Variational problems with inequality constraints (original) (raw)
Variational Analysis, Optimization and Applications: Preface
Journal of Optimization Theory and Applications, 2013
This Special Issue of JOTA is mainly based on selected invited papers presented at the Tenth International Seminar in Optimization and Related Areas (ISORA), which was held in Lima (Perú) on October 3-7, 2011. Perú, the old Inca's country, had not long traditions in mathematical research and in mathematical conferences before the first ISORA meeting organized in 1993. It was dedicated to the memory of Eugen Blum, a Swiss mathematician who spent the last twenty years of his life in Perú. His area of research was Mathematical Optimization to which he made fundamental contributions. His seminal paper with Werner Oettli entitled "From optimization and variational inequalities to equilibrium problems", published in 1994, has initiated equilibrium problems in optimization theory, and then has been quoted by many researchers. According to MathSciNet, it was cited 404 times up to July 2013. Eugen Blum started the research on optimization theory in Lima. His students and collaborators have later continued these lines of research and have succeeded to build a strong research team that includes a number of very promising young mathematicians.
Theory and Algorithms of Variational Inequality and Equilibrium Problems, and Their Applications
Abstract and Applied Analysis, 2014
The variational inequality problem is a general problem formulation that encompasses many mathematical problems, among others, including nonlinear equations, optimization problems, complementarity problems, and fixed point problems. Variational inequality is developed as a tool for the study of certain classes of partial deferential equations, economic equilibrium problems, and the pricing model of the option.
Preface to “Optimization, Convex and Variational Analysis”
Set-Valued and Variational Analysis, 2021
This collection of works in the honor of professor Terry Rockafellar is a follow-up of the "Workshop on Optimization and Variational Analysis", dedicated to Terry's 85th birthday. The meeting, jointly organized by the CMM Center for Mathematical Modeling of the University of Chile (Chile) and the University of Perpignan (France), was held in Santiago on January 20-21, 2020. That workshop was one of the last meetings we could attend physically, before Coronavirus changed our lives in so many ways. Globetrotting has become virtual since then. Suddenly, the beauty of the world found itself flattened to a screen. Fortunately, some things have not changed: our admiration and appreciation for Terry's unique career has remained intact, as has the momentum to duly celebrate his birthday, through the edition of this special volume. We are very grateful to the authors and referees for their valuable contributions and careful work. The two volumes that make up the special issue in Terry's honor sample the tremendous breadth of subjects where Terry has made fruitful contributions. This special issue is a modest gift for someone who has gifted us with seminal textbooks, whose content has marked generations of researchers, influencing the way of doing mathematics when it involves "variations", regarding not only theory or analysis in optimization but also applications.
Journal of Optimization Theory and Applications
This special issue is comprised of eighteen articles that investigate various aspects of variational calculus. Contributions focus on both the abstract and numerical point of view with major applications in optimization theory and methods. Potentially, we may cite, for instance, the advancement of the general theory of regularity and stationarity that have recently emerged, like that of metric regularity or error bounds. These regularity concepts are appropriate for studying stability of solutions to optimization problems, particularly those of semi-infinite optimization and programs with equilibrium constraints, when standard assumptions are not satisfied. We may also provide in a large part of this issue some contributions establishing new connections between the theoretical estimates for several regularity properties within the field of variational analysis and convergence analysis of computational algorithms, enhancing the existing applied models, computational algorithms and facilitating the post-optimal analysis of solutions of variational systems. The range of applicability of this results is wide, for example, we mention applications in management (facility location, energy policy), engineering (optimal design, robotics), environmental sciences, as well as in many fields of economics (oligopolistic markets, network design, electric power pricing) and in emerging areas such as signal processing, finance, risk and statistical learning.
The Pre-Variational Problems and the Constrained Mathematical Programming Problem
OPSEARCH, 2002
In this paper relationships between the solutions of the Variational-Like and Pre-Variational Problems with those of the Constrained Mathematical Programming Problems have been established. Sufficient conditions are obtained for Kuhn-Tucker points to be minima of the Constrained Mathematical Programming Problem, through the solutions of the Pre-Variational Problems.
Optimization, Convex and Variational Analysis – Volume II
Set-Valued and Variational Analysis, 2022
This collection of works in the honor of professor Terry Rockafellar is a follow-up of the "Workshop on Optimization and Variational Analysis", dedicated to Terry's 85th birthday. The meeting, jointly organized by the CMM Center for Mathematical Modeling of the University of Chile (Chile) and the University of Perpignan (France), was held in Santiago on January 20-21, 2020. That workshop was one of the last meetings we could attend physically, before Coronavirus changed our lives in so many ways. Globetrotting has become virtual since then. Suddenly, the beauty of the world found itself flattened to a screen. Fortunately, some things have not changed: our admiration and appreciation for Terry's unique career has remained intact, as has the momentum to duly celebrate his birthday, through the edition of this special volume. We are very grateful to the authors and referees for their valuable contributions and careful work. The two volumes that make up the special issue in Terry's honor sample the tremendous breadth of subjects where Terry has made fruitful contributions. This special issue is a modest
On the optimal control of variational–hemivariational inequalities
Journal of Mathematical Analysis and Applications, 2019
The present paper represents a continuation of [23]. There, a continuous dependence result for the solution of an elliptic variational-hemivariational inequality was obtained and then used to prove the existence of optimal pairs for two associated optimal control problems. In the current paper we complete this study with more general results. Indeed, we prove the continuous dependence of the solution with respect to a parameter which appears in all the data of the problem, including the set of constraints, the nonlinear operator and the two functionals which govern the variational-hemivariational inequality. This allows us to consider a general associated optimal control problem for which we prove the existence of optimal pairs, together with a new convergence result. The mathematical tools developed in this paper are useful in the analysis and control of a large class of boundary value problems which, in a weak formulation, lead to elliptic variational-hemivariational inequalities. To provide an example, we illustrate our results in the study of an inequality which describes the equilibrium of an elastic body in frictional contact with a foundation made of a rigid body covered by a layer of soft material.
Foundations of the Calculus of Variations and Optimal Control
International Series in Operations Research & Management Science, 2010
In this chapter, we treat time as a continuum and derive optimality conditions for the extremization of certain functionals. We consider both variational calculus problems that are not expressed as optimal control problems and optimal control problems themselves. In this chapter, we relie on the classical notion of the variation of a functional. This classical perspective is the fastest way to obtain useful results that allow simple example problems to be solved that bolster one's understanding of continuous-time dynamic optimization.