Optimization Problems on Affine Differential Manifolds

Under the red thread "Optimization Problems on Affine Differential Manifolds", our results were distributed in four Chapters: bilevel disjunctive optimization on affine differential manifolds, barrier algorithm for optimization on affine differential manifolds, evolution of Tzitzeica hypersurfaces, and dynamic of evolutive optimization problems. The Topics researched in Chapter I are structured as follows: affine convex functions, optimizations with auto-parallel restrictions, affine convexity of posynomial functions, bilevel disjunctive problem, models of bilevel disjunctive programming problems, properties of minimum functions, and multi-time perspective of a function. Chapter II presents a barrier algorithm, specific to affine differential manifolds, for solving optimization problems with nonlinear constraints. Chapter III specifies and outlines some evolutions implemented for processing the dynamics that may occur in nonlinear optimization problems. Chapter IV contains a possible dynamic of evolutive optimization problems: evolution of an optimization problem, evolution of catastrophe manifold associated to a Lagrange potential, and evolution of an optimal control problem.