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In this thesis, we consider the solution of high-dimensional optimization problems with an underlying low-rank tensor structure. Due to the exponentially increasing computational complexity in the number of dimensionsâ€”the so-called curse of dimensionalityâ€”they present a considerable computational challenge and become infeasible even for moderate problem sizes.
Multilinear algebra and tensor numerical methods have a wide range of applications in the fields of data science and scientific computing. Due to the typically large problem sizes in practical settings, efficient methods, which exploit low-rank structures, are essential. In this thesis, we consider an application each in both of these fields.
Tensor completion, or imputation of unknown values in partially known multiway data is an important problem, which appears in statistics, mathematical imaging science and data science. Under the assumption of redundancy in the underlying data, this is a well-defined problem and methods of mathematical optimization can be applied to it.
Due to the fact that tensors of fixed rank form a Riemannian submanifold of the ambient high-dimensional tensor space, Riemannian optimization is a natural framework for these problems, which is both mathematically rigorous and computationally efficient.
We present a novel Riemannian trust-region scheme, which compares favourably with the state of the art on selected application cases and outperforms known methods on some test problems.
Optimization problems governed by partial differential equations form an area of scientific computing which has applications in a variety of areas, ranging from physics to financial mathematics. Due to the inherent high dimensionality of optimization problems arising from discretized differential equations, these problems present computational challenges, especially in the case of three or more dimensions. An even more challenging class of optimization problems has operators of integral instead of differential type in the constraint. These operators are nonlocal, and therefore lead to large, dense discrete systems of equations. We present a novel solution method, based on separation of spatial dimensions and provably low-rank approximation of the nonlocal operator. Our approach allows the solution of multidimensional problems with a complexity which is only slightly larger than linear in the univariate grid size; this improves the state of the art for a particular test problem problem by at least two orders of magnitude.

Many combinatorial optimization problems on finite graphs can be formulated as conic convex programs, e.g. the stable set problem, the maximum clique problem or the maximum cut problem. Especially NP-hard problems can be written as copositive programs. In this case the complexity is moved entirely into the copositivity constraint.
Copositive programming is a quite new topic in optimization. It deals with optimization over the so-called copositive cone, a superset of the positive semidefinite cone, where the quadratic form x^T Ax has to be nonnegative for only the nonnegative vectors x. Its dual cone is the cone of completely positive matrices, which includes all matrices that can be decomposed as a sum of nonnegative symmetric vector-vector-products.
The related optimization problems are linear programs with matrix variables and cone constraints.
However, some optimization problems can be formulated as combinatorial problems on infinite graphs. For example, the kissing number problem can be formulated as a stable set problem on a circle.
In this thesis we will discuss how the theory of copositive optimization can be lifted up to infinite dimension. For some special cases we will give applications in combinatorial optimization.