Set-valued Inclusions Research Articles

Dynamical Analysis of Full-Range Cellular Neural Networks by Exploiting Differential Variational Inequalities

The paper considers the full-range (FR) model of cellular neural networks (CNNs) in the case where the neuron nonlinearities are ideal hard-comparator functions with two vertical straight segments. The dynamics of FR-CNNs, which is described by a differential inclusion, is rigorously analyzed by means of theoretical tools from set-valued analysis and differential inclusions. The fundamental property proved in the paper is that FR-CNNs are equivalent to a special class of differential inclusions termed differential variational inequalities. A sound foundation to the dynamics of FR-CNNs is then given by establishing the existence and uniqueness of the solution starting at a given point, and the existence of equilibrium points. Moreover, a fundamental result on trajectory convergence towards equilibrium points (complete stability) for reciprocal standard CNNs is extended to reciprocal FR-CNNs by using a generalized Lyapunov approach. As a consequence, it is shown that the study of the ideal case with vertical straight segments in the neuron nonlinearities is able to give a clear picture and analytic characterization of the salient features of motion, such as the sliding modes along the boundary of the hypercube defined by the hard-comparator nonlinearities. Finally, it is proved that the solutions of the ideal FR model are the uniform limit as the slope tends to infinity of the solutions of a model where the vertical segments in the nonlinearities are approximated by segments with finite slope.

Perturbed Iterative Algorithms for Generalized Nonlinear Set-Valued Quasivariational Inclusions Involving Generalized m-Accretive Mappings

A new class of generalized nonlinear set-valued quasivariational inclusions involving generalized -accretive mappings in Banach spaces are studied, which included many variational inclusions studied by others in recent years. By using the properties of the resolvent operator associated with generalized -accretive mappings, we established the equivalence between the generalized nonlinear set-valued quasi-variational inclusions and the fixed point problems, and some new perturbed iterative algorithms, proved that its proximate solution converges strongly to its exact solution in real Banach spaces.

Sensitivity analysis of generalized set-valued quasi-variational inclusion in Banach spaces

The sensitivity analysis for a class of generalized set-valued quasi-variational inclusion problems is investigated in the setting of Banach spaces. By using the resolvent operator technique, without assuming the differentiability and monotonicity of the given data, equivalence of these problems to the class of generalized resolvent equations is established.

Globally and Superlinearly Convergent Algorithm for Minimizing a Normal Merit Function

In this paper we present two new concepts related to the solution of systems of nonsmooth equations (NE) and variational inequalities (VI). The first concept is that of a normal merit function, which summarizes the simple basic properties shared by various known merit functions. In general, normal merit functions are locally Lipschitz, but not differentiable. The second concept is that of a Newtonian operator, whose values generalize the concept of the Hessian for normal merit functions. These two concepts are then used to generalize the nonsmooth Newton method for solving the equation $\nabla f(x) = 0$, where f is a normal merit function with $f\in C^1$, to the case where f is only locally Lipschitz and the set-valued inclusion $0 \in \partial f(x)$ needs to be solved. Combining the resulting generalized Newton method with certain first-order methods, we obtain a globally and superlinearly convergent algorithm for minimizing normal merit functions.