Abstract

The concept of monotonicity is at the heart of this research project, and will serve us as the guiding line of this proposal. Throughout this work, the monotonicity property will manifest in different forms: - as a convexity property, through the study of convex optimization problems as well as convex Bolza problems in the calculus of variations. These problems involves convex functions either in the objective or in the constraints parts; - as a convexity property, through the study of bi- level/multilevel optimization problems; - as a hidden convexity, through the study of quadratic optimization problems and the use of Frank-Wolfe's or Dines' like-theorems which assert the convexity of the image of quadratic functions. Such a phenomenon will also be studied when we will deal with non-convex functional integrals--in this case, Lyapunov's convexity-like theorem would be the key tool to go back to the convex framework, - as a useful property of operators defined on Banach spaces; in this case, we will study many nice consequences of the monotonicity regarding the representability by a Fitzpatrick functions of a given operator, the maximality, and the validity of Minty's surjectivity theorem in reflexive and non-reflexive Banach spaces, - as a consistent tool for the study of proximal algorithms for general equations/inclusions governed by monotone operators, as well as for the study of (continuous) dynamics.