Roberto Lucchetti – författare

This book deals mainly with the study of convex functions and their behavior from the point of view of stability with respect to perturbations. We shall consider convex functions from the most modern point of view: a function is de?ned to be convex whenever its epigraph, the set of the points lying above the graph, is a convex set. Thus many of its properties can be seen also as properties of a certain convex set related to it. Moreover, we shall consider extended real valued functions, i. e. , functions taking possibly the values?? and +?. The reason for considering the value +? is the powerful device of including the constraint set of a constrained minimum problem into the objective function itself (by rede?ning it as +? outside the constraint set). Except for trivial cases, the minimum value must be taken at a point where the function is not +?, hence at a point in the constraint set. And the value ?? is allowed because useful operations, such as the inf-convolution, can give rise to functions valued?? even when the primitive objects are real valued. Observe that de?ning the objective function to be +? outside the closed constraint set preserves lower semicontinuity, which is the pivotal and mi- mal continuity assumption one needs when dealing with minimum problems. Variational calculus is usually based on derivatives.

This book deals mainly with the study of convex functions and their behavior from the point of view of stability with respect to perturbations. We shall consider convex functions from the most modern point of view: a function is de?ned to be convex whenever its epigraph, the set of the points lying above the graph, is a convex set. Thus many of its properties can be seen also as properties of a certain convex set related to it. Moreover, we shall consider extended real valued functions, i. e. , functions taking possibly the values?? and +?. The reason for considering the value +? is the powerful device of including the constraint set of a constrained minimum problem into the objective function itself (by rede?ning it as +? outside the constraint set). Except for trivial cases, the minimum value must be taken at a point where the function is not +?, hence at a point in the constraint set. And the value ?? is allowed because useful operations, such as the inf-convolution, can give rise to functions valued?? even when the primitive objects are real valued. Observe that de?ning the objective function to be +? outside the closed constraint set preserves lower semicontinuity, which is the pivotal and mi- mal continuity assumption one needs when dealing with minimum problems. Variational calculus is usually based on derivatives.

The increasing complexity of mathematical models, and the related need to introduce simplifying assumptions and numerical approximations, has led to the need to consider approximate solutions. When dealing with any mathematical model, some of the basic questions to be asked are whether the solution is stable to perturbations, what the approximate solutions are, and if the set of approximate solutions is close to the original solution set. The interrelationships between these aspects are also of theoretical interest. Such concepts are described in the present volume, which emphasizes the concepts of approximate solution, well-posedness and stability in optimization, calculus of variations, optimal control, and the mathematics of conflict (e.g. game theory and vector optimization). The most recent developments are covered.

This book deals mainly with the study of convex functions and their behavior from the point of view of stability with respect to perturbations. We shall consider convex functions from the most modern point of view: a function is de?ned to be convex whenever its epigraph, the set of the points lying above the graph, is a convex set. Thus many of its properties can be seen also as properties of a certain convex set related to it. Moreover, we shall consider extended real valued functions, i. e. , functions taking possibly the values?? and +?. The reason for considering the value +? is the powerful device of including the constraint set of a constrained minimum problem into the objective function itself (by rede?ning it as +? outside the constraint set). Except for trivial cases, the minimum value must be taken at a point where the function is not +?, hence at a point in the constraint set. And the value ?? is allowed because useful operations, such as the inf-convolution, can give rise to functions valued?? even when the primitive objects are real valued. Observe that de?ning the objective function to be +? outside the closed constraint set preserves lower semicontinuity, which is the pivotal and mi- mal continuity assumption one needs when dealing with minimum problems. Variational calculus is usually based on derivatives.

Contents: I. Ekeland: Some Variational Methods Arising from Mathematical Economics.- A. Mas-Colell: Four Lectures on the Differentiable Approach to General Equilibrium Theory.- J. Scheinkman: Dynamic General Equilibrium Models.- S. Zamir: Topics in Non Cooperative Game Theory.

Steps forward in mathematics often reverberate in other scientific disciplines, and give rise to innovative conceptual developments or find surprising technological applications. This volume brings to the forefront some of the proponents of the mathematics of the twentieth century, who have put at our disposal new and powerful instruments for investigating the reality around us. The portraits present people who have impressive charisma and wide-ranging cultural interests, who are passionate about defending the importance of their own research, are sensitive to beauty, and attentive to the social and political problems of their times. What we have sought to document is mathematics’ central position in the culture of our day. Space has been made not only for the great mathematicians but also for literary texts, including contributions by two apparent interlopers, Robert Musil and Raymond Queneau, for whom mathematical concepts represented a valuable tool for resolving the struggle between ‘soul and precision.’

Steps forward in mathematics often reverberate in other scientific disciplines, and give rise to innovative conceptual developments or find surprising technological applications. This volume brings to the forefront some of the proponents of the mathematics of the twentieth century, who have put at our disposal new and powerful instruments for investigating the reality around us. The portraits present people who have impressive charisma and wide-ranging cultural interests, who are passionate about defending the importance of their own research, are sensitive to beauty, and attentive to the social and political problems of their times. What we have sought to document is mathematics’ central position in the culture of our day. Space has been made not only for the great mathematicians but also for literary texts, including contributions by two apparent interlopers, Robert Musil and Raymond Queneau, for whom mathematical concepts represented a valuable tool for resolving the struggle between ‘soul and precision.’

Steps forward in mathematics often reverberate in other scientific disciplines, and give rise to innovative conceptual developments or find surprising technological applications. This volume brings to the forefront some of the proponents of the mathematics of the twentieth century, who have put at our disposal new and powerful instruments for investigating the reality around us. The portraits present people who have impressive charisma and wide-ranging cultural interests, who are passionate about defending the importance of their own research, are sensitive to beauty, and attentive to the social and political problems of their times. What we have sought to document is mathematics’ central position in the culture of our day. Space has been made not only for the great mathematicians but also for literary texts, including contributions by two apparent interlopers, Robert Musil and Raymond Queneau, for whom mathematical concepts represented a valuable tool for resolving the struggle between ‘soul and precision.’

È convinzione generale che la matematica sia una materia difficile da capire, che usa simboli esoterici e un linguaggio poco comprensibile, che sia soprattutto calcolo. Certamente, è una materia particolare, che ha bisogno di formule e che necessita di un linguaggio formale a volte molto sofisticato. Tuttavia, è anche una scienza piena di idee, che non hanno solo la funzione di progredire in una qualche teoria o di servire altre scienze per i loro modelli quantitativi. Come la filosofia, come la letteratura, la matematica è utile all’uomo per cercare di capire un po’ meglio il mondo che lo circonda, e soprattutto se stesso. Convinto profondamente di questo, l’autore propone alcuni argomenti, che sono particolarmente adatti a mettere in luce questo aspetto della matematica. L’autore utilizza, a volte, un linguaggio più matematico per completare il ragionamento, ma è del tutto convinto che il lettore interessato possa seguire tutti i suoi ragionamenti perché, parafrasando un grande matematico del secolo scorso, "chi non ha dimestichezza con le tecniche matematiche si renderà conto di potersela cavare senza problemi ignorandole del tutto" (J.F.Nash, jr).

La varietà e l’interesse dei contributi alla Lettera Matematica Pristem hanno spinto i curatori, d’accordo con il comitato di redazione, a proporre in questa raccolta alcuni articoli apparsi sulla rivista dalla sua fondazione ad oggi. atica Pristem è una rivista, edita dalla Springer-Verlag Italia, che affronta temi legati alla ricerca matematica, ai fondamenti di questa disciplina, alla sua storia e alle sue applicazioni negli ambiti più vari. La sfida è quella di interessare e divertire il lettore, riuscendo allo stesso tempo a convincerlo che la matematica, di solito ritenuta affare di pochi iniziati, lontana dagli interessi della gente comune, arido esercizio di astrusi calcoli, è invece fondamentale nella nostra vita quotidiana, ed è davvero dappertutto attorno a noi: nelle carte di credito, nella posta elettronica, in internet, nell’arte, nei giochi, nelle scelte (anche di tipo etico) che facciamo in situazioni conflittuali, e perfino in politica.

La varietà e l’interesse dei contributi alla Lettera Matematica Pristem hanno spinto i curatori, d’accordo con il comitato di redazione, a proporre in questa raccolta alcuni articoli apparsi sulla rivista dalla sua fondazione ad oggi. atica Pristem è una rivista, edita dalla Springer-Verlag Italia, che affronta temi legati alla ricerca matematica, ai fondamenti di questa disciplina, alla sua storia e alle sue applicazioni negli ambiti più vari. La sfida è quella di interessare e divertire il lettore, riuscendo allo stesso tempo a convincerlo che la matematica, di solito ritenuta affare di pochi iniziati, lontana dagli interessi della gente comune, arido esercizio di astrusi calcoli, è invece fondamentale nella nostra vita quotidiana, ed è davvero dappertutto attorno a noi: nelle carte di credito, nella posta elettronica, in internet, nell’arte, nei giochi, nelle scelte (anche di tipo etico) che facciamo in situazioni conflittuali, e perfino in politica.

This volume contains several surveys focused on the ideas of approximate solutions, well-posedness and stability of problems in scalar and vector optimization, game theory and calculus of variations. These concepts are of particular interest in many fields of mathematics. The idea of stability goes back at least to J. Hadamard who introduced it in the setting of differential equations; the concept of well-posedness for minimum problems is more recent (the mid-sixties) and originates with A.N. Tykhonov. It turns out that there are connections between the two properties in the sense that a well-posed problem which, at least in principle, is "easy to solve", has a solution set that does not vary too much under perturbation of the data of the problem, i.e. it is "stable". These themes have been studied in depth for minimum problems and now we have a general picture of the related phenomena in this case. But, of course, the same concepts can be studied in other more complicated situations as, e.g. vector optimization, game theory and variational inequalities. Let us mention that in several of these new areas there is not even a unique idea of what should be called approximate solution, and the latter is at the basis of the definition of well­ posed problem.