Invited Session Thu.2.MA 004

Thursday, 13:15 - 14:45 h, Room: MA 004

Cluster 20: Robust optimization [...]

Multistage robustness


Chair: Ulf Lorenz



Thursday, 13:15 - 13:40 h, Room: MA 004, Talk 1

Jan Wolf
Accelerating nested Benders decomposition with game-tree search techniques to solve quantified linear programs

Coauthor: Ulf Lorenz


Quantified linear programs (QLPs) are linear programs with variables being either existentially or universally quantified. The problem is similar to two-person zero-sum games with perfect information, like, e.g., chess, where an existential and a universal player have to play against each other. At the same time, a QLP is a variant of a linear program with a polyhedral solution space. On the one hand it has strong similarities to multi-stage stochastic linear programs with variable right-hand side. On the other hand it is a special case of a multi-stage robust optimization problem where the variables that are affected by uncertainties are assumed to be fixed. In this paper we show how the problem's ambiguity of being a two-person zero-sum game, and simultaneously being a convex multistage decision problem, can be used to combine linear programming techniques with solution techniques from game theory. Therefore, we propose an extension of the Nested Benders Decomposition algorithm with two techniques that are successfully used in game-tree search - the αβ-heuristic and move-ordering.



Thursday, 13:45 - 14:10 h, Room: MA 004, Talk 2

Kai Habermehl
Robust design of active trusses via mixed integer nonlinear semidefinite programming

Coauthor: Stefan Ulbrich


This work is an extension of Ben-Tal and Nemirovski’s approach on robust truss topology design to active trusses. Active trusses may use active components (e.g., piezo-actuators) to react on uncertain loads. The aim is to find a load-carrying structure with minimal worst-case compliance, when actuators may be used to react on uncertain loadings. This problem leads to a min-max-min formulation.
The approach is based on a semidefinite program formulation, which is a well-known optimization approach for robust truss topology design. By introducing actors into the model, it becomes a nonlinear semidefinite program with binary variables. We use a sequential semidefinite programming approach within a branch-and-bound-framework to solve these problems.
Different uncertainty sets are analyzed for the robust optimization approach - mainly polyhedral and ellipsoidal uncertainty sets. These different approaches have their specific advantages and disadvantages. A combined approach seems to be the best way to deal with active elements in robust truss topology design.
Several solution methods (e.g., Cascading techniques, projection approaches) and numerical results will be presented.



Thursday, 14:15 - 14:40 h, Room: MA 004, Talk 3

Marc Goerigk
A geometric approach to recovery robustness

Coauthors: Emilio Carrizosa, Anita Schöbel


Finding robust solutions of an optimization problem is an important issue in practice, as solutions to optimization problems may become infeasible if the exact model parameters are not known exactly. Roughly speaking, the goal in robust optimization is to find solutions which are still valid if the input data changes, thus increasing the practical applicability of optimization algorithms in real-world problems.
Various concepts on how to define robustness have been suggested. A recent model follows the idea of recovery robustness. Here, one looks for a first-stage solution which is recoverable to a feasible one for any possible scenario in the second stage. Unfortunately, finding recovery robust solutions is in many cases computationally hard.
In this talk we propose the concept of "recovery to feasibility'', a variation of recovery robustness based on geometric ideas, that is applicable for a wide range of problems. In particular, an optimal solution can be determined efficiently for linear programming problems and problems with quasiconvex constraints for different types of uncertainties. For more complex settings reduction approaches are proposed.


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