A regularization method for constrained nonlinear least squares

摘要

We propose a regularization method for nonlinear least-squares problems with equality constraints. Our approach is modeled after those of Arreckx and Orban (SIAM J Optim 28(2):1613-1639, 2018. 10.1137/16M1088570) and Dehghani et al. (INFOR Inf Syst Oper Res, 2019. 10.1080/03155986.2018.1559428) and applies a selective regularization scheme that may be viewed as a reformulation of an augmented Lagrangian. Our formulation avoids the occurrence of the operatorA(x)(T)A(x), whereAis the Jacobian of the nonlinear residual, which typically contributes to the density and ill conditioning of subproblems. Under boundedness of the derivatives, we establish global convergence to a KKT point or a stationary point of an infeasibility measure. If second derivatives are Lipschitz continuous and a second-order sufficient condition is satisfied, we establish superlinear convergence without requiring a constraint qualification to hold. The convergence rate is determined by a Dennis-More-type condition. We describe our implementation in the Julia language, which supports multiple floating-point systems. We illustrate a simple progressive scheme to obtain solutions in quadruple precision. Because our approach is similar to applying an SQP method with an exact merit function on a related problem, we show that our implementation compares favorably to IPOPT in IEEE double precision.