Optimization methods have become an established paradigm to address most Computer Vision challenges including thereconstruction of three-dimensional objects from multiple images, or the tracking of a deformable shape over time. Yet, it hasbeen largely overlooked that optimization approaches are practically useless if they do not come with efficient algorithms tocompute minimizers of respective energies. Most existing formulations give rise to non-convex energies. As a consequence,solutions highly depend on the choice of minimization scheme and implementational (initialization, time step sizes, etc.), withlittle or no guarantees regarding the quality of computed solutions and their robustness to perturbations of the input data.In the proposed research project, we plan to develop optimization methods for Computer Vision which allow to efficientlycompute globally optimal solutions. Preliminary results indicate that this will drastically leverage the power of optimizationmethods and their applicability in a substantially broader context. Specifically we will focus on three lines of research: 1) Wewill develop convex formulations for a variety of challenges. While convex formulations are currently being developed forlow-level problems such as image segmentation, our main effort will focus on carrying convex optimization to higher levelproblems of image understanding and scene interpretation. 2) We will investigate alternative strategies of global optimizationby means of discrete graph theoretic methods. We will characterize advantages and drawbacks of continuous and discretemethods and thereby develop novel algorithms combining the advantages of both approaches. 3) We will go beyond convexformulations, developing relaxation schemes that compute near-optimal solutions for problems that cannot be expressed byconvex functionals.