Inner Parallel Sets in Mixed-Integer Optimization

Abstract

This thesis contains an extensive study of inner parallel sets in mixed-integer optimization. Inner parallel sets are a recent idea in this context and offer a possibility to relax the difficulties imposed by integrality constraints by guaranteeing feasibility of roundings of their (continuous) elements. To be able to use inner parallel sets algorithmically, various modifications, such as their enlargements and inner and outer approximations, are helpful and sometimes even necessary. Such ideas are introduced and investigated in this thesis, both theoretically as well as computationally. From our theoretical study of inner parallel sets emerge a number of feasible rounding approaches which mainly focus on the computation of good feasible points for mixed-integer linear and nonlinear minimization problems. Good feasible points are useful in the context of solving these problems by providing tight upper bounds on the objective value. In especially difficult cases, feasible rounding approaches may also be considered as an alternative to solving a problem. The contributions of this thesis include a thorough discussion of possibilities to enlarge inner parallel sets in the linear as well as in the nonlinear setting. Moreover, we introduce a novel cutting plane method based on inner parallel sets for mixed-integer convex minimization problems. This method, in addition to computing a good feasible point, also provides a lower bound on the objective value which is another important ingredient for solving such minimization problems. We study the possibility of dealing with equality constraints on integer variables which at first glance seem to prevent a nonempty inner parallel set. Under the occurrence of such constraints, we show that inner parallel sets can be nonempty in a reduced variable space, which allows the application of feasible rounding approaches. Finally, we investigate the behavior of inner parallel sets when integrated into search trees. Our study gives rise to a novel diving method which turns out to be a major improvement over standalone feasible rounding approaches. We test the introduced methods on standard libraries for mixed-integer linear, convex and nonconvex minimization problems separately in several computational studies. The computational results illustrate the potential of our ideas.