Least squares problems with inequality constraints as quadratic constraints

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, we formulate the box constraints as quadratic constraints, and solve the corresponding unconstrained regularized least squares problem. Box constraints as quadratic constraints is an efficient approach because the optimization problem has a known unique solution. The effectiveness of the proposed algorithm is investigated through solving three benchmark problems and a hydrological application. Results are compared with solutions found by lsqlin, and the quadraticaly constrained formulation is solved using the L-curve, maximum a posteriori estimation (MAP), and the χ2 regularization methods. The χ2 regularization method with quadratic constraints is the most effective method for solving least squares problems with box constraints.