From local to global and back : a closed walk in mathematical programming and its applications
An overview of my research contributions in optimization is provided, passing through mixed-integer nonlinear optimization, nonlinear continuous local optimization and network (graph) clustering. The first chapter deals with mixed-integer nonlinear programming and deterministic global optimization, and presents contributions concerning theoretical investigations as well as applications to real-world problems. We mainly discuss about convex relaxations and automatic reformulations of mathematical programming problems, aiming at enhancing the efficiency of Branch-and-Bound algorithms. Focusing on polynomial programming, we investigate tight convex relaxations for multilinear monomials and the generation of compact relaxations of polynomial problems based on a special reformulation-linearization technique. Among applications, a special attention is devoted to real-life problems arising in Air Traffic Management. We propose new mathematical models and solution approaches from mixed-integer optimization on the one hand, and optimal control on the other hand. A few topics in nonlinear continuous optimization are described in the second chapter. Interior point methods for quadratic programming and their linear algebra kernels (KKT systems) are first discussed. The focus is on iterative methods for the KKT systems and related issues, such as preconditioning techniques and convergence properties. The other discussed topic relates, again, to air traffic problems. This concerns the mentioned optimal control-based approaches that lead to nonlinear problems. The third chapter presents my main results in the area of network clustering. The problem of identifying clusters in networks can be formulated using mathematical programming and usually leads to a combinatorial optimization problem. My contributions concern clustering criteria and corresponding clustering methods. A special attention is devoted to exact methods, used either to solve the whole optimization problem or, locally, subproblems arising in hierarchical heuristics, or to refine solutions previously obtained by other methods.