Instance-specific linear relaxations of semidefinite optimization problems

We present a simple yet flexible systematic way to significantly improve linear relaxations of semidefinite optimization programs (SDPs) using instance-specific information. We get inspiration for our approach by studying the celebrated SDP relaxation for max cut (GW) due to Poljak and Rendl [1995] and analyzed by Goemans and Williamson [1995]. Using the instance at hand, we provide a pair of closely related compact linear programs that sandwich the optimal value of the GW semidefinite program. The instance specific information allows us to trivially avoid hardness of approximation results of semidefinite programs using linear ones, such as Braun et al. [2015] and Kothari et al. [2021]. We give sufficient conditions that guarantee that the optimal value of both our programs match the optimal value of the semidefinite relaxation. We use these conditions to prove that our two bounds give the same value as the GW SDP for max cut on distance-regular graphs, and in particular, strongly regular graphs such as cycles and the Petersen graph. Further, we show that the approximation of the upper bounding problem is strictly better than the eigenvalue bounds of Mohar and Poljak [1990], Alon and Sudakov [2000] for max cut. To the best of our knowledge, no previously proposed linear program proposed for max cut has this guarantee. We extensively test our methodology on synthetic and real graphs, and show how our ideas perform in practice. Even though our methodology is inspired by the SDP relaxation for max cut, we show experimentally that our ideas can be applied successfully to obtain good solutions to other computationally hard SDPs such as sparse PCA and the Lovasz Theta number

Quantum machine learning: a classical perspective

Recently, increased computational power and data availability, as well as algorithmic advances, have led machine learning techniques to impressive results in regression, classification, data-generation and reinforcement learning tasks. Despite these successes, the proximity to the physical limits of chip fabrication alongside the increasing size of datasets are motivating a growing number of researchers to explore the possibility of harnessing the power of quantum computation to speed-up classical machine learning algorithms. Here we review the literature in quantum machine learning and discuss perspectives for a mixed readership of classical machine learning and quantum computation experts. Particular emphasis will be placed on clarifying the limitations of quantum algorithms, how they compare with their best classical counterparts and why quantum resources are expected to provide advantages for learning problems. Learning in the presence of noise and certain computationally hard problems in machine learning are identified as promising directions for the field. Practical questions, like how to upload classical data into quantum form, will also be addressed.Comment: v3 33 pages; typos corrected and references adde

Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes