  In this work, the author presents a novel method for finding descent
directions shared by two or more differentiable functions defined on the same
unconstrained domain space. Then, the author illustrates an alternative
Multiple-Gradient Descent procedure for Multi-Objective Optimization problems
that is based on this new method. In particular, the proposed method consists
in finding the shared descent direction solving a relatively cheap Linear
Programming (LP) problem, where the LP's objective function and the constraints
are defined by the gradients of the objective functions of the Multi-Objective
Optimization problem. More precisely, the formulation of the LP problem is such
that, if a shared descent direction does not exist for the objective functions,
but a non-ascent direction for all the objectives does, the LP problem returns
the latter. Moreover, the author defines a new backtracking strategy for
Multiple-Gradient Descent methods such that, if the proposed LP is used for
computing the direction, the ability to reach and/or explore the Pareto set and
the Pareto front is improved. A theoretical analysis of the properties of the
new methods is performed, and tests on classic Multi-Objective Optimization
problems are proposed to assess their goodness.
