Convergent descent methods for smooth multiobjective optimization

The thesis concerns the development of algorithms for smooth MultiObjective Optimization (MOO), where two or more objectives have to be simultaneously minimized. MOO plays a crucial role in several real-life applications like, for instance, portfolio selection problems, vehicle routing problems, traffic equilibrium problems. The main purpose of the work is to develop (in a rigorous way from a mathematical programming point of view) effective and efficient algorithms for smooth MOO problems based on a sound convergence theory. The first part of the thesis regards the design of algorithms for unconstrained MOO problems. Since the classical steepest descent algorithm generates a single Pareto-stationary point, a framework for the approximation of the Pareto front, using the steepest descent direction, is proposed. Convergence properties of the proposed algorithm are stated. The results of computational experiments performed on unconstrained MOO test problems have been reported. In the second part of the thesis, sparse MOO problems are considered, i.e. problems where one of the objectives is the so-called zero-norm. The proposed approach is that of approximating the zero-norm by a smooth concave function. Equivalence properties between the original and the approximated problem have been stated and an algorithm based on the steepest descent framework has been designed, implemented and tested on sparse portfolio selection problems. Finally, in the third part of the work, derivative-free MOO problems with box constraints have been considered. A method, that is a nontrivial extension of the well-known Implicit Filtering algorithm to the MOO case, is proposed. Global convergence results are stated under smooth assumptions on the objective functions. Numerical results on a set of test problems show the effectiveness of the MOO Implicit Filtering algorithm both when used to find a single Pareto solution of the problem and when used to reconstruct the whole Pareto front.