35 research outputs found
Centered solutions for uncertain linear equations
Our contribution is twofold. Firstly, for a system of uncertain linear equations where the uncertainties are column-wise and reside in general convex sets, we derive convex representations for united and tolerable solution sets. Secondly, to obtain centered solutions for uncertain linear equations, we develop a new method based on adjustable robust optimization (ARO) techniques to compute the maximum size inscribed convex body (MCB) of the set of the solutions. In general, the obtained MCB is an inner approximation of the solution set, and its center is a potential solution to the system. We use recent results from ARO to characterize for which convex bodies the obtained MCB is optimal. We compare our method both theoretically and numerically with an existing method that minimizes the worst-case violation. Applications to the input–output model, Colley’s Matrix Rankings and Article Influence Scores demonstrate the advantages of the new method
Pareto Adaptive Robust Optimality via a Fourier-Motzkin Elimination Lens
We introduce the concept of Pareto Adaptive Robust Optimality (PARO) for
linear Adaptive Robust Optimization (ARO) problems. A worst-case optimal
solution pair of here-and-now decisions and wait-and-see decisions is PARO if
it cannot be Pareto dominated by another solution, i.e., there does not exist
another such pair that performs at least as good in all scenarios in the
uncertainty set and strictly better in at least one scenario. We argue that,
unlike PARO, extant solution approaches -- including those that adopt Pareto
Robust Optimality from static robust optimization -- could fail in ARO and
yield solutions that can be Pareto dominated. The latter could lead to
inefficiencies and suboptimal performance in practice. We prove the existence
of PARO solutions, and present particular approaches for finding and
approximating such solutions. We present numerical results for a facility
location problem that demonstrate the practical value of PARO solutions. Our
analysis of PARO relies on an application of Fourier-Motzkin Elimination as a
proof technique. We demonstrate how this technique can be valuable in the
analysis of ARO problems, besides PARO. In particular, we employ it to devise
more concise and more insightful proofs of known results on (worst-case)
optimality of decision rule structures.Comment: Revised version. 38 pages, 2 figure
Biologically-based radiation therapy planning and adjustable robust optimization
Radiation therapy is one of the main treatment modalities for various different cancer types. One of the core components of personalized treatment planning is the inclusion of patient-specific biological information in the treatment planning process. Using biological response models, treatment parameters such as the treatment length and dose distribution can be tailored, and mid treatment biomarker information can be used to adapt the treatment during its course. These additional degrees of freedom in treatment planning lead to new mathematical optimization problems. This thesis studies various optimization aspects of biologically-based treatment planning, and focuses on the influence of uncertainty. Adjustable robust optimization is the main technique used to study these problems, and is also studied independently of radiation therapy applications
Minkowski Centers via Robust Optimization: Computation and Applications
Centers of convex sets are geometric objects that have received extensive attention in the mathematical and optimization literature, both from a theoretical and practical standpoint. For instance, they serve as initialization points for many algorithms such as interior-point, hit-and-run, or cutting-planes methods. First, we observe that computing a Minkowski center of a convex set can be formulated as the solution of a robust optimization problem. As such, we can derive tractable formulations for computing Minkowski centers of polyhedra and convex hulls. Computationally, we illustrate that using Minkowski centers, instead of analytic or Chebyshev centers, improves the convergence of hit-and-run and cutting-plane algorithms. We also provide efficient numerical strategies for computing centers of the projection of polyhedra and of the intersection of two ellipsoids
A Machine Learning Approach to Two-Stage Adaptive Robust Optimization
We propose an approach based on machine learning to solve two-stage linear
adaptive robust optimization (ARO) problems with binary here-and-now variables
and polyhedral uncertainty sets. We encode the optimal here-and-now decisions,
the worst-case scenarios associated with the optimal here-and-now decisions,
and the optimal wait-and-see decisions into what we denote as the strategy. We
solve multiple similar ARO instances in advance using the column and constraint
generation algorithm and extract the optimal strategies to generate a training
set. We train a machine learning model that predicts high-quality strategies
for the here-and-now decisions, the worst-case scenarios associated with the
optimal here-and-now decisions, and the wait-and-see decisions. We also
introduce an algorithm to reduce the number of different target classes the
machine learning algorithm needs to be trained on. We apply the proposed
approach to the facility location, the multi-item inventory control and the
unit commitment problems. Our approach solves ARO problems drastically faster
than the state-of-the-art algorithms with high accuracy