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Liste des exposés
I. Nouvelles tendances en mécanique des fluides
II. Problèmes à frontière libre
III. Modèles mathèmatiques et méthodes numériques en mécanique des milieux continus
VI. Analyse et contrôle des EDP
VIII. Analyse non-lisse et optimisation
Analyse non-lisse et optimisation
(cette liste est en processus d'actualisation)
List des exposés
1.
BAGDASAR Ovidiu o.bagdasar@derby.ac.uk
University of Derby, United Kingdom
Titre: Extremal properties of explicitly quasiconvex vector functions (details)
University of Derby, United Kingdom
Titre: Extremal properties of explicitly quasiconvex vector functions (details)
Résumé:
It is known that any local maximizer of an explicitly quasiconvex real-valued function is a global minimizer, whenever it belongs to the intrinsic core of the function's domain. Jointly with Nicolae Popovici we extend this result for a special class of semistrictly quasiconvex vector-valued functions by means of a general concept of optimality, proposed by F. Flores-Baz\'an and E. Hern\'andez (2011)
It is known that any local maximizer of an explicitly quasiconvex real-valued function is a global minimizer, whenever it belongs to the intrinsic core of the function's domain. Jointly with Nicolae Popovici we extend this result for a special class of semistrictly quasiconvex vector-valued functions by means of a general concept of optimality, proposed by F. Flores-Baz\'an and E. Hern\'andez (2011)
2.
BONNEL Henri henribonnel@icloud.com
University of New Caledonia & Curtin University, Perth, Australia, France
Titre: Post Pareto Analysis for multiobjective parabolic control systems (details)
University of New Caledonia & Curtin University, Perth, Australia, France
Titre: Post Pareto Analysis for multiobjective parabolic control systems (details)
Résumé:
This talk deals with the optimization of a functional over a Pareto con- trol set associated with a convex multiobjective optimal control problem in Hilbert spaces, namely parabolic system. This approach generalizes for multiobjective optimal control of the dynamical systems governed by PDEs some results obtained in the case of multiobjective optimal control for the systems governed by ODEs. Two examples will be presented. General op- timality results will be given, and a special attention will be paid to the linear-quadratic multiobjective parabolic system.
This talk deals with the optimization of a functional over a Pareto con- trol set associated with a convex multiobjective optimal control problem in Hilbert spaces, namely parabolic system. This approach generalizes for multiobjective optimal control of the dynamical systems governed by PDEs some results obtained in the case of multiobjective optimal control for the systems governed by ODEs. Two examples will be presented. General op- timality results will be given, and a special attention will be paid to the linear-quadratic multiobjective parabolic system.
3.
COSTEA Nicusor nicusorcostea@yahoo.com
Department of Mathematics and Computer Science, POLITEHNICA University, Bucharest and SIMION STOILOW Institute of Mathematics of the Romanian Academy, Bucharest, Romania
Titre: Bounded saddle point methods for locally Lipschitz functionals (details)
Department of Mathematics and Computer Science, POLITEHNICA University, Bucharest and SIMION STOILOW Institute of Mathematics of the Romanian Academy, Bucharest, Romania
Titre: Bounded saddle point methods for locally Lipschitz functionals (details)
Résumé:
In the framework of critical point theory for locally Lipschitz functionals defined on a bounded closed ball of a real reflexive Banach space we show how deformation results can be used to obtain saddle-point type results even in the case when the usual Palais-Smale condition fails. Applications to differential inclusions are also discussed.
In the framework of critical point theory for locally Lipschitz functionals defined on a bounded closed ball of a real reflexive Banach space we show how deformation results can be used to obtain saddle-point type results even in the case when the usual Palais-Smale condition fails. Applications to differential inclusions are also discussed.
4.
FLORESCU Liviu lflo@uaic.ro
Alexandru Ioan Cuza University, Romania
Titre: Sur la continuit\' e des fonctionnelles int\' egrales (details)
Alexandru Ioan Cuza University, Romania
Titre: Sur la continuit\' e des fonctionnelles int\' egrales (details)
Résumé:
Nous \'{e}tudions la continuit\'{e} de la fonctionnelle \(u\mapsto\int_{\Omega} f(t,u(t))dt\) sur des sous-ensembles de l'espace \(\mathscr L^{1}(\Omega,\mathbb R^{s})\) muni avec la topologie de la convergence en mesure. Les r\'{e}sultats sont appliqu\'{e}s sur les espaces de Sobolev \(\mathscr W^{1,1} (\Omega,\mathbb R^{m})\hookrightarrow\mathscr L^{1} (\Omega,\mathbb R^{m + m\times d})\).
Nous \'{e}tudions la continuit\'{e} de la fonctionnelle \(u\mapsto\int_{\Omega} f(t,u(t))dt\) sur des sous-ensembles de l'espace \(\mathscr L^{1}(\Omega,\mathbb R^{s})\) muni avec la topologie de la convergence en mesure. Les r\'{e}sultats sont appliqu\'{e}s sur les espaces de Sobolev \(\mathscr W^{1,1} (\Omega,\mathbb R^{m})\hookrightarrow\mathscr L^{1} (\Omega,\mathbb R^{m + m\times d})\).
5.
GüNTHER Christian Christian.Guenther@mathematik.uni-halle.de
Martin Luther University Halle-Wittenberg, Institute for Mathematics, Germany
Titre: Relationships between constrained and unconstrained multiobjective optimization (details)
Martin Luther University Halle-Wittenberg, Institute for Mathematics, Germany
Titre: Relationships between constrained and unconstrained multiobjective optimization (details)
Résumé:
In this talk we investigate relationships between constrained and unconstrained multi-objective optimization problems. We mainly focus on generalized convex multi-objective optimization problems, i.e., the objective function is a componentwise generalized convex (e.g., quasi-convex or semi-strictly quasi-convex) function and the feasible domain is a convex set. Beside the field of location theory the assumptions of generalized convexity are found in several branches of Economics. We derive a characterization of the set of efficient solutions of a constrained multi-objective optimization problem using characterizations of the sets of efficient solutions of unconstrained multi-objective optimization problems. We demonstrate the usefulness of the results by applying it on constrained multi-objective location problems. Using our new results we show that special classes of constrained multi-objective location problems (e.g., point-objective location problems, Weber location problems and center location problems) can be completely solved with the help of algorithms for the unconstrained case. At the end of the talk, we present some information about the current development of the MATLAB-based software tool "Facility Location Optimizer" (see www.project-flo.de).
In this talk we investigate relationships between constrained and unconstrained multi-objective optimization problems. We mainly focus on generalized convex multi-objective optimization problems, i.e., the objective function is a componentwise generalized convex (e.g., quasi-convex or semi-strictly quasi-convex) function and the feasible domain is a convex set. Beside the field of location theory the assumptions of generalized convexity are found in several branches of Economics. We derive a characterization of the set of efficient solutions of a constrained multi-objective optimization problem using characterizations of the sets of efficient solutions of unconstrained multi-objective optimization problems. We demonstrate the usefulness of the results by applying it on constrained multi-objective location problems. Using our new results we show that special classes of constrained multi-objective location problems (e.g., point-objective location problems, Weber location problems and center location problems) can be completely solved with the help of algorithms for the unconstrained case. At the end of the talk, we present some information about the current development of the MATLAB-based software tool "Facility Location Optimizer" (see www.project-flo.de).
6.
GRAD Anca ancagrad@math.ubbcluj.ro
Babes-Bolyai University of Cluj-Napoca, Romania
Titre: Optimiality conditions by means of generalized interiors (details)
Babes-Bolyai University of Cluj-Napoca, Romania
Titre: Optimiality conditions by means of generalized interiors (details)
Résumé:
We present several optimality notions defined by means of the quasi interior and of the quasi relative interior of a convex cone. In the case of set-valued optimization problems we present a comparison between the so-called vector criterion and the set-criterion. A set-valued Lagrange dual problem is introduced and optimality conditions are formulated for it.
We present several optimality notions defined by means of the quasi interior and of the quasi relative interior of a convex cone. In the case of set-valued optimization problems we present a comparison between the so-called vector criterion and the set-criterion. A set-valued Lagrange dual problem is introduced and optimality conditions are formulated for it.
7.
JOURANI Abderrahim abderrahim.jourani@u-bourgogne.fr
Université de Bourgogne Franche-Comté, France
Titre: A venir (details)
Université de Bourgogne Franche-Comté, France
Titre: A venir (details)
Résumé:
A venir
A venir
8.
JOURANI Abderrahim abdou.jourani@free.fr
Université de Bourgogne Franche-Comté, France
Titre: Favorables classes for radiality and semismoothness (details)
Université de Bourgogne Franche-Comté, France
Titre: Favorables classes for radiality and semismoothness (details)
Résumé:
We provide sufficient conditions for radiality and semismoothness. In general Banach spaces, we show that calmness ensures Dini-radiality as well as Dini-convexity of solution set to inequality systems. In finite dimensional spaces, we introduce the concept of Clarke-radiality and semismoothness of order \(m\) and show that each subanalytic set satisfies these properties. Similar properties are obtained for locally Lipschitzian subanalytic functions.
We provide sufficient conditions for radiality and semismoothness. In general Banach spaces, we show that calmness ensures Dini-radiality as well as Dini-convexity of solution set to inequality systems. In finite dimensional spaces, we introduce the concept of Clarke-radiality and semismoothness of order \(m\) and show that each subanalytic set satisfies these properties. Similar properties are obtained for locally Lipschitzian subanalytic functions.
9.
KHANH Phan Quoc pqkhanh@hcmiu.edu.vn
International University, Vietnam National University Hochiminh City, Vietnam
Titre: Variational convergence of bifunctions on nonrectangular domains and approximations of quasivariational problems (details)
International University, Vietnam National University Hochiminh City, Vietnam
Titre: Variational convergence of bifunctions on nonrectangular domains and approximations of quasivariational problems (details)
Résumé:
\documentclass[12pt,a4paper,oneside]{article} \usepackage{amsmath} \usepackage{amscd} \usepackage{amsfonts} \topmargin=-0.6cm \textwidth=14.5cm \textheight=23cm \headheight=2.5ex \headsep=0.85cm \oddsidemargin=0.8cm \evensidemargin=-.4cm \parskip=0.7ex plus0.5ex minus 0.5ex \baselineskip=17pt plus2pt minus2pt \newlength{\defbaselineskip} \setlength{\defbaselineskip}{\baselineskip} \newcommand{\setlinespacing}[1]% {\setlength{\baselineskip}{#1 \defbaselineskip}} \date{} \begin{document} \setlinespacing{1.5} \centerline{\bf 13th French-Romanian Symposium on Applied Mathematics} \centerline{\bf Iasi, Romania, August 25-29, 2016} \vskip0.6cm \centerline{\large\bf Variational convergence of bifunctions on nonrectangular} \centerline{\large\bf domains and approximations of quasivariational problems} \vspace*{0.5cm}\centerline{\bf Phan Quoc Khanh} \vskip0.2cm \centerline{International University, Vietnam national University Hochiminh City} \vskip0.7cm \noindent{\bf Abstract} \,Variational convergence is a common terminology for kinds of convergence which preserve variational properties such as those about infima, minimizers, infsup-values, minsup-points, saddle values and points, etc. In 2009 Jofre and Wets considered variational convergence of finite-valued bifunctions defined on rectangles instead of defined on the entire product spaces with extended-real-values. Such bifunctions have been proved to be crucial in expressing many variational models in terms of finding minsup- (or maxinf-) points. However, in practice quasivariational problems, i.e., problems with constraint sets depending on their variables, are frequently met. The aim of this paper is to develop epi/hypo and lopsided convergence, the main kinds of variational convergence of bifunctions, to the case of bifunctions defined on nonrectangular domains in order to deal with quasivatiational models. Their basic characterizations are established. Variational properties are proved to be preserved for the limit bifunctions when the bifunctions epi/hypo or lopsided converge (possibly under some additional assumptions). These results are applied to approximations of some typical quasivariational problems. The obtained results are new and, in the special rectangular case, also improve some known results. \end{document}
\documentclass[12pt,a4paper,oneside]{article} \usepackage{amsmath} \usepackage{amscd} \usepackage{amsfonts} \topmargin=-0.6cm \textwidth=14.5cm \textheight=23cm \headheight=2.5ex \headsep=0.85cm \oddsidemargin=0.8cm \evensidemargin=-.4cm \parskip=0.7ex plus0.5ex minus 0.5ex \baselineskip=17pt plus2pt minus2pt \newlength{\defbaselineskip} \setlength{\defbaselineskip}{\baselineskip} \newcommand{\setlinespacing}[1]% {\setlength{\baselineskip}{#1 \defbaselineskip}} \date{} \begin{document} \setlinespacing{1.5} \centerline{\bf 13th French-Romanian Symposium on Applied Mathematics} \centerline{\bf Iasi, Romania, August 25-29, 2016} \vskip0.6cm \centerline{\large\bf Variational convergence of bifunctions on nonrectangular} \centerline{\large\bf domains and approximations of quasivariational problems} \vspace*{0.5cm}\centerline{\bf Phan Quoc Khanh} \vskip0.2cm \centerline{International University, Vietnam national University Hochiminh City} \vskip0.7cm \noindent{\bf Abstract} \,Variational convergence is a common terminology for kinds of convergence which preserve variational properties such as those about infima, minimizers, infsup-values, minsup-points, saddle values and points, etc. In 2009 Jofre and Wets considered variational convergence of finite-valued bifunctions defined on rectangles instead of defined on the entire product spaces with extended-real-values. Such bifunctions have been proved to be crucial in expressing many variational models in terms of finding minsup- (or maxinf-) points. However, in practice quasivariational problems, i.e., problems with constraint sets depending on their variables, are frequently met. The aim of this paper is to develop epi/hypo and lopsided convergence, the main kinds of variational convergence of bifunctions, to the case of bifunctions defined on nonrectangular domains in order to deal with quasivatiational models. Their basic characterizations are established. Variational properties are proved to be preserved for the limit bifunctions when the bifunctions epi/hypo or lopsided converge (possibly under some additional assumptions). These results are applied to approximations of some typical quasivariational problems. The obtained results are new and, in the special rectangular case, also improve some known results. \end{document}
10.
LáSZLó Szilárd laszlosziszi@yahoo.com
Technical University of Cluj-Napoca, Romania
Titre: Minimax results on dense sets (details)
Technical University of Cluj-Napoca, Romania
Titre: Minimax results on dense sets (details)
Résumé:
Recall that a minimax theorem deals with sufficient conditions under which the equality \(\inf\nolimits_{x\in X}\sup\nolimits_{y\in Y} f(x, y) =\sup\nolimits_{y\in Y}\inf\nolimits_{x\in X} f(x, y)\) holds, where \(X\) and \(Y\) are arbitrary sets and \(f:X\times Y\To\R\) is a given bifunction. The most general minimax results are due to Fan and Sion, and both assume the compactness of \(X.\) As a matter of fact, minimax results on dense sets, (that is \(X\) is dense in a subset of a topological vector space), are absent in the literature. In this paper we show that the general results of Fan and Sion cannot be extended on usual dense sets. Nevertheless, we obtain some new minimax results on a special type of dense set that we call self-segment-dense. We apply our results in order to obtain denseness of some family of functionals in function spaces.
Recall that a minimax theorem deals with sufficient conditions under which the equality \(\inf\nolimits_{x\in X}\sup\nolimits_{y\in Y} f(x, y) =\sup\nolimits_{y\in Y}\inf\nolimits_{x\in X} f(x, y)\) holds, where \(X\) and \(Y\) are arbitrary sets and \(f:X\times Y\To\R\) is a given bifunction. The most general minimax results are due to Fan and Sion, and both assume the compactness of \(X.\) As a matter of fact, minimax results on dense sets, (that is \(X\) is dense in a subset of a topological vector space), are absent in the literature. In this paper we show that the general results of Fan and Sion cannot be extended on usual dense sets. Nevertheless, we obtain some new minimax results on a special type of dense set that we call self-segment-dense. We apply our results in order to obtain denseness of some family of functionals in function spaces.
11.
NECOARA Ion ion.necoara@acse.pub.ro
University Politehnica Bucharest, Romania
Titre: Linear convergence of gradient type methods for non-strongly convex optimization (details)
University Politehnica Bucharest, Romania
Titre: Linear convergence of gradient type methods for non-strongly convex optimization (details)
Résumé:
In general gradient type methods are converging linearly for optimization problems with smooth and strongly convex objective function. However, these conditions are conservative, especially the strong convexity assumption, and usually do not hold in many real applications. In this talk we relax the strong convexity assumption and we provide several relaxed non-strong convexity conditions that still allow us to prove linear convergence for several gradient type methods. Moreover, we show that our relaxed non-strong convexity conditions cover many mathematical optimization models that appear in several important real-world applications.
In general gradient type methods are converging linearly for optimization problems with smooth and strongly convex objective function. However, these conditions are conservative, especially the strong convexity assumption, and usually do not hold in many real applications. In this talk we relax the strong convexity assumption and we provide several relaxed non-strong convexity conditions that still allow us to prove linear convergence for several gradient type methods. Moreover, we show that our relaxed non-strong convexity conditions cover many mathematical optimization models that appear in several important real-world applications.
12.
NICULESCU Constantin cpniculescu@gmail.com
University of Craiova, Romania
Titre: Old and new on 2d-increasing functions (details)
University of Craiova, Romania
Titre: Old and new on 2d-increasing functions (details)
Résumé:
An overview of the theory of 2d-increasing functions is presented, outlining the connection of this classical subject with many areas of interest today: convex functions and optimization, rearrangements and transport theory, copulas and comparative statistics, Abel's inequality and many more.
An overview of the theory of 2d-increasing functions is presented, outlining the connection of this classical subject with many areas of interest today: convex functions and optimization, rearrangements and transport theory, copulas and comparative statistics, Abel's inequality and many more.
13.
NICULESCU Cristian crnicul@fmi.unibuc.ro
Department of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest, Romania
Titre: Hölder Continuity of Solutions of Generalized Ky Fan Inequalities (details)
Department of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest, Romania
Titre: Hölder Continuity of Solutions of Generalized Ky Fan Inequalities (details)
Résumé:
We establish new sufficient conditions for the local uniqueness and Hölder continuity of the solutions of two generalized Ky Fan inequalities. First, we consider a Ky Fan Inequality perturbed in the objective function and in the feasible set. The objective function has real values. Next, using the nonlinear scalarization approach by virtue of the nonlinear scalarization function, commonly known as Gerstewitz function in the theory of vector optimization, we study a generalized Ky Fan inequality in which the objective function is vector valued and the inequalities are with respect to cones. Keywords: Generalized Ky Fan inequalities, Hölder continuity, Lipschitz property, nonlinear scalarization, solution mapping, uniqueness, strong convexity Mathematics Subject Clasification 49K40; 90C31; 47J20
We establish new sufficient conditions for the local uniqueness and Hölder continuity of the solutions of two generalized Ky Fan inequalities. First, we consider a Ky Fan Inequality perturbed in the objective function and in the feasible set. The objective function has real values. Next, using the nonlinear scalarization approach by virtue of the nonlinear scalarization function, commonly known as Gerstewitz function in the theory of vector optimization, we study a generalized Ky Fan inequality in which the objective function is vector valued and the inequalities are with respect to cones. Keywords: Generalized Ky Fan inequalities, Hölder continuity, Lipschitz property, nonlinear scalarization, solution mapping, uniqueness, strong convexity Mathematics Subject Clasification 49K40; 90C31; 47J20
14.
NITICA Viorel vnitica@wcupa.edu
West Chester University of Pennsylvania, USA
Titre: Topological transitivity of extensions of hyperbolic systems (details)
West Chester University of Pennsylvania, USA
Titre: Topological transitivity of extensions of hyperbolic systems (details)
Résumé:
There is great interest in proving topological transitivity of various classes of dynamical systems. The methods used are quite diverse and related to the class of dynamical systems under consideration. We aim to review the state of the art for the class of Hölder extensions of hyperbolic systems with non-compact connected Lie group fiber. In particular, we address the stability and genericity of topological transitivity in large classes of such transformations. We also discuss the generalization of these results to systems with infinite dimensional fiber.
There is great interest in proving topological transitivity of various classes of dynamical systems. The methods used are quite diverse and related to the class of dynamical systems under consideration. We aim to review the state of the art for the class of Hölder extensions of hyperbolic systems with non-compact connected Lie group fiber. In particular, we address the stability and genericity of topological transitivity in large classes of such transformations. We also discuss the generalization of these results to systems with infinite dimensional fiber.
15.
PATRICHE Monica monica.patriche@yahoo.com
Universitatea din Bucuresti, Romania
Titre: Existence of equilibrium for generalized games in choice form and applications (details)
Universitatea din Bucuresti, Romania
Titre: Existence of equilibrium for generalized games in choice form and applications (details)
Résumé:
This paper has two central aims: first, to provide simple conditions under which the generalized games in choice form and, consequently, the abstract economies, admit equilibrium; second, to study the solvability of several types of systems of vector quasi-equilibrium problems as an application. Our work outlines that there still is much to be gained from using the results concerning the existence of equilibrium of games as tools of research for other optimization problems.
This paper has two central aims: first, to provide simple conditions under which the generalized games in choice form and, consequently, the abstract economies, admit equilibrium; second, to study the solvability of several types of systems of vector quasi-equilibrium problems as an application. Our work outlines that there still is much to be gained from using the results concerning the existence of equilibrium of games as tools of research for other optimization problems.
16.
PINTEA Cornel cpintea@math.ubbcluj.ro
'Babes-Bolyai' University, Romania
Titre: Global injectivity results for some classes of operators and applications (details)
'Babes-Bolyai' University, Romania
Titre: Global injectivity results for some classes of operators and applications (details)
Résumé:
We provide estimates for the parameters of monotonicity of two composed operators and for the sum of two operators. The estimates are given in terms of the parameters of monotonicity of the involved operators and they produce examples of h-monotone operators which are not Minty-Browder monotone. As an application to the estimates for the inferior parameter of monotonicity associated to the sum of two operators we prove a global injectivity result.
We provide estimates for the parameters of monotonicity of two composed operators and for the sum of two operators. The estimates are given in terms of the parameters of monotonicity of the involved operators and they produce examples of h-monotone operators which are not Minty-Browder monotone. As an application to the estimates for the inferior parameter of monotonicity associated to the sum of two operators we prove a global injectivity result.
17.
POPOVICI Nicolae popovici@math.ubbcluj.ro
Babes-Bolyai University, Cluj-Napoca, Romania
Titre: A decomposition approach to vector optimization and related variational problems (details)
Babes-Bolyai University, Cluj-Napoca, Romania
Titre: A decomposition approach to vector optimization and related variational problems (details)
Résumé:
We study certain classes of vector optimization problems, vector variational inequalities, and vector equilibrium problems. Under appropriate assumptions, we show that every weakly efficient solution of the original problem is a strongly/properly efficient solution of at least one subproblem.
We study certain classes of vector optimization problems, vector variational inequalities, and vector equilibrium problems. Under appropriate assumptions, we show that every weakly efficient solution of the original problem is a strongly/properly efficient solution of at least one subproblem.
18.
PREDA Vasile vasilepreda0@gmail.com
University of Bucharest, Romania
Titre: On Holder Continuity of Generalized Ky Fan Inequalities Set of Solutions (details)
University of Bucharest, Romania
Titre: On Holder Continuity of Generalized Ky Fan Inequalities Set of Solutions (details)
Résumé:
We establish new sufficient conditions for the local uniqueness and H\"{o}% lder continuity of the solutions of two generalized Ky Fan inequalities. First, we consider a Ky Fan Inequality perturbed in the objective function and in the feasible set. The objective function has real values. Next, using the nonlinear scalarization approach by virtue of the nonlinear scalarization function, commonly known as Gerstewitz function in the theory of vector optimization, we study a generalized Ky Fan inequality in which the objective function is vector valued and the inequalities are with respect to cones.
We establish new sufficient conditions for the local uniqueness and H\"{o}% lder continuity of the solutions of two generalized Ky Fan inequalities. First, we consider a Ky Fan Inequality perturbed in the objective function and in the feasible set. The objective function has real values. Next, using the nonlinear scalarization approach by virtue of the nonlinear scalarization function, commonly known as Gerstewitz function in the theory of vector optimization, we study a generalized Ky Fan inequality in which the objective function is vector valued and the inequalities are with respect to cones.
19.
REVALSKI Julian revalski@math.bas.bg
Bulgarian Academy of Sciences, Bulgaria
Titre: Uniform-like properties of the norm and optimization problems in Banach spaces (details)
Bulgarian Academy of Sciences, Bulgaria
Titre: Uniform-like properties of the norm and optimization problems in Banach spaces (details)
Résumé:
It is well known that the uniform properties of the norm are very useful in the study of the existence, uniqueness and stability of the solution of several types of optimization problems in Banach spaces. In this talk we will demonstrate first, the usefulness of special geometric objects in Banach spaces -lenses- in the characterization of various types of uniform convexity of the norm in Banach spaces, as well as their use to define new properties of this type. The relations between the different kinds of uniform properties are studied as well. We show further that these properties still keep some of the above mentioned features of the classical uniform convexity related to the existence and stability of solutions to various variational problems, in particular to certain convex optimization problems and best approximation problems.
It is well known that the uniform properties of the norm are very useful in the study of the existence, uniqueness and stability of the solution of several types of optimization problems in Banach spaces. In this talk we will demonstrate first, the usefulness of special geometric objects in Banach spaces -lenses- in the characterization of various types of uniform convexity of the norm in Banach spaces, as well as their use to define new properties of this type. The relations between the different kinds of uniform properties are studied as well. We show further that these properties still keep some of the above mentioned features of the classical uniform convexity related to the existence and stability of solutions to various variational problems, in particular to certain convex optimization problems and best approximation problems.
20.
SEREA Oana Silvia oserea@yahoo.fr
Univ. Perpignan Via Domitia, France
Titre: On control problems assocaited with sweeping processes (details)
Univ. Perpignan Via Domitia, France
Titre: On control problems assocaited with sweeping processes (details)
Résumé:
We study different types of control problem associated to sweeping processes. The corresponding value function will be characterized via PDEs of Hamilton Jacobi type. Optimality conditions are also provided.
We study different types of control problem associated to sweeping processes. The corresponding value function will be characterized via PDEs of Hamilton Jacobi type. Optimality conditions are also provided.
21.
STRUGARIU Radu rstrugariu@tuiasi.ro
Gheorghe Asachi Technical University of Iasi, Romania
Titre: A new type of directional regularity for multifunctions with applications to optimization (details)
Gheorghe Asachi Technical University of Iasi, Romania
Titre: A new type of directional regularity for multifunctions with applications to optimization (details)
Résumé:
We introduce a new type of directional regularity for multifunctions, constructed by the use of a minimal time function studied by the authors in a previous work. We present necessary and sufficient conditions for directional regularity, formulated in terms of generalized differentiation objects of Fréchet type. A new directional Ekeland Variational Principle is also obtained. Finally, applications to necessary and sufficient optimality conditions for Pareto minima of sets and multifunctions are provided, making use by the regularity concepts analyzed before.
We introduce a new type of directional regularity for multifunctions, constructed by the use of a minimal time function studied by the authors in a previous work. We present necessary and sufficient conditions for directional regularity, formulated in terms of generalized differentiation objects of Fréchet type. A new directional Ekeland Variational Principle is also obtained. Finally, applications to necessary and sufficient optimality conditions for Pareto minima of sets and multifunctions are provided, making use by the regularity concepts analyzed before.
22.
THERA Michel michel.thera@unilim.fr
University of Limoges, France
Titre: An overview on the implicit (multifunction ) theorem from I. Newton to nowadays. (details)
University of Limoges, France
Titre: An overview on the implicit (multifunction ) theorem from I. Newton to nowadays. (details)
Résumé:
The implicit function theorem is an essential component of modern variational analysis and a device to solving nonlinear equations. The implicit function theorem or its variants known as the inverse function theorem or the rank theorem have been established originally in Euclidean spaces and then extended to the Banach space setting. Outside the setting of Banach spaces, for instance in Fréchet spaces, it is known that the inverse function theorem generally fails. In Fréchet spaces, inverse theorems of Nash-Moser type have been proved for map allowing smoothing operator. Recently, Ekeland produced a new result in the same framework. Many applied problems can be modeled as differential inclusions or more generally as generalized equations, that is, inclusions governed by a set-valued mapping. For these problems which are the analogous of nonlinear equations, there is a need to use implicit multifunction theorems. In this lecture we intend to give an overview of implicit function theorems from Newton to nowadays. In particular, motivated by the recent work by I. Ekeland, we will give, in the context of graded Fréchet spaces, a new implicit multifunction theorem for set-valued operator, as well as some corollaries. This work is a joint collaboration with Huynh Van Ngai.
The implicit function theorem is an essential component of modern variational analysis and a device to solving nonlinear equations. The implicit function theorem or its variants known as the inverse function theorem or the rank theorem have been established originally in Euclidean spaces and then extended to the Banach space setting. Outside the setting of Banach spaces, for instance in Fréchet spaces, it is known that the inverse function theorem generally fails. In Fréchet spaces, inverse theorems of Nash-Moser type have been proved for map allowing smoothing operator. Recently, Ekeland produced a new result in the same framework. Many applied problems can be modeled as differential inclusions or more generally as generalized equations, that is, inclusions governed by a set-valued mapping. For these problems which are the analogous of nonlinear equations, there is a need to use implicit multifunction theorems. In this lecture we intend to give an overview of implicit function theorems from Newton to nowadays. In particular, motivated by the recent work by I. Ekeland, we will give, in the context of graded Fréchet spaces, a new implicit multifunction theorem for set-valued operator, as well as some corollaries. This work is a joint collaboration with Huynh Van Ngai.
23.
VILCHES Emilio emilio.vilches@u-bourgogne.fr
University of Burgundy - University of Chile, France - Chile
Titre: On a generalized perturbed sweeping process with nonregular sets (details)
University of Burgundy - University of Chile, France - Chile
Titre: On a generalized perturbed sweeping process with nonregular sets (details)
Résumé:
The purpose of this talk is to show existence of solutions for the following differential inclusion: \begin{equation}\label{GSP} \left\{ \begin{aligned} -\dot{u}(t)&=Bv(t) & \textrm{ a.e. } t\in [T_0,T];\\ -\dot{v}(t)&\in N\left(C(t,u(t),v(t));v(t)\right)+F(t,u(t),v(t))+Au(t) & \textrm{ a.e. } t\in [T_0,T];\\ u(T_0)&=u_0, v(T_0)=v_0\in C(T_0,u_0,v_0), \end{aligned} \right. \end{equation} where \(A,B\colon H \to H\) are two bounded linear operators, \(N(S;\cdot)\) denotes the Clarke normal cone of a closed set \(S\subset H\) and \(F\colon [T_0,T]\times H \times H \rightrightarrows H\) is a set-valued mapping with nonempty closed and convex values which satisfies some appropriate conditions. The sets \(C(\cdot,\cdot,\cdot)\) are equi-uniformly subsmooths or positively \(\alpha\)-far satisfying some appropriate compactness condition. The differential inclusion (\ref{GSP}) includes the classical sweeping process, the state-dependent sweeping process, some variants of sweeping process and second order sweeping process among others. We show that under mild conditions there exists at least one solution of (\ref{GSP}) and then we explore some consequences.
The purpose of this talk is to show existence of solutions for the following differential inclusion: \begin{equation}\label{GSP} \left\{ \begin{aligned} -\dot{u}(t)&=Bv(t) & \textrm{ a.e. } t\in [T_0,T];\\ -\dot{v}(t)&\in N\left(C(t,u(t),v(t));v(t)\right)+F(t,u(t),v(t))+Au(t) & \textrm{ a.e. } t\in [T_0,T];\\ u(T_0)&=u_0, v(T_0)=v_0\in C(T_0,u_0,v_0), \end{aligned} \right. \end{equation} where \(A,B\colon H \to H\) are two bounded linear operators, \(N(S;\cdot)\) denotes the Clarke normal cone of a closed set \(S\subset H\) and \(F\colon [T_0,T]\times H \times H \rightrightarrows H\) is a set-valued mapping with nonempty closed and convex values which satisfies some appropriate conditions. The sets \(C(\cdot,\cdot,\cdot)\) are equi-uniformly subsmooths or positively \(\alpha\)-far satisfying some appropriate compactness condition. The differential inclusion (\ref{GSP}) includes the classical sweeping process, the state-dependent sweeping process, some variants of sweeping process and second order sweeping process among others. We show that under mild conditions there exists at least one solution of (\ref{GSP}) and then we explore some consequences.
24.
ZAGRODNY Dariusz d.zagrodny@uksw.edu.pl
Cardinal Stefan Wyszyński University, Poland
Titre: Rregularity and Lipschitz-like properties of subdifferential (details)
Cardinal Stefan Wyszyński University, Poland
Titre: Rregularity and Lipschitz-like properties of subdifferential (details)
Résumé:
It is known that the subdifferential of a lower semicontinuous convex function f over a Banach space X determines this function up to an additive constant in the sense that another function of the same type g whose subdifferential coincides with that of f at every point is equal to f plus a constant, i.e., g = f + c for some real constant c. Recently, Thibault and Zagrodny introduced a large class of directionally essentially smooth functions for which the subdifferential determination still holds. More generally, for extended real-valued functions in that class, they provided a detailed analysis of the enlarged inclusion ∂g(x) ⊂ ∂f(x) + γB for all x ∈ X, where γ is a nonnegative real number and B is the closed unit ball of the topological dual space. The aim of the talk is to show how results concerning such an enlarged inclusion of subdifferentials allow us to establish the C1 or C1,ω(·) property of an essentially directionally smooth function f whose subdifferential set-valued mapping admits a continuous or H¨older continuous selection. The C1,ω(·)-property is also obtained under a natural H¨older-like behaviour of the set-valued mapping ∂f. Similar results are also proved for another class of functions that we call ∂1,ϕ(·)-subregular functions. When X is a Hilbert space, the latter class contains prox-regular functions and hence our results extend old and recent results in the literature
It is known that the subdifferential of a lower semicontinuous convex function f over a Banach space X determines this function up to an additive constant in the sense that another function of the same type g whose subdifferential coincides with that of f at every point is equal to f plus a constant, i.e., g = f + c for some real constant c. Recently, Thibault and Zagrodny introduced a large class of directionally essentially smooth functions for which the subdifferential determination still holds. More generally, for extended real-valued functions in that class, they provided a detailed analysis of the enlarged inclusion ∂g(x) ⊂ ∂f(x) + γB for all x ∈ X, where γ is a nonnegative real number and B is the closed unit ball of the topological dual space. The aim of the talk is to show how results concerning such an enlarged inclusion of subdifferentials allow us to establish the C1 or C1,ω(·) property of an essentially directionally smooth function f whose subdifferential set-valued mapping admits a continuous or H¨older continuous selection. The C1,ω(·)-property is also obtained under a natural H¨older-like behaviour of the set-valued mapping ∂f. Similar results are also proved for another class of functions that we call ∂1,ϕ(·)-subregular functions. When X is a Hilbert space, the latter class contains prox-regular functions and hence our results extend old and recent results in the literature
Participants
1.
APETRII Marius mapetrii@uaic.ro
Universitatea "Al. I. Cuza" din Iasi, Romania
Universitatea "Al. I. Cuza" din Iasi, Romania
2.
EUGENIU Gârlă eugeniugarla@yahoo.com
ASEM, Republica Moldova
ASEM, Republica Moldova