The branch-and-bound algorithm is used by solvers to efficiently find the optimal solution of discrete optimisation problems. It does so by sequentially partitioning parts of the search space based on the solution to the linear relaxation of the problem. This sequential decision-making is performed by the variable selection and node selection heuristics. The sequential nature of these heuristics makes them suitable for trajectory-based learning in the form of imitation learning and reinforcement learning. Learning problem-specific heuristics in this way has become increasingly popular in recent years. Despite their similarities, the two heuristics have very different dynamics during learning, and success has mainly been achieved for variable selection. In this work, we evaluate the node selection problem and formulate a learning to select paradigm for both imitation and reinforcement learning. We find that learning to select is generally more difficult due to the small margin of possible improvement over the current baselines, and the lack of informative features to distinguish nodes during ranking. These challenges are exacerbated by focusing on sibling comparisons, which are generally the most difficult due to the high similarity between the nodes. Sibling comparisons are also arguably the most important in node selection, however, due to the importance of plunging to reduce context switching overhead. The results indicate that both approaches fail to learn meaningful decision-making policies based on the limited fixed-size feature representation of the nodes. A code repository for reproducing and extending the experiments is publicly available at https://github.com/jgroenheide/rl2select.

In 2022, Golse and Paul defined a pseudometric for quantum optimal transport that extends the classical Wasserstein distance. They proved that the pseudometric satisfies the triangle inequality in certain cases. This thesis reviews their proof in the case where the middle point is a classical density. Motivated by that proof, we formulate the optimal transport problem and propose the quantum Wasserstein distance on the noncommutative 2-torus. This thesis also proves that the proposed quantum Wasserstein distance satisfies the triangle inequality in the case where the middle point is a classical density on the 2-torus.

Normalizing flows are a probabilistic method to estimate the underlying density of data samples. The method is flow based and non-parametric, with the aim of being flexible, but still computationally manageable. This report aims to explain the process of normalizing flows by the underlying principles of this probabilistic method. Both the method itself and the underlying principles are explained and reduced. An example of a normalizing flow is then implemented to showcase the method and support the underlying principles.

We will study the Clebsch-Gordan coefficients of the modular double of the quantum group Uq(sl(2, R)). This will be done by studying and taking a good look at how B. Ponsot and J. Teschner showed how to compute the Clebsch-Gordan coefficients [1]. Moreover, we will also take an introductory look at the concept of quantum groups by looking at some general theory on Hopf ∗-algebras and their representations. The Clebsch-Gordan coefficients can roughly be described as a relation between a basis of a tensor product U ⊗V of two simple Uq(sl(2, R))-modules and a basis of the decomposition of U ⊗V into simple modules. We will show that this relation can be explicitly described by an integral transformation. Since this describes a relation between modules of a quantum group, the first part of this thesis will give the necessary information to introduce the reader to the concept of quantum groups and their modules. This will be done by introducing Hopf algebras and their modules and then look at their quantum deformations. This first part will also introduce several examples of algebras, Hopf algebras and quantum groups to make the reader get used to the concept of Hopf algebras and quantum groups.

In this thesis we will for a quantum Markov semi-group (Φt)t≥0 on a finite von Neumann algebra N with a trace τ , investigate the property of the semi-group being gradient-Sp for some p ∈ [1, ∞]. This property was introduced in [12] (see also [9]) and has been studied in [9, 10, 12] for quantum Markov semi-groups on compact quantum groups and on q-Gaussian algebras. Beyond these classes the property gradient-Sp has not been studied; in particular for groups and their operator algebras no (non-trivial) examples were known before this thesis. The main aim of this thesis is therefore to construct interesting examples of quantum Markov semi-groups that possess the gradient-Sp property. The reason why we are interested in constructing such semi-groups, is because they can be used to obtain properties like the Akemann-Ostrand property (AO+) and strong solidity for the underlying von Neumann algebra. Over the last decade, these properties have become a topic of interest and have been studied for several von Neumann algebras, see [3, 8, 9, 10, 12, 23, 32, 33, 37, 41]. In this thesis we shall focus on group von Neumann algebras (L(Γ), τ ) for certain discrete groups Γ that possess the Haagerup property. Namely, for such groups there exists a proper, conditionally negative definite function ψ on Γ. We can then define an unbounded operator ∆ψ on the GNS-Hilbert space L2(L(Γ), τ ) as ∆ψ(λv) = ψ(v)λv and consider the corresponding quantum Markov semi-group (e−t∆ )t≥0. For this semi-group we can investigate for what p it has the gradient-Sp property. In particular we will be considering group von Neumann algebras of Coxeter groups. Namely, a Coxeter group W possesses the Haagerup property by [4], and a proper conditionally negative function ψ on W is given by the minimal word length ψ(w) = |w| w.r.t some set of generators. We will ‘almost completely’ characterize for what types of Coxeter systems the semi-group corresponding to the word length is gradient-Sp. Moreover, in the cases that we get the gradient-S2 property, we obtain the Akemann-Ostand property (AO+) and strong solidity for L(W). Hereafter, we will also consider other quantum Markov semi-groups on L(W). We consider word lengths that arise by putting different weights on the generators, and consider the semi-groups associated to these proper, conditionally negative functions. From this we obtain (AO+) and strong solidity for L(W) for some other cases. Thereafter, we will generalize some of our results obtained for L(W) to the Hecke algebras Nq(W), which are q-deformations of L(W). For the case of group von Neumann algebras L(Γ) for general groups, we shall examine for semi-groups induced by a proper, conditionally negative function ψ, how the gradient-Sp property of the semi-group (Φt)t≥0 := (e−t∆ )t≥0 relates to the gradient-Sq property of the semi-group that is generated by the αth-root ∆α of the generator. Last, we will also show a method that allows us, for right-angled word hyperbolic Coxeter groups, to obtain (AO+) and strong solidity for L(W) without building a gradient-Sp quantum Markov semi-group, but by using a slightly different method.