Invited Session Tue.1.MA 004

Tuesday, 10:30 - 12:00 h, Room: MA 004

Cluster 20: Robust optimization [...]

Dynamic optimization and its applications


Chair: Vineet Goyal



Tuesday, 10:30 - 10:55 h, Room: MA 004, Talk 1

Dan Andrei Iancu
Supermodularity and dynamic robust optimization

Coauthors: Mayank Sharma, Maxim Sviridenko


We consider two classical paradigms for solving Dynamic Robust Optimization (DRO) problems: (1) Dynamic Programming (DP), and (2) policies parameterized in model uncertainties (i.e., decision rules), obtained by solving tractable convex optimization problems.
We provide a set of unifying conditions (based on the interplay between the convexity and supermodularity of the DP value functions, and the lattice structure of the uncertainty sets) that guarantee the optimality of the class of affine decision rules. We also derive conditions under which such rules can be recovered by optimizing simple (e.g., affine) functions over the uncertainty sets. Our results suggest new modeling paradigms for robust optimization, and our proofs, bringing together ideas from three areas of optimization typically studied separately (robust, combinatorial - lattice programming and supermodularity, and global - the theory of concave envelopes), may be of independent interest.
We exemplify our results in an application concerning the design of flexible contracts in a two-echelon supply chain, where all optimal contractual pre-commitments and optimal ordering policies can be found by solving a small LP.



Tuesday, 11:00 - 11:25 h, Room: MA 004, Talk 2

Omid Nohadani
Robust evolution-based optimization in radiation therapy


The treatment of solid cancer tumors with external radiation is typically planned based on information that is collected during the initial examination. The overall goal of the treatment is to eliminate the tumor, hence a certain evolution is anticipated. However, current optimized radiation delivery strategies do not vary over the course of the treatment, which typically spans over four to six weeks. We present novel methods that address this issue by taking the changes in the tumor into account and exploiting its evolution to both enhance the recovery and increase the success of the therapy. We demonstrate the performance of the method based on clinical cases, where a) the geometric shape of the tumor varies or b) the cell sensitivity to radiation and its effect changes over time. Moreover, the presented treatment plans are intrinsically robust to deviations from the assumes evolution path.



Tuesday, 11:30 - 11:55 h, Room: MA 004, Talk 3

Vineet Goyal
Static vs. dynamic robust optimization

Coauthor: Dimitris Bertsimas


Most real world problems require optimization models that handle uncertain parameters. In a dynamic robust optimization framework, uncertainty is modeled as a set and we optimize over the worst-case realization of the uncertain parameters. We can compute an optimal fully-adjustable solution via a classical DP approach but this is often intractable. Another solution paradigm is to construct a static solution that is feasible for all future uncertainty realizations. This is a tractable approach but is often perceived to be highly conservative. We compare the performance of static solutions with optimal fully adjustable solutions and show that it is a good approximation for the dynamic robust optimization problem under fairly general conditions. In particular, we consider problems with linear constraints and linear objective under uncertainty and relate the performance of static solutions with the properties of uncertainty set. Our analysis also provides important insights about constructing good uncertainty sets in dynamic robust optimization problems.


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