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. https://doi.org/10.1137/16M1088570) and Dehghani et al. (INFOR Inf Syst Oper Res, 2019. https://doi.org/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 operator \(A(x)^T A(x)\), where A is 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–Moré-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.

中文翻译：

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约束非线性最小二乘的正则化方法

我们提出了一种具有等式约束的非线性最小二乘问题的正则化方法。我们的方法以Arreckx和Orban（SIAM J Optim 28（2）：1613-1639，2018. https://doi.org/10.1137/16M1088570）和Dehghani等人的方法为模型。（INFOR Inf Syst Oper Res，2019.https：//doi.org/10.1080/03155986.2018.1559428）并应用了选择性正则化方案，可以将其视为增强拉格朗日的重新表述。我们的公式避免了运算符\（A（x）^ TA（x）\）的出现，其中A是非线性残差的雅可比行列式，通常有助于子问题的密度和不良条件。在导数有界的情况下，我们建立了到不可行度量的KKT点或固定点的全局收敛。如果二阶导数是Lipschitz连续且满足二阶充分条件，则我们将建立超线性收敛而无需约束条件成立。收敛速度取决于Dennis-Moré型条件。我们用支持多种浮点系统的Julia语言描述实现。我们说明了一种简单的渐进方案，以四倍的精度获得解决方案。由于我们的方法类似于在相关问题上应用具有精确绩效函数的SQP方法，