For constrained equations with nonisolated solutions and a certain family of Newton-type methods, it was previously shown that if the equation mapping is 2-regular at a given solution with respect to a direction which is interior feasible and which is in the null space of the Jacobian, then there is an associated large (not asymptotically thin) domain of starting points from which the iterates are well defined and converge to the specific solution in question. Under these assumptions, the constrained local Lipschitzian error bound does not hold, unlike the common settings of convergence and rate of convergence analyses. In this work, we complement those previous results by considering the case when the equation mapping is 2-regular with respect to a direction in the null space of the Jacobian which is in the tangent cone to the set, but need not be interior feasible. Under some further conditions, we still show linear convergence of order 1/2 from a large domain around the solution (despite degeneracy, and despite that there may exist other solutions nearby). Our results apply to constrained variants of the Gauss–Newton and Levenberg–Marquardt methods, and to the LP-Newton method. An illustration for a smooth constrained reformulation of the nonlinear complementarity problem is also provided.

对于具有非孤立解的约束方程和某一族牛顿型方法，之前已经表明，如果方程映射在给定解上相对于内部可行且位于以下零空间中的方向是 2-正则雅可比行列式，则存在一个相关的大（非渐近薄）起始点域，从该起始点迭代被很好地定义并收敛到所讨论的特定解决方案。在这些假设下，与收敛和收敛率分析的常见设置不同，受约束的局部 Lipschitzian 误差界不成立。在这项工作中，我们通过考虑以下情况来补充以前的结果：方程映射相对于雅可比行列式零空间中的方向为 2-正则，该方向位于集合的切锥中，但不必是内部可行的。在一些进一步的条件下，我们仍然显示解周围大域的 1/2 阶线性收敛（尽管存在简并性，并且尽管附近可能存在其他解）。我们的结果适用于高斯-牛顿法和莱文伯格-马夸特法的约束变体，以及 LP-牛顿法。还提供了非线性互补问题的平滑约束重构的说明。