6. Conclusion

The main aim of this synthesis has been to show that, despite its simple formalism, Problem 1 models a wide variety of concrete problems, and that it can be solved by an algorithm alternating a gradient step on its smooth function and a proximal step on its non-differentiable function: the proximate gradient method. The asymptotic properties of this algorithm were studied and several of its applications were described. Finally, we have seen that, through dual reformulations or in product spaces, the scope of the proximate gradient method can be extended to optimization problems beyond the initial scope of Problem 1.