Newton’s Method Based on Generalized Derivatives for Nonsmooth Functions: Convergence Analysis

Semismooth SQP method for equality-constrained optimization problems with an application to the lifted reformulation of mathematical programs with complementarity constraints

“…Therefore, local convergence properties of semismooth SQP follow from the corresponding general results for SNM (see [17,18,25,27] and [6,Chapter 7]). Specifically, primal-dual local superlinear convergence of semismooth SQP is obtained under the following assumptions:…”

“…As is well-known, semismooth SQP can be interpreted as the semismooth Newton method (see [17,18,25,27] and [6,Chapter 7]) applied to the Lagrange optimality system (4). Specifically, define Φ :…”

“…It should be noted that, in local analysis, semismooth SQP for (1) is just the semismooth Newton method (SNM) applied to the Lagrange optimality system of (1). For generic systems of nonlinear equations, such methods were developed in [17,18,25,27], with local superlinear convergence properties established under reasonable regularity assumptions (see the discussion in Section 2.1 below). Moreover, local convergence and rate of convergence properties of semismooth SQP for more general optimization problems with additional inequality constraints were studied in [9,26].…”

The semismooth Newton method for the solution of reactive transport problems including mineral precipitation-dissolution reactions

“…Related material can also be found in Kummer [29,30]. Similar to the classical Newton method these methods can be shown to be locally quadratically convergent under moderate assumptions that are somewhat similar to the assumptions that the classical method requires.…”

A further result on an implicit function theorem for locally Lipschitz functions

“…Condition (1.3) and the nonsingularity of any matrix in ∂F (x) (or ∂ B F (x) or an variant) are two key conditions for superlinear convergence of nonsmooth Newton methods, e.g., Kummer [4] and [7,8,11].…”