Semidefinite Programming Approach for the Quadratic Assignment Problem with a Sparse Graph

The matching problem between two adjacency matrices can be formulated as the NP-hard quadratic assignment problem (QAP). Previous work on semidefinite programming (SDP) relaxations to the QAP have produced solutions that are often tight in practice, but such SDPs typically scale badly, involving matrix variables of dimension $n^2$ where n is the number of nodes. To achieve a speed up, we propose a further relaxation of the SDP involving a number of positive semidefinite matrices of dimension $\mathcal{O}(n)$ no greater than the number of edges in one of the graphs. The relaxation can be further strengthened by considering cliques in the graph, instead of edges. The dual problem of this novel relaxation has a natural three-block structure that can be solved via a convergent Augmented Direction Method of Multipliers (ADMM) in a distributed manner, where the most expensive step per iteration is computing the eigendecomposition of matrices of dimension $\mathcal{O}(n)$. The new SDP relaxation produces strong bounds on quadratic assignment problems where one of the graphs is sparse with reduced computational complexity and running times, and can be used in the context of nuclear magnetic resonance spectroscopy (NMR) to tackle the assignment problem.Comment: 31 page

Global Optimisation for Energy System

The goal of global optimisation is to find globally optimal solutions, avoiding local optima and other stationary points. The aim of this thesis is to provide more efficient global optimisation tools for energy systems planning and operation. Due to the ongoing increasing of complexity and decentralisation of power systems, the use of advanced mathematical techniques that produce reliable solutions becomes necessary. The task of developing such methods is complicated by the fact that most energy-related problems are nonconvex due to the nonlinear Alternating Current Power Flow equations and the existence of discrete elements. In some cases, the computational challenges arising from the presence of non-convexities can be tackled by relaxing the definition of convexity and identifying classes of problems that can be solved to global optimality by polynomial time algorithms. One such property is known as invexity and is defined by every stationary point of a problem being a global optimum. This thesis investigates how the relation between the objective function and the structure of the feasible set is connected to invexity and presents necessary conditions for invexity in the general case and necessary and sufficient conditions for problems with two degrees of freedom. However, nonconvex problems often do not possess any provable convenient properties, and specialised methods are necessary for providing global optimality guarantees. A widely used technique is solving convex relaxations in order to find a bound on the optimal solution. Semidefinite Programming relaxations can provide good quality bounds, but they suffer from a lack of scalability. We tackle this issue by proposing an algorithm that combines decomposition and linearisation approaches. In addition to continuous non-convexities, many problems in Energy Systems model discrete decisions and are expressed as mixed-integer nonlinear programs (MINLPs). The formulation of a MINLP is of significant importance since it affects the quality of dual bounds. In this thesis we investigate algebraic characterisations of on/off constraints and develop a strengthened version of the Quadratic Convex relaxation of the Optimal Transmission Switching problem. All presented methods were implemented in mathematical modelling and optimisation frameworks PowerTools and Gravity

Calculating Sparse and Dense Correspondences for Near-Isometric Shapes

Comparing and analysing digital models are basic techniques of geometric shape processing. These techniques have a variety of applications, such as extracting the domain knowledge contained in the growing number of digital models to simplify shape modelling. Another example application is the analysis of real-world objects, which itself has a variety of applications, such as medical examinations, medical and agricultural research, and infrastructure maintenance. As methods to digitalize physical objects mature, any advances in the analysis of digital shapes lead to progress in the analysis of real-world objects. Global shape properties, like volume and surface area, are simple to compare but contain only very limited information. Much more information is contained in local shape differences, such as where and how a plant grew. Sadly the computation of local shape differences is hard as it requires knowledge of corresponding point pairs, i.e. points on both shapes that correspond to each other. The following article thesis (cumulative dissertation) discusses several recent publications for the computation of corresponding points: - Geodesic distances between points, i.e. distances along the surface, are fundamental for several shape processing tasks as well as several shape matching techniques. Chapter 3 introduces and analyses fast and accurate bounds on geodesic distances. - When building a shape space on a set of shapes, misaligned correspondences lead to points moving along the surfaces and finally to a larger shape space. Chapter 4 shows that this also works the other way around, that is good correspondences are obtain by optimizing them to generate a compact shape space. - Representing correspondences with a “functional map” has a variety of advantages. Chapter 5 shows that representing the correspondence map as an alignment of Green’s functions of the Laplace operator has similar advantages, but is much less dependent on the number of eigenvectors used for the computations. - Quadratic assignment problems were recently shown to reliably yield sparse correspondences. Chapter 6 compares state-of-the-art convex relaxations of graphics and vision with methods from discrete optimization on typical quadratic assignment problems emerging in shape matching

Theoretical and computational advances in finite-size facility placement and assignment problems

The goal of this research is to develop fundamental theory and exact solution methods for the optimal placement of multiple, finite-size, rectangular facilities in presence of existing rectangular facilities, in a plane. Applications of this research can be found in facility layout (re)design in manufacturing, distribution systems, services (retail outlets, hospital floors, etc.), and printed circuit board design; where designing an efficient layout can save millions of dollars in operational costs. Main difficulty of this optimization problem lies in its continuous non-convex/non-concave feasible space, which makes it tough to escape local optimality. Through this research, novel approaches will be proposed which can be used to distill this continuous space into a finite set of candidate solutions, making it amenable to search for the global optimum. The first two parts of this research deal with establishing a unified theory for the finite-size facility placement problem and establishing the theory of dominance for pruning the sub-optimal solutions. Traditionally, the facility location/layout problems are modeled as the Quadratic Assignment Problem (QAP), which is strongly NP-hard. Also, for getting strong lower bounds in the dominance procedure, we may need to solve an instance of the NP-hard Quadratic Semi-Assignment Problem (QSAP). To this end, the third part of this research deals with investigating parallel and High Performance Computing (HPC) methods for solving the Linear Assignment Problem (LAP), which is an important sub-problem of the QAP. The final part of this research deals with investigating parallel and HPC methods for obtaining strong lower bounds and possibly solving large QAPs. Since QAP is known to be a computationally intensive problem, it should be noted that large in this context means problem instances with up to 30 facilities and locations

Doctor of Philosophy

dissertationWith the tremendous growth of data produced in the recent years, it is impossible to identify patterns or test hypotheses without reducing data size. Data mining is an area of science that extracts useful information from the data by discovering patterns and structures present in the data. In this dissertation, we will largely focus on clustering which is often the first step in any exploratory data mining task, where items that are similar to each other are grouped together, making downstream data analysis robust. Different clustering techniques have different strengths, and the resulting groupings provide different perspectives on the data. Due to the unsupervised nature i.e., the lack of domain experts who can label the data, validation of results is very difficult. While there are measures that compute "goodness" scores for clustering solutions as a whole, there are few methods that validate the assignment of individual data items to their clusters. To address these challenges we focus on developing a framework that can generate, compare, combine, and evaluate different solutions to make more robust and significant statements about the data. In the first part of this dissertation, we present fast and efficient techniques to generate and combine different clustering solutions. We build on some recent ideas on efficient representations of clusters of partitions to develop a well founded metric that is spatially aware to compare clusterings. With the ability to compare clusterings, we describe a heuristic to combine different solutions to produce a single high quality clustering. We also introduce a Markov chain Monte Carlo approach to sample different clusterings from the entire landscape to provide the users with a variety of choices. In the second part of this dissertation, we build certificates for individual data items and study their influence on effective data reduction. We present a geometric approach by defining regions of influence for data items and clusters and use this to develop adaptive sampling techniques to speedup machine learning algorithms. This dissertation is therefore a systematic approach to study the landscape of clusterings in an attempt to provide a better understanding of the data

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions

Recent Advances in Robust Control

Robust control has been a topic of active research in the last three decades culminating in H_2/H_\infty and \mu design methods followed by research on parametric robustness, initially motivated by Kharitonov's theorem, the extension to non-linear time delay systems, and other more recent methods. The two volumes of Recent Advances in Robust Control give a selective overview of recent theoretical developments and present selected application examples. The volumes comprise 39 contributions covering various theoretical aspects as well as different application areas. The first volume covers selected problems in the theory of robust control and its application to robotic and electromechanical systems. The second volume is dedicated to special topics in robust control and problem specific solutions. Recent Advances in Robust Control will be a valuable reference for those interested in the recent theoretical advances and for researchers working in the broad field of robotics and mechatronics

Proceedings of the 3rd Annual Conference on Aerospace Computational Control, volume 1

Conference topics included definition of tool requirements, advanced multibody component representation descriptions, model reduction, parallel computation, real time simulation, control design and analysis software, user interface issues, testing and verification, and applications to spacecraft, robotics, and aircraft