Title: Advancing Parametric Optimization for the Benefit of Multiobjective Optimization
Abstract:
Scalarization methods, which reformulate the original multiobjective optimization problem (MOP) into a single-objective optimization problem (SOP) by means of scalarizing parameters such as weights, right-hand-side values, reference points, etc., make up a common and well-established methodology for computing efficient sets to MOPs. It is expected that the optimal solutions to the SOP are efficient to the MOP. When solving the SOP, the scalarizing parameters may be known and assume specific values for which specific efficient solutions are computed. Otherwise, when these parameters are unknown, the efficient set is parametrized and represented as a set of efficient solutions in the form of functions of these parameters. Independently of the scalarizing parameters, other parameters can be included in MOPs to model uncertain data. In this case, the resulting parametric MOP (parMOP) is solved to obtain a parametrized collection of efficient sets, as opposed to a specific efficient set.
We put forward the premise that parametrization of the efficient set can naturally be combined with solving parMOPs because the algorithms performing the former can also be used to achieve the latter. Based on our premise, building on classical scalarizations, we first propose new formulations of scalarizations. We then present algorithmic developments for parametric convex quadratic SOPs, that are solved exactly, and parametric jointly convex or biconvex nonlinear SOPs that are solved approximately. Given the new theory and methodology, we apply them to MOPs and parMOPs to compute parametric efficient solutions or parametrized collections of such solutions. The performance of the algorithms is examined on synthetic instances. The algorithms are also applied to decision-making problems in portfolio optimization and engineering design, which carry both conflict and uncertainty.
Bio: Margaret M. Wiecek is Professor of Mathematical Sciences in the School of Mathematical and Statistical Sciences and has held joint professorship with the Department of Mechanical Engineering at Clemson University in Clemson, South Carolina. She has obtained a Ph.D. degree in Systems Engineering from the AGH University of Science and Technology in Krakow, Poland. Her research area includes theory, methodology, and applications of mathematical programming with special interest in multiobjective optimization and decision-making. Part of her work is interdisciplinary since she has introduced new multiobjective optimization concepts and methods into engineering optimization to enrich the field of automotive and structural design. She has served as the President of the MCDM Section of INFORMS. She serves as an area editor for Journal of Multi-Criteria Decision Analysis, an associate editor for Journal of Optimization Theory and Applications, and is a member of the editorial boards for International Journal of Multicriteria Decision Making and Decision Making in Manufacturing and Services. In recognition of her lifetime contributions, she has been awarded the MCDM Gold Medal by the International Society on MCDM.
