The Josephy–newton Method for Semismooth Generalized Equations and Semismooth Sqp for Optimization
نویسندگان
چکیده
While generalized equations with differentiable single-valued base mappings and the associated Josephy–Newton method have been studied extensively, the setting with semismooth base mapping had not been previously considered (apart from the two special cases of usual nonlinear equations and of Karush-Kuhn-Tucker optimality systems). We introduce for the general semismooth case appropriate notions of solution regularity and prove local convergence of the corresponding Josephy–Newton method. As an application, we immediately recover the known primal-dual local convergence properties of semismooth SQP, but also obtain some new results that complete the analysis of the SQP primal rate of convergence, including its quasi-Newton variant.
منابع مشابه
Optimization Methods in Banach Spaces
In this chapter we present a selection of important algorithms for optimization problems with partial differential equations. The development and analysis of these methods is carried out in a Banach space setting. We begin by introducing a general framework for achieving global convergence. Then, several variants of generalized Newton methods are derived and analyzed. In particular, necessary a...
متن کاملA SQP-Semismooth Newton-type Algorithm applied to Control of the instationary Navier--Stokes System Subject to Control Constraints
Sequential quadratic programming (SQP) methods for the optimal control of the instationary Navier-Stokes equations with pointwise constraints on the control are considered. Due to the presence of the constraints the quadratic subproblems (QP) of SQP require a more sophisticated solver compared to the unconstrained case. In this paper, a semismooth Newton method is proposed for efficiently solvi...
متن کاملGlobal and Local Superlinear Convergence Analysis of Newton - TypeMethods for Semismooth Equations with Smooth Least
The local superlinear convergence of the generalized Newton method for solving systems of nonsmooth equations has been proved by Qi and Sun under the semismooth condition and nonsingularity of the generalized Jacobian at the solution. Unlike the Newton method for systems of smooth equations, globalization of the generalized Newton method seems dif-cult to achieve in general. However, we show th...
متن کاملA semismooth Newton method for tensor eigenvalue complementarity problem
In this paper, we consider the tensor eigenvalue complementarity problem which is closely related to the optimality conditions for polynomial optimization, as well as a class of differential inclusions with nonconvex processes. By introducing an NCP-function, we reformulate the tensor eigenvalue complementarity problem as a system of nonlinear equations. We show that this function is strongly s...
متن کاملA nonmonotone semismooth inexact Newton method
In this work we propose a variant of the inexact Newton method for the solution of semismooth nonlinear systems of equations. We introduce a nonmonotone scheme, which couples the inexact features with the nonmonotone strategies. For the nonmonotone scheme, we present the convergence theorems. Finally, we show how we can apply these strategies in the variational inequalities context and we prese...
متن کامل