Newton's Method for Solving Inclusions Using Set-Valued Approximations

Abstract

International audienceResults on stability of both local and global metric regularity under set-valued perturbations are presented. As an application, we study (super)linear convergence of a Newton- type iterative process for solving generalized equations. We investigate several iterative schemes such as the inexact Newton’s method, the nonsmooth Newton’s method for semismooth functions, the inexact proximal point algorithm, etc. Moreover, we also cover a forward-backward splitting algorithm for finding a zero of the sum of two multivalued (not necessarily monotone) operators. Finally, a globalization of the Newton’s method is discussed

Existence and continuity of solution trajectories of generalized equations with application in electronics

“…Then, make slight modifications by considering the additional condition of single-valuedness 3 In the graphical analysis of the circuit, we plot two maps on the same i − v plane: the i − v characteristic of the diode, and the ordered equation (with respect to i) of the circuit gained from KVL. The solution can then be obtained as the equilibrium point, that is the coordinates of the intersection point of the two graphs.…”

On semiregularity of mappings

“…First, the mapping F is said to be metrically regular 5 around (x,ȳ) whenȳ ∈ F (x) and there is a constant κ > 0 along with a neighborhood U × V of (x,ȳ) in X × Y such that (1) dist x, F −1 (y) ≤ κ dist y, F (x) for every (x, y) ∈ U × V, where dist(u, C) is the distance from a point u to a set C and the space X × Y is equipped with the product (box) topology. The infimum of κ > 0 for which there exists a neighborhood U × V of (x,ȳ) in X × Y such that (1) holds is called the regularity modulus of F around (x,ȳ) and is denoted by reg F (x,ȳ). Second, the mapping F is called open with a linear rate 6 around (x,ȳ) whenȳ ∈ F (x) and there are positive constants c and ε along with a neighborhood U × V of (x,ȳ) in X × Y such that (2) IB[y, ct] ⊂ F (IB[x, t]) whenever (x, y) ∈ U × V, y ∈ F (x) and t ∈ (0, ε),…”

“…A fundamental well-known fact is that (4) sur F (x,ȳ) · reg F (x,ȳ) = 1 and reg F (x,ȳ) = lip F −1 (ȳ,x), under the convention that 0·∞ = ∞·0 = 1, inf ∅ = ∞, and, as we work with nonnegative quantities, that sup ∅ = 0. Fixing one of the components of (x, y) in (1), that is letting either x :=x or y :=ȳ, one gets two different, weaker than regularity, concepts. Of course, one can reformulate both of them in terms of openness and continuity of the inverse, respectively.…”

“…Theorem 4.1 therein for iteration (67) is quite similar to Theorem 5.1 above with one important difference. We assume that all the "partial linearizations" G A in (54) are regular around (x, 0), while in [1] the mapping f + F is assumed to be such. Clearly, our assumption is weaker.…”

Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods

“…where f : X → Y is a single-valued mapping while F : X ⇒ Y is a set-valued mapping between arbitrary Banach spaces. It is well recognized that the general model (1) has been used to describe a vast variety of problems in a unified way, including equations and most notably variational inequalities, constraint systems, and optimality conditions in mathematical programming and optimal control.…”