Inexact Newton methods are constructed by combining Newton’s method with another iterative method that is used to solve the Newton equations inexactly. In this paper, we establish two semilocal convergence theorems for the inexact Newton methods. When these two theorems are specified to Newton’s method, we obtain a different Newton-Kantorovich theorem about Newton’s method. When the iterative m...

We propose and analyze an inexact version of the modified subgradient (MSG) algorithm, which we call the IMSG algorithm, for nonsmooth and nonconvex optimization over a compact set. We prove that under an approximate, i.e. inexact, minimization of the sharp augmented Lagrangian, the main convergence properties of the MSG algorithm are preserved for the IMSG algorithm. Inexact minimization may a...

Inexact Newton methods are constructed by combining Newton’s method with another iterative method that is used to solve the Newton equations inexactly. In this paper, we establish two semilocal convergence theorems for the inexact Newton methods. When these two theorems are specified to Newton’s method, we obtain a different Newton-Kantorovich theorem about Newton’s method. When the iterative m...

In this paper we present the convergence analysis of the inexact infeasible path-following (IIPF) interior point algorithm. In this algorithm the preconditioned conjugate gradient method is used to solve the reduced KKT system (the augmented system). The augmented system is preconditioned by using a block triangular matrix. The KKT system is solved approximately. Therefore, it becomes necessary...

We present the convergence analysis of the inexact infeasible pathfollowing (IIPF) interior-point algorithm. In this algorithm, the preconditioned conjugate gradient method is used to solve the reduced KKT system (the augmented system). The augmented system is preconditioned by using a block triangular matrix. The KKT system is solved approximately. Therefore, it becomes necessary to study the ...

We compare the linear rate of convergence estimates for two inexact proximal point methods. The first one is the classical inexact scheme introduced by Rockafellar, for which we obtain a slightly better estimate than the one given in [16]. The second one is the hybrid inexact proximal point approach introduced in [25, 22]. The advantage of the hybrid methods is that they use more constructive a...

For a class of nonconvex nonsmooth functions, we consider the problem of computing an approximate critical point, in the case of inexact oracles. The latter means that only an inexact function value and an inexact subgradient are available, at any given point. We assume that the errors in function and subgradient evaluations are merely bounded, and in principle need not vanish in the limit. We ...