Computer vision concerns itself with understanding the real world through the analysis of images. Typical problems are object recognition, medical image segmentation, geometric reconstruction problems and navigation of autonomous vehicles. Such problems often lead to complicated optimization problems with a mixture of discrete and continuous variables, or even infinite dimensional variables in terms of curves and surfaces. Today, state-of-the-art in solving these problems generally relies on heuristic methods that generate only local optima of various qualities. During the last few years, work by the applicant, co-workers, and others has opened new possibilities. This research project builds on this. We will in this project focus on developing new global optimization methods for computing high-quality solutions for a broad class of problems. A guiding principle will be to relax the original, complicated problem to an approximate, simpler one to which globally optimal solutions can more easily be computed. Technically, this relaxed problem often is convex. A crucial point in this approach is to estimate the quality of the exact solution of the approximate problem compared to the (unknown) global optimum of the original problem. Preliminary results have been well received by the research community and we now wish to extend this work to more difficult and more general problem settings, resulting in thorough re-examination of algorithms used widely in different and trans-disciplinary fields. This project is to be considered as a basic research project with relevance to industry. The expected outcome is new knowledge spread to a wide community through scientific papers published at international journals and conferences as well as publicly available software.