Uncertainty is an inevitable feature of many decision making environments. On a regular basis, engineers, economists, investment professionals, and others need to make decisions to optimize a system with incomplete information and considerable uncertainty. Understanding, quantifying, and managing uncertainty is critical to the success and effectiveness of their decisions. Robust optimization refers to the modeling of optimization problems with data uncertainty with the objective of obtaining a solution that is guaranteed to perform well under all possible realizations of the uncertain parameters.
This project will introduce novel approaches for the characterization of robustness in uncertain systems and develop efficient and reliable optimization algorithms for identification of robust solutions. In particular, the project will focus on (i) developing methods to identify solutions that are ``relative robust'', i.e., solutions that are good in each scenario, where the quality of the solution is measured relative to the best solution of each scenario; (ii) developing new models and methods for ``adjustable robust optimization'', i.e., for problems with uncertainty where a subset of the decision variables can be adjusted after some of the uncertain variables are observed.
In many cases, robust optimization formulations are expressed as the problem of optimizing a linear function over a set defined by linear equalities and cone membership constraints, a so-called conic optimization problem. Implementation efforts will include the development of structure-exploiting algorithms for the solution of large-scale conic optimization problems. In particular, problems with fixed variables, cone intersection constraints, and quadratic objectives will be studied. The resulting algorithms will be carefully implemented and the software will be freely distributed.
Robust and conic optimization formulations are used to model a wide array of applications in many diverse fields. For example, robust optimization problems arise when some of the input parameters of a given optimization problem are unknown or uncertain, which is common in finance, economics, and engineering models. Conic optimization problems are encountered in truss design, control and system theory, statistics, eigenvalue optimization, antenna array weight design, and mathematical finance. Given their wide applicability, the progress achieved through this project in solving these problem classes more efficiently and effectively, will benefit activities in many of the disciplines listed above. In addition, the project work will be integrated into the PI's ongoing teaching, training, and outreach activities through (i) the CMU Summer Institute, (ii) the teaching of a project-based course that benefits local industries and organizations, (iii) "NEC: Give a Day, Make a Difference" program.
