## Invited Session Mon.2.H 1012

#### Monday, 13:15 - 14:45 h, Room: H 1012

**Cluster 17: Nonsmooth optimization** [...]

### Constrained variational inequalities: Approximation and numerical resolution

**Chair: Juan Peypouquet**

**Monday, 13:15 - 13:40 h, Room: H 1012, Talk 1**

**Juan Peypouquet**

Lagrangian-penalization algorithm for constrained optimization and variational inequalities

**Coauthor: Pierre Frankel**

**Abstract:**

Let *X,Y* be real Hilbert spaces. Consider a bounded linear operator *A:X → Y* and a nonempty closed convex set *C ⊂ Y*. In this paper we propose an inexact proximal-type algorithm to solve constrained optimization problems

* ∈ f {f(x) : Ax ∈ C},*

where *f* is a proper lower-semicontinuous convex function on *X*; and variational inequalities

*0 ∈ M x+A*^{*}N_{C}(Ax),

where *M:X rightarrows X* is a maximal monotone operator and *N*_{C} denotes the normal cone to the set *C*. Our method combines a penalization procedure involving a bounded sequence of parameters, with the predictor corrector proximal multiplier method. Under suitable assumptions the sequences generated by our algorithm are proved to converge weakly to solutions of the aforementioned problems. As applications, we describe how the algorithm can be used to find sparse solutions of linear inequality systems and solve partial differential equations by domain decomposition.

**Monday, 13:45 - 14:10 h, Room: H 1012, Talk 2**

**Yboon Victoria Garcia Ramos**

Representable monotone operators and limits of sequences of maximal monotone operators

**Coauthor: Marc Lassonde**

**Abstract:**

We show that the lower limit of a sequence of maximal monotone operators on a reflexive Banach space is a representable monotone operator. As a consequence, we obtain that the variational sum of maximal monotone operators

and the variational composition of a maximal monotone operator with a linear continuous operator are both representable monotone operators.

**Monday, 14:15 - 14:40 h, Room: H 1012, Talk 3**

**Felipe Alvarez**

A strictly feasible Bundle method for solving convex nondifferentiable minimization problems under second-order constraints

**Coauthor: Julio Lopez**

**Abstract:**

We will describe a bundle proximal method with variable metric for solving nonsmooth convex optimization problems under positivity and second-order cone constraints. The proposed algorithm relies on a local variable metric which is induced by the Hessian of the log barrier. An appropriate choice of a regularization parameter ensures the well-definedness of the algorithm and forces the iterates to belong to the relative interior of the feasible set. Also, under suitable but fairly general assumptions, we will show that the limit points of the sequence generated by the algorithm are optimal solutions. Finally, we will report some computational results on several test nonsmooth problems.