ROM-Based Multiobjective Optimization with PDE Constraints

Abstract

In many optimization problems arising from applications, several objectives are present. Thus, if these objectives are conflicting with each other, there is no canonical solution to these problems. This leads to the notion of multiobjective optimization, in which the goal is to compute the set of all optimal compromises (the Pareto set or Pareto front) between the conflicting objectives. Multiobjective optimization problems with PDE constraints are especially relevant in applications, since many occurring systems can be modeled by a PDE. This thesis is concerned with investigating methods for efficiently solving PDE-constrained multiobjective optimization problems with a potentially arbitrary number of objectives.Scalarization methods are a popular tool for solving general multiobjective optimization problems. Their underlying idea is to transform the problem into a series of scalar optimization problems, which can then be solved by using well-known techniques from scalar optimization. For two specific methods -- the Euclidean reference point method and the Pascoletti-Serafini method -- a hierarchical approach of computing the Pareto front based on iteratively solving subproblems is presented and rigorously analyzed. The resulting algorithms are tested by a linear-quadratic multiobjective optimal control problem and a non-convex multiobjective parameter optimization problem.However, solving these problems implies a high computational effort due to the high-dimensional equation systems arising from a discretization of the PDE. Model-order reduction techniques, which replace the high-dimensional model of the PDE with a low-dimensional reduced-order model, are a popular tool for reducing the computational effort in these cases. One issue in this context is finding a good balance between reducing the computational effort and the resulting approximation error. To this end, it is shown how a-posteriori error estimates and a Trust-Region method can be used to guarantee a desired error tolerance while still gaining a sufficient decrease of the computational effort. Numerical experiments on the two afore-mentioned examples are used to verify these approaches.