Investigation on Approximation of Pareto Set and Structure Constraints

Multiobjective optimization problems naturally arise in many real-world applications, where more than one objective is needed to optimize. Multitask learning and many engineering design problems can be treated as multiobjective optimization problems. Very often, no single solution can optimize all the objectives simultaneously. In the past two decades, many multiobjective optimization algorithms have been proposed to find a single Pareto solution or a finite set of Pareto solutions with different trade-off preferences among all objectives. However, many multiobjective optimization problems have an infinite number of Pareto optimal solutions, and a finite set cannot approximate them very well. This project will study how to learn the whole Pareto set in the decision space by a single math model. Using the learned model, decision makers can obtain their preferred Pareto solutions with specific trade-offs in real time. In recent multitask learning and some engineering applications, it is highly desirable that the Pareto optimal solutions for different preferences have common components. For example, when we build one neural network for multiple tasks with different preferences, for the sake of robustness and cost-effectiveness, we require that some weights in the neural network are the same for all the preferences, and others vary from preference to preference. We call it the structure constraint. In multiobjective modular design, such constraints can arise naturally as well. No effort has been made to study it. We will investigate this constraint and develop algorithms for finding the best trade-off solution set subject to the structure constraint.The proposed algorithms in this project can be used in a wide range of real-world applications, such as flexible multitask learning with real-time adjustment, robust multitask network building, as well as personalized manufacturing and robust modular design in industry engineering applications. The outcome of this project will bridge multitask learning and multiobjective optimization, and lead to a breakthrough in these areas. We will use the proposed algorithms to design multiobjective recommendation systems and base station antennas.