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Abstract of keynote speakers

Stephen Dempe: Transformation of bilevel optimization problems into single level ones

Bilevel optimization problems are hierarchical problems where the second (or lower level) problem is a parametric optimization problem. For solving it we need to transform it into a single level problem. This can be realized using various approaches: we can replace the lower level problem using its Karush-Kuhn-Tucker optimality conditions, apply Lagrange, Wolfe as well as Mond-Weir duality or formulate a new constraint involving its optimal value function. Topic of the presentation are relations between the bilevel optimization problem and its single level transformations, and also properties of the latter problems.

Ivana Ljubic: Bilevel Optimization Under Uncertainty

Significant algorithmic advances in the field of computational bilevel optimization allow us to solve much larger and also more complicated problems today compared to what was possible two decades ago. This results in more and more challenging bilevel problems that researchers try to solve today. In this talk, we will focus on one of these more challenging classes of bilevel problems: bilevel optimization under uncertainty. We will discuss classical ways of addressing uncertainties in bilevel optimization using stochastic or robust techniques. Moreover, the sources of uncertainty in bilevel optimization can be much richer than for usual, single-level problems, since not only the problem's data can be uncertain but also the (observation of the) decisions of the two players can be subject to uncertainty. Thus, we will also discuss bilevel optimization under limited observability, the area of problems considering only near-optimal decisions, and intermediate solution concepts between the optimistic and pessimistic cases.

The talk is based on joint work with Yasmine Beck and Martin Schmidt.

Martin Schmidt: Matchmaking Bilevel and Γ-Robust Optimization

Robust optimization is a prominent approach in optimization to deal with uncertainties in the data of the optimization problem by hedging against the worst-case realization of the uncertain event. Doing this
usually leads to a multilevel structure of the mathematical formulation that is very similar to what we are used to consider in bilevel optimization. Hence, these two fields are closely related but the study of their combination is still in its infancy. In this talk, we show how branch-and-cut methods can be derived for solving discrete bilevel problems in which the follower tackles uncertainties in a Γ-robust way. Moreover, we discuss structural
results showing that the Γ-robust bilevel problem can be solved by solving a polynomial set of nominal, i.e., certain, bilevel problems. By doing so, we generalize the famous result by Bertsimas and Sim for combinatorial optimization to combinatorial bilevel optimization.

The talk is based on joint work with Yasmine Beck and Ivana Ljubić.

Jane J. Ye: Recent developments in solving bilevel programming problems

A bilevel programming problem is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. It can be used to model a two-level hierarchical system where the two decision makers have different objectives and make their decisions on different levels of hierarchy. Recently, more and more applications including those in machine learning have been modelled as bilevel optimization problems. In this talk, I will report some recent developments in optimality conditions and numerical algorithms for solving this class of very difficult optimization problems.

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