Computation of generalized solution spaces

Solution spaces are applied in distributed design processes. They enable an independent and robust development of the components of a target design. A solution space is a region which contains only good designs and lies in a potentially high-dimensional design space. By finding an appropriate solution space, the design processes for individual components can be decoupled from each other. This increases the efficiency of the overall design process and saves valuable resources.
An established method to find solution spaces is the box optimization algorithm. It provides solution spaces which are products of intervals and take on the shape of a high-dimensional, axis-parallel box. We review this method and give a detailed account of how different parameter settings affect the outcome of the algorithm.
The box optimization algorithm yields sometimes intervals that are too small. To this end, we develop the rotated box optimization algorithm. It couples specific pairs of components and rotates the corresponding box. Thus, it is able to find boxes with a larger volume and increases the amount of available good designs.
An algorithm which might yield even larger solution spaces is the polytope optimization algorithm. Instead of trying to find boxes which are as large as possible, it maximizes the volume of polytopes. Because polytopes have a much more flexible shape than boxes, this gives rise to larger solution spaces compared to the previous algorithms. However, the algorithm is more complex and requires additional steps to handle the polytopes.
We compare these algorithms by applying them to several high-dimensional optimization problems. Our results show that, indeed, the polytope optimization algorithm yields the solution spaces with the largest volume.