Topic: An Overview of Online Multiobjective Optimization
Abstract: In online optimization, algorithms learn as they continue to optimize. In every iteration, at the time of computing a solution, the outcome of this solution is unknown because the objective function evaluating this solution has not been revealed. Since the computed solution is not optimal, it incurs loss. As the process continues, subsequent solutions are computed based on the knowledge gained in the past and the loss accumulates.
In online single-objective optimization, the overall loss is measured by the scalar regret being the difference between the total cost of the computed solutions and the cost of an optimal solution computed in hindsight. In online multiobjective optimization, the computed solutions are evaluated by a vector-valued objective function and the accumulated loss of not computing Pareto solutions is also vector-valued.
It is shown that in the multiobjective case, the scalar or vector-valued regret can be used. For the former, the multiobjective optimization problem (MOP) is scalarized into a single objective optimization problem (SOP) by means of a scalarizing parameter. Under some conditions and for a given parameter value, an optimal solution to the SOP is also Pareto for the MOP. The scalar regret can be computed for every SOP, or equivalently, individually for each Pareto solution originating from solving the SOP. In effect, this scalar regret provides point-wise information of not computing specific Pareto solutions.
To assess the regret of not computing the overall Pareto set of the offline MOP, a concept of set-based regret is introduced and approaches to its computation are proposed. In particular, it is shown that the set-based regret computed using the hypervolume and the generational distance associated with the offline Pareto set and the nondominated online outcome set achieves similar upper bounds to those obtained by the scalar regret. Numerical examples are included.
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 ClemsonUniversity 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 forInternational 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.
