Uniformity and inexact version of a proximal method for metrically regular mappings

We study stability properties of a proximal point algorithm for solving the inclusion 0∈T(x) when T is a set-valued mapping that is not necessarily monotone. More precisely we show that the convergence of our algorithm is uniform, in the sense that it is stable under small perturbations whenever the set-valued mapping T is metrically regular at a given solution. We present also an inexact proximal point method for strongly metrically subregular mappings and show that it is super-linearly convergent to a solution to the inclusion 0∈T(x).     

An inertial proximal scheme for nonmonotone mappings

We present an inertial proximal method for solving an inclusion involving a nonmonotone set-valued mapping enjoying some regularity properties. More precisely, we investigate the local convergence of an implicit scheme for solving inclusions of the type T(x)∋0 where T is a set-valued mapping acting from a Banach space into itself. We consider subsequently the case when T is strongly metrically subregular, metrically regular and strongly regular around a solution to the inclusion. Finally, we study the convergence of our algorithm under variational perturbations.     

The radius of metric regularity

Metric regularity is a central concept in variational analysis for the study of solution mappings associated with ``generalized equations', including variational inequalities and parameterized constraint systems. Here it is employed to characterize the distance to irregularity or infeasibility with respect to perturbations of the system structure. Generalizations of the Eckart-Young theorem in numerical analysis are obtained in particular.

     

Realization of the hybrid method for Mann iterations

The Mann iterations for nonexpansive mappings have only weak convergence even in a Hilbert space H. In order to overcome this weakness, Nakajo and Takahashi proposed the hybrid method for Mann’s iteration process:

     

Strong metric subregularity of mappings in variational analysis and optimization

Although the property of strong metric subregularity of set-valued mappings has been present in the literature under various names and with various (equivalent) definitions for more than two decades, it has attracted much less attention than its older “siblings”, the metric regularity and the strong (metric) regularity. The purpose of this paper is to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications. We show that the strong metric subregularity of a mapping F acting between metric spaces is stable under perturbations of the form $f+F$, where f is a function with a small calmness constant. This result is parallel to the Lyusternik–Graves theorem for metric regularity and to the Robinson theorem for strong regularity, where the perturbations are represented by a function f with a small Lipschitz constant. Then we study perturbation stability of the same kind for mappings acting between Banach spaces, where f is not necessarily differentiable but admits a set-valued derivative-like approximation. Strong metric q-subregularity is also considered, where q is a positive real constant appearing as exponent in the definition. Rockafellar's criterion for strong metric subregularity involving injectivity of the graphical derivative is extended to mappings acting in infinite-dimensional spaces. A sufficient condition for strong metric subregularity is established in terms of surjectivity of the Fréchet coderivative, and it is shown by a counterexample that surjectivity of the limiting coderivative is not a sufficient condition for this property, in general. Then various versions of Newton's method for solving generalized equations are considered including inexact and semismooth methods, for which superlinear convergence is shown under strong metric subregularity. As applications to optimization, a characterization of the strong metric subregularity of the KKT mapping is obtained, as well as a radius theorem for the optimality mapping of a nonlinear programming problem. Finally, an error estimate is derived for a discrete approximation in optimal control under strong metric subregularity of the mapping involved in the Pontryagin principle.     

Robinson metric regularity of parametric variational systems

In this paper, the Robinson metric regularity of a parametric variational system is investigated. Some applications to the contingent derivative of parametric variational system and to the Robinson metric regularity of a parametric vector optimization problem are then studied.     

Values on regular games under Kirchhoff’s laws

The Shapley value is a central notion defining a rational way to share the total worth of a cooperative game among players. We address a general framework leading to applications to games with communication graphs, where the feasible coalitions form a poset whose all maximal chains have the same length. Considering a new way to define the symmetry among players, we propose an axiomatization of the Shapley value of these games. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the efficiency axiom correspond to the two Kirchhoff’s laws in the circuit associated to the Hasse diagram of feasible coalitions.     

Constrained versions of Sauer’s lemma

Let [n]={1,…,n}. For a function h:[n]→{0,1}, x[n] and y{0,1} define by the width ω_h(x,y) of h at x the largest nonnegative integer a such that h(z)=y on x−a≤z≤x+a. We consider finite VC-dimension classes of functions h constrained to have a width ω_h(x_i,y_i) which is larger than N for all points in a sample or a width no larger than N over the whole domain [n]. Extending Sauer’s lemma, a tight upper bound with closed-form estimates is obtained on the cardinality of several such classes.     

On Perturbation Stability of Metric Regularity

We introduce a measure of closeness for set-valued mappings from a metric space to a Banach space which allows to quantatively estimate the effect of perturbation of a set-valued mapping on its metric regularity and covering characteristics.     

On the equivalence of the Mizoguchi–Takahashi fixed point theorem to Nadler’s theorem

In this work, we prove that on a metrically convex complete metric space, the Mizoguchi–Takahashi theorem is equivalent to Nadler’s theorem. Also, we obtain its equivalence on a compact metric space.     

A note on kernels and Sperner’s Lemma

The kernel-solvability of perfect graphs was first proved by Boros and Gurvich, and later Aharoni and Holzman gave a shorter proof. Both proofs were based on Scarf’s Lemma. In this note we show that a very simple proof can be given using a polyhedral version of Sperner’s Lemma. In addition, we extend the Boros–Gurvich theorem to h-perfect graphs and to a more general setting.     

Some variant of Newton’s method with third-order convergence

We present a new modification of the Newton’s method which produces iterative methods with order of convergence three. A general error analysis providing the higher order of convergence is given, and the best efficiency, in term of function evaluations, of two of this new methods is provided.     

Lyapunov’s inequality on timescales

The purpose of this work is to establish the timescale version of Lyapunov’s inequality as follows: Let $x\left(t\right)$ be a nontrivial solution of ${\left(r\left(t\right)x^{\Delta }\left(t\right)\right)}^{\Delta }+p\left(t\right)x^{\sigma }\left(t\right)=0\text{on }[a,b]$ satisfying $x\left(a\right)=x\left(b\right)=0$. Then, under suitable conditions on $p$, $r$, $a$ and $b$, we have ${\int }_a^b{p}_{+}\left(t\right)\Delta t\ge \{\begin{array}{cc}\frac{r\left(a\right)}{r\left(b\right)}\frac{b-a}{f\left(d\right)}\text{,}& \text{if }r\text{ is increasing}\text{,}\\ \frac{r\left(b\right)}{r\left(a\right)}\frac{b-a}{f\left(d\right)}\text{,}& \text{if }r\text{ is decreasing}\text{,}\end{array}$ where $p_{+}\left(t\right)=max\{p\left(t\right),0\},f\left(t\right)=(t-a)(b-t)$ and $d\in \mathbb{T}$ satisfies $|\frac{a+b}{2}-d|=min\{|\frac{a+b}{2}-s|\mid s\in [a,b]\cap \mathbb{T}\}$ if $\frac{a+b}{2}\in \mathbb{T}$. Here $\mathbb{T}$ is a timescale (see below).     

Iterative solving of variational inclusions under Wijsman perturbations

We prove that the metric regularity of set-valued mappings is stable under some Wijsman-type perturbations. Then, we solve a variational inclusion viewed as a limit-problem using assumptions on a sequence of associated problems. Finally, we apply our results to classical methods for solving variational inclusions.      

A note on the fractional-order Volta’s system

This paper deals with the fractional-order Volta’s system. It is based on the concept of chaotic system, where the mathematical model of system contains fractional order derivatives. This system has simple structure and can display a double-scroll attractor. The behavior and stability analysis of the integer-order and the fractional commensurate and non-commensurate order Volta’s system with total order less than 3 which exhibits chaos are presented as well.     

A general metric regularity in asplund banach spaces

This paper establishes a simple and easily-applied criterion for determining whether a multivalued mapping is metrically regular relatively to a subset in the range space.     

A note on travelling-wave solutions to Lax’s seventh-order KdV equation

Ganji and Abdollahzadeh [D.D. Ganji, M. Abdollahzadeh, Appl. Math. Comput. 206 (2008) 438–444] derived three supposedly new travelling-wave solutions to Lax’s seventh-order KdV equation. Each solution was obtained by a different method. It is shown that any two of the solutions may be obtained trivially from the remaining solution. Furthermore it is noted that one of the solutions has been known for many years.     

A new generalization of Aczél’s inequality and its applications to an improvement of Bellman’s inequality

A new generalized version of Aczél’s inequality is proved. This is a unified generalization of some known results. Moreover, the result is applied to the improvement of the well-known Bellman’s inequality.