TY - JOUR

T1 - Least squares problems with inequality constraints as quadratic constraints

AU - Mead, Jodi L.

AU - Renaut, Rosemary

N1 - Funding Information:
Corresponding author. Tel.: +1 208 426 2432; fax: +1 208 426 1354. E-mail addresses: [email protected] (J.L. Mead), [email protected] (R.A. Renaut). 1 Supported by NSF grant EPS 0447689. 2 Supported by NSF grants DMS 0513214 and DMS 0421846. 3 Tel.: +1 480 965 3795; fax: +1 480 965 4160.

PY - 2010/4/1

Y1 - 2010/4/1

N2 - Linear least squares problems with box constraints are commonly solved to find model parameters within bounds based on physical considerations. Common algorithms include Bounded Variable Least Squares (BVLS) and the Matlab function lsqlin. Here, the goal is to find solutions to ill-posed inverse problems that lie within box constraints. To do this, we formulate the box constraints as quadratic constraints, and solve the corresponding unconstrained regularized least squares problem. Using box constraints as quadratic constraints is an efficient approach because the optimization problem has a closed form solution. The effectiveness of the proposed algorithm is investigated through solving three benchmark problems and one from a hydrological application. Results are compared with solutions found by lsqlin, and the quadratically constrained formulation is solved using the L-curve, maximum a posteriori estimation (MAP), and the χ2 regularization method. The χ2 regularization method with quadratic constraints is the most effective method for solving least squares problems with box constraints.

AB - Linear least squares problems with box constraints are commonly solved to find model parameters within bounds based on physical considerations. Common algorithms include Bounded Variable Least Squares (BVLS) and the Matlab function lsqlin. Here, the goal is to find solutions to ill-posed inverse problems that lie within box constraints. To do this, we formulate the box constraints as quadratic constraints, and solve the corresponding unconstrained regularized least squares problem. Using box constraints as quadratic constraints is an efficient approach because the optimization problem has a closed form solution. The effectiveness of the proposed algorithm is investigated through solving three benchmark problems and one from a hydrological application. Results are compared with solutions found by lsqlin, and the quadratically constrained formulation is solved using the L-curve, maximum a posteriori estimation (MAP), and the χ2 regularization method. The χ2 regularization method with quadratic constraints is the most effective method for solving least squares problems with box constraints.

KW - Box constraints

KW - Linear least squares

KW - Regularization

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U2 - 10.1016/j.laa.2009.04.017

DO - 10.1016/j.laa.2009.04.017

M3 - Article

AN - SCOPUS:75749099675

SN - 0024-3795

VL - 432

SP - 1936

EP - 1949

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 8

ER -