International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Resource Management for Distributed Estimation via Sparsity-Promoting Regularization

Recent advances in wireless communications and electronics have enabled the development of low-cost, low-power, multifunctional sensor nodes that are small in size and communicate untethered in a sensor network. These sensor nodes can sense, measure, and gather information from the environment and, based on some local processing, they transmit the sensed data to a fusion center that is responsible for making the global inference. Sensor networks are often tasked to perform parameter estimation; example applications include battlefield surveillance, medical monitoring, and navigation. However, under limited resources, such as limited communication bandwidth and sensor battery power, it is important to design an energy-efficient estimation architecture. The goal of this thesis is to provide a fundamental understanding and characterization of the optimal tradeoffs between estimation accuracy and resource usage in sensor networks. In the thesis, two basic issues of resource management are studied, sensor selection/scheduling and sensor collaboration for distributed estimation, where the former refers to finding the best subset of sensors to activate for data acquisition in order to minimize the estimation error subject to a constraint on the number of activations, and the latter refers to seeking the optimal inter-sensor communication topology and energy allocation scheme for distributed estimation systems. Most research on resource management so far has been based on several key assumptions, a) independence of observation, b) strict resource constraints, and c) absence of inter-sensor communication, which lend analytical tractability to the problem but are often found lacking in practice. This thesis introduces novel techniques to relax these assumptions and provide new insights into addressing resource management problems. The thesis analyzes how noise correlation affects solutions of sensor selection problems, and proposes both a convex relaxation approach and a greedy algorithm to find these solutions. Compared to the existing sensor selection approaches that are limited to the case of uncorrelated noise or weakly correlated noise, the methodology proposed in this thesis is valid for any arbitrary noise correlation regime. Moreover, this thesis shows a correspondence between active sensors and the nonzero columns of an estimator gain matrix. Based on this association, a sparsity-promoting optimization framework is established, where the desire to reduce the number of selected sensors is characterized by a sparsity-promoting penalty term in the objective function. Instead of placing a hard constraint on sensor activations, the promotion of sparsity leads to trade-offs between estimation performance and the number of selected sensors. To account for the individual power constraint of each sensor, a novel sparsity-promoting penalty function is presented to avoid scenarios in which the same sensors are successively selected. For solving the proposed optimization problem, we employ the alternating direction method of multipliers (ADMM), which allows the optimization problem to be decomposed into subproblems that can be solved analytically to obtain exact solutions. The problem of sensor collaboration arises when inter-sensor communication is incorporated in sensor networks, where sensors are allowed to update their measurements by taking a linear combination of the measurements of those they interact with prior to transmission to a fusion center. In this thesis, a sparsity-aware optimization framework is presented for the joint design of optimal sensor collaboration and selection schemes, where the cost of sensor collaboration is associated with the number of nonzero entries of a collaboration matrix, and the cost of sensor selection is characterized by the number of nonzero rows of the collaboration matrix. It is shown that a) the presence of sensor collaboration smooths out the observation noise, thereby improving the quality of the signal and eventual estimation performance, and b) there exists a trade-off between sensor selection and sensor collaboration. This thesis further addresses the problem of sensor collaboration for the estimation of time-varying parameters in dynamic networks that involve, for example, time-varying observation gains and channel gains. Impact of parameter correlation and temporal dynamics of sensor networks on estimation performance is illustrated from both theoretical and practical points of view. Last but not least, optimal energy allocation and storage control polices are designed in sensor networks with energy-harvesting nodes. We show that the resulting optimization problem can be solved as a special nonconvex problem, where the only source of nonconvexity can be isolated to a constraint that contains the difference of convex functions. This specific problem structure enables the use of a convex-concave procedure to obtain a near-optimal solution

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

Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs

Laplacian mixture models identify overlapping regions of influence in unlabeled graph and network data in a scalable and computationally efficient way, yielding useful low-dimensional representations. By combining Laplacian eigenspace and finite mixture modeling methods, they provide probabilistic or fuzzy dimensionality reductions or domain decompositions for a variety of input data types, including mixture distributions, feature vectors, and graphs or networks. Provable optimal recovery using the algorithm is analytically shown for a nontrivial class of cluster graphs. Heuristic approximations for scalable high-performance implementations are described and empirically tested. Connections to PageRank and community detection in network analysis demonstrate the wide applicability of this approach. The origins of fuzzy spectral methods, beginning with generalized heat or diffusion equations in physics, are reviewed and summarized. Comparisons to other dimensionality reduction and clustering methods for challenging unsupervised machine learning problems are also discussed.Comment: 13 figures, 35 reference