On the convergence of quasi-Newton methods for nonsmooth problems
نویسنده
چکیده
We develop a theory of quasi-Newton and least-change update methods for solving systems of nonlinear equations F (x) = 0. In this theory, no diierentiability conditions are necessary. Instead, we assume that F can be approximated, in a weak sense, by an aane function in a neighborhood of a solution. Using this assumption, we prove local and ideal convergence. Our theory can be applied to B-diierentiable functions and to partially diierentiable functions.
منابع مشابه
A parameterized Newton method and a quasi-Newton method for nonsmooth equations
This paper presents a parameterized Newton method using generalized Jacobians and a Broyden-like method for solving nonsmooth equations. The former ensures that the method is well-deened even when the generalized Jacobian is singular. The latter is constructed by using an approximation function which can be formed for nonsmooth equations arising from partial diierential equations and nonlinear ...
متن کاملOn the Behavior of Damped Quasi-Newton Methods for Unconstrained Optimization
We consider a family of damped quasi-Newton methods for solving unconstrained optimization problems. This family resembles that of Broyden with line searches, except that the change in gradients is replaced by a certain hybrid vector before updating the current Hessian approximation. This damped technique modifies the Hessian approximations so that they are maintained sufficiently positive defi...
متن کاملNewton and Quasi-Newton Methods for a Class of Nonsmooth Equations and Related Problems
The paper presents concrete realizations of quasi-Newton methods for solving several standard problems including complementarity problems, special variational inequality problems, and the Karush–Kuhn–Tucker (KKT) system of nonlinear programming. A new approximation idea is introduced in this paper. The Q-superlinear convergence of the Newton method and the quasiNewton method are established und...
متن کاملModify the linear search formula in the BFGS method to achieve global convergence.
<span style="color: #333333; font-family: Calibri, sans-serif; font-size: 13.3333px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: justify; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: #ffffff; text-dec...
متن کاملA Distributed Quasi-Newton Algorithm for Empirical Risk Minimization with Nonsmooth Regularization
In this paper, we propose a communicationand computationefficient distributed optimization algorithm using second-order information for solving ERM problems with a nonsmooth regularization term. Current second-order and quasi-Newton methods for this problem either do not work well in the distributed setting or work only for specific regularizers. Our algorithm uses successive quadratic approxim...
متن کامل