Mixed Variational Inequality Interval-valued Problem: Theorems of Existence of Solutions

In this article, our efforts focus on finding the conditions for the existence of solutions of Mixed Stampacchia Variational Inequality Interval-valued Problem on Hadamard manifolds with monotonicity assumption by using KKM mappings. Conditions that allow us to prove the existence of equilibrium points in a market of perfect competition. We will identify solutions of Stampacchia variational problem and optimization problem with the interval-valued convex objective function, improving on previous results in the literature. We will illustrate the main results obtained with some examples and numerical results

Generalized modularity matrices

Various modularity matrices appeared in the recent literature on network analysis and algebraic graph theory. Their purpose is to allow writing as quadratic forms certain combinatorial functions appearing in the framework of graph clustering problems. In this paper we put in evidence certain common traits of various modularity matrices and shed light on their spectral properties that are at the basis of various theoretical results and practical spectral-type algorithms for community detection

Advances in Optimization and Nonlinear Analysis

The present book focuses on that part of calculus of variations, optimization, nonlinear analysis and related applications which combines tools and methods from partial differential equations with geometrical techniques. More precisely, this work is devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The book is a valuable guide for researchers, engineers and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science and economics

Formulation, existence, and computation of boundedly rational dynamic user equilibrium with fixed or endogenous user tolerance

This paper analyzes dynamic user equilibrium (DUE) that incorporates the notion of boundedly rational (BR) user behavior in the selection of departure times and routes. Intrinsically, the boundedly rational dynamic user equilibrium (BR-DUE) model we present assumes that travelers do not always seek the least costly route-and-departure-time choice. Rather, their perception of travel cost is affected by an indifference band describing travelers’ tolerance of the difference between their experienced travel costs and the minimum travel cost. An extension of the BR-DUE problem is the so-called variable tolerance dynamic user equilibrium (VT-BR-DUE) wherein endogenously determined tolerances may depend not only on paths, but also on the established path departure rates. This paper presents a unified approach for modeling both BR-DUE and VT-BR-DUE, which makes significant contributions to the model formulation, analysis of existence, solution characterization, and numerical computation of such problems. The VT-BR-DUE problem, together with the BR-DUE problem as a special case, is formulated as a variational inequality. We provide a very general existence result for VT-BR-DUE and BR-DUE that relies on assumptions weaker than those required for normal DUE models. Moreover, a characterization of the solution set is provided based on rigorous topological analysis. Finally, three computational algorithms with convergence results are proposed based on the VI and DVI formulations. Numerical studies are conducted to assess the proposed algorithms in terms of solution quality, convergence, and computational efficiency

Nonlinear Analysis and Optimization with Applications

Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world