# Mix algorithms for structured convex non-differentiable optimization

Friday, June 30th, 2017, 12:00h.,
Central Meeting Room at DeustoTech

#### Martin Lazar

University of Dubrovnik, Dubrovnik, Croatia

#### Cesare Molinari

Universidad Técnica Federico Santa María, Valparaíso, Chile.

Abstract:
In this talk, we propose a new iterative algorithm for solving structured variational inclusions in the product of Hilbert spaces, and where the corresponding set-valued map is asymmetrical in terms of regularity. This setting covers the convex constrained optimization problem
$$\min_{\left(x,y\right) \in X \times Y} \left\{ f(x) + g(y): \ Ax+By \in C \right\},$$
where $f$ is smooth, $g$ is closed, $A$ and $B$ are bounded linear operators and $C$ is non-empty, closed and convex. This multiplier method is inspired by [1,2] and combines Lagrangian techniques and a penalization scheme with bounded parameters, with parallel forward-backward iterations. Conveniently combined, these techniques allow us to take advantage of the particular structure of the problem. We prove the weak convergence of the sequence generated by this scheme, allowing errors in the computation of the implicit steps. We also analyze a case in which the convergence is (strong and) linear. Several applications are discussed and some computational experiments are reported.

###### References

[1] G. Chen, M. Teboulle A proximal-based decomposition method for convex minimization problems Mathematical Programming 64, 81-101 (1994).
[2] P. Frankel, J. Peypouquet Lagrangian-penalization algorithm for constrained optimization and variational inequalities Set-Valued and Variational Analysis 20, no. 2, 169-185 (2012).
[3] J. Peypouquet Convex Optimization in Normed Spaces; Theory, Methods and Examples SpringerBriefs in Optimization (2015).
[4] H. Raguet, J. Fadili, G. Peyré Generalized Forward-Backward Splitting SIAM Journal on Imaging Sciences, Vol. 6(3), pp. 1199-1226, (2013).