Decision Space Decomposition for Multiobjective Programs

Being inspired by the parametric decomposition theorem for multiobjective optimization problems (MOPs) of Cuenca and Miguel (2017), and by the block- coordinate descent for single objective optimization problems, we present a decom- position theorem for computing the set of minimal elements of a partially ordered set. This set is decomposed into subsets whose minimal elements are used to retrieve the overall minimal elements. We apply this approach to strictly convex MOPs de- composing their decision space into lines. The line decomposition benefits from the fact that a multiobjective line search problem is equivalent to solving a collection of single objective line search problems. In the presence of one objective function, no modifications of the method are needed. We implement this decomposition algorithm in Python for bi-objective and single-objective programs with bounded variables. We prove the convergence of this algorithm and provide preliminary error analysis for an implementation in Rn.