## Invited Session Wed.1.H 2035

#### Wednesday, 10:30 - 12:00 h, Room: H 2035

**Cluster 24: Variational analysis** [...]

### Nonsmooth variational inequalities: Theory and algorithms

**Chair: Russell Luke**

**Wednesday, 10:30 - 10:55 h, Room: H 2035, Talk 1**

**Russell Luke**

Constraint qualifications for nonconvex feasibility problems

**Coauthors: Heinz Bauschke, Hung Phan, Xianfu Wang**

**Abstract:**

The current convergence theory for the method of alternating projections applied to nonconvex feasibility problems does not entirely cover the convex case as one might expect. We propose a restricted normal cone and the attendant sufficient set intersection qualifications that guarantee local linear convergence of nonconvex alternating projections. This generalization recovers all of the theory for consistent convex feasibility problems and yields new convergence results for nonconvex problems involving sparsity constraints.

**Wednesday, 11:00 - 11:25 h, Room: H 2035, Talk 2**

**Shoham Sabach**

A first order method for finding minimal norm-like solutions of convex optimization problems

**Coauthor: Amir Beck**

**Abstract:**

We consider a general class of convex optimization problems in which one seeks to minimize a strongly convex function over a closed and convex set which is by itself an optimal set of another convex problem. We introduce a gradient-based method, called the minimal norm gradient method, for solving this class of problems, and establish the convergence of the sequence generated by the algorithm as well as a rate of convergence of the sequence of function values. A portfolio optimization example is given in order to illustrate our results.

**Wednesday, 11:30 - 11:55 h, Room: H 2035, Talk 3**

**Charitha Cherugondi**

A descent method for solving an equilibrium problem based on generalized D-gap function

**Abstract:**

The gap function approach for solving equilibrium problems has been investigated by many authors in the recent past. As in the case of variational inequalities, (EP) can be formulated as an unconstrained minimization problem through the D-gap function. We present a descent type algorithm for solving (EP) based on the generalized D-gap function. The convergence properties of the proposed algorithm under suitable assumptions has been discussed while supporting our approach with appropriate examples. We construct a global error bound for the equilibrium problem in terms of the generalized D-gap function. This error bound generalizes most of the existing error bounds for (EP) in the literature.