## Invited Session Thu.1.MA 313

#### Thursday, 10:30 - 12:00 h, Room: MA 313

**Cluster 3: Complementarity & variational inequalities** [...]

### Bilevel programs and MPECs

**Chair: Jane Ye**

**Thursday, 10:30 - 10:55 h, Room: MA 313, Talk 1**

**Chao Ding**

First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints

**Coauthors: Defeng Sun, Jane Ye**

**Abstract:**

In this talk we consider a mathematical program with semidefinite cone complementarity constraints (SDCMPCC). Such a problem is a matrix analogue of the mathematical program with (vector) complementarity constraints (MPCC) and includes MPCC as a special case. We derive explicit expressions for the strong-, Mordukhovich- and Clarke- (S-, M- and C-)stationary conditions and give constraint qualifications under which a local solution of SDCMPCC is a S-, M- and C-stationary point.

**Thursday, 11:00 - 11:25 h, Room: MA 313, Talk 2**

**Stephan Dempe**

Optimality conditions for bilevel programming problems

**Coauthor: Alain B. Zemkoho**

**Abstract:**

Bilevel programming problems are hierarchical optimization problems where the feasible region is (in part) restricted to the graph of the solution set mapping of a second parametric optimization problem. To solve them and to derive optimality conditions for these problems this parametric optimization problem needs to be replaced with its (necessary) optimality conditions. This results in a (one-level) optimization problem. In the talk different approaches to transform the bilevel programming will be suggested, and relations between the original bilevel problem and the one replacing it will be investigated. Necessary optimality conditions being based

on these transformations will be formulated.

**Thursday, 11:30 - 11:55 h, Room: MA 313, Talk 3**

**Jane Ye**

On solving bilevel programs with a nonconvex lower level program

**Coauthors: Guihua Lin, Mengwei Xu**

**Abstract:**

By using the value function of the lower level program, we reformulate a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint as a single level optimization problem with a nonsmooth inequality constraint and a convex set constraint. To deal with such a nonsmooth and nonconvex optimization problem, we design a smoothing projected gradient algorithm for a general optimization problem with a nonsmooth inequality constraint and a convex set constraint. We show that, if either the sequence of the penalty parameters is bounded or the extended Mangasarian-Fromovitz constraint qualification holds at the accumulation point of the iteration points, any accumulation point

is a stationary point of the nonsmooth optimization problem. We apply the smoothing projected gradient algorithm to the bilevel program if the calmness condition holds and to an approximate bilevel program otherwise.