Adaptive discretization methods for the global solution of semi-infinite optimization problems

Semi-infinite optimization problems occur naturally in different applications, particularly in the context of optimization under uncertainty. Semi-infinite optimization problems are optimization problems containing a finite number of degrees of freedom but an infinite number of constraints. This thesis considers adaptive discretization-based methods for their solution, both extending and applying existing methods. First, the adaptive discretization-based approach is applied to a problem maximizing flexibility in power transmission grids. Specifically, adjustable grid parameters are optimized in order to maximize the range of permissible deviations from predicted loads where safe grid operation can be maintained through appropriate control actions. As the model also considers control actions, an existing algorithm for solving existence-constrained semi-infinite problems is extended, specialized and applied. Based on the experience gained in this practically motivated problem, areas of improvements are identified and some of them are addressed in this thesis. To improve the accessibility and applicability of the discussed discretization-based methods, a software framework is presented that enables users to model hierarchical optimization problems and apply existing adaptive discretization-based algorithms. The implemented solvers, as well as different choice of important hyperparameters, are compared on a large set of test problems, compiled from the literature. In addition to easy access to the already implemented algorithms, the open-source framework facilitates the development of new or specialized adaptive discretization-based algorithms. Consequently, generalized semi-infinite problems with coupling equality constraints are considered. Previous work had already proposed solutions based on discretization under a uniqueness assumption. A proposal is developed to relax this assumption in favor of assuming local stability of the solutions to the coupling equality constraints. The abstract proposal is concretized into variants for the cases where i) a Lipschitz constant regarding the solution set of the equality constraints is known, ii) full rank of the Jacobian of the coupling equality is assumed, or iii) the coupling equality constraints are linear. These variants are implemented and successfully applied in numerical case studies. Finally, the utility of using derivative information of the semi-infinite constraint is investigated. To this end, an approach is first developed to integrate derivation information into the lower-bounding subproblem. This subproblem is then used in a modified discretization algorithm and compared with the traditional method, which does not use derivative information, as well as with existing approaches that take derivative information into account.