; it is sufficiently time saving by inverting an m×m positive definite matrix at each iteration step, where m≪min(n,q). The properties of the DOA are investigated and the estimation of residual error is provided. The residual norms are proven to be strictly decreasing in the iterations; hence, the DOA is absolutely convergent. Numerical tests reveal the efficiency of the DOA for solving least-squares problems. The DOA is applicable to least-squares problems regardless of whether q<n or q>n. The Moore–Penrose inverse matrix is also addressed by adopting the DOA; the accuracy and efficiency of the proposed method are proven. The m+1-dimensional VAKS is different from the traditional m-dimensional affine Krylov subspace used in the conjugate gradient (CG)-type iterative algorithms CGNR (or CGLS) and CGRE (or Craig method) for solving least-squares problems with q>n. We propose a variant of the Karush–Kuhn–Tucker equation, and then we apply the partial pivoting Gaussian elimination method to solve the variant, which is better than the original Karush–Kuhn–Tucker equation, the CGNR and the CGNE for solving over-determined linear systems. Our main contribution is developing a double-optimization-based iterative algorithm in a varying affine Krylov subspace for effectively and accurately solving least-squares problems, even for a dense and ill-conditioned matrix A with q≪n or q≫n.

show abstract