Inner Parallel Sets in Mixed-Integer Optimization

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

Bounds on the Objective Value of Feasible Roundings

For mixed-integer linear and nonlinear optimization problems we study the objective value of feasible points which are constructed by the feasible rounding approaches from Neumann et al. (Comput. Optim. Appl. 72, 309–337, 2019; J. Optim. Theory Appl. 184, 433–465, 2020). We provide a-priori bounds on the deviation of such objective values from the optimal value and apply them to explain and quantify the positive effect of finer grids of integer feasible points on the performance of the feasible rounding approaches. Computational results for large scale knapsack problems illustrate our theoretical findings

Multi-Criteria Optimal Planning for Energy Policies in CLP

In the policy making process a number of disparate and diverse issues such as economic development, environmental aspects, as well as the social acceptance of the policy, need to be considered. A single person might not have all the required expertises, and decision support systems featuring optimization components can help to assess policies. Leveraging on previous work on Strategic Environmental Assessment, we developed a fully-fledged system that is able to provide optimal plans with respect to a given objective, to perform multi-objective optimization and provide sets of Pareto optimal plans, and to visually compare them. Each plan is environmentally assessed and its footprint is evaluated. The heart of the system is an application developed in a popular Constraint Logic Programming system on the Reals sort. It has been equipped with a web service module that can be queried through standard interfaces, and an intuitive graphic user interface.Comment: Accepted at ICLP2014 Conference as Technical Communication, due to appear in Theory and Practice of Logic Programming (TPLP

On modelling effects in the battery and thermal storage scheduling problem

The growing use of intermittent renewable energy sources requires an increased amount of storage capacity to match uncertain generation with uncertain demand. A possible solution is the use of thermal and electrical storages. This paper compares several model formulations: mixed integer linear programs (MILPs), nonlinear programs (NLPs), mixed integer nonlinear programs (MINLPs) for optimizing the operation of a multi-modal home energy system comprising heating and electricity subsystems. The respective optimization problems are then resolved within a model predictive control scheme and the final solutions are compared in terms of runtime and optimality. The results indicate that a thermocline-based thermal storage model leads to the overall lowest costs while not significantly impeding computing times. Additionally, the results show that a continuous heat pump model leads to reduced computing times without affecting the modelling accuracy