In the literature, error bound conditions have been widely used for studying the linear convergence rates of various first-order algorithms and the majority of literature focuses on how to sufficiently ensure these error bound conditions, usually posing more assumptions on the model under discussion. In this thesis, we focus on the alternating direction method of multipliers (ADMM), and show that the known error bound conditions for studying ADMM's linear convergence, can indeed be further weakened if the error bound is studied over the specific iterative sequence generated by ADMM. A so-called partial error bound condition, which is tailored for the specific ADMM's iterative scheme and weaker than known error bound conditions in the literature, is thus proposed to derive the linear convergence of ADMM. We further show that this partial error bound condition theoretically justifies the difference if the two primal variables are updated in different orders in implementing ADMM, which had been empirically observed in the literature yet no theory is known so far.

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University of Macau / Faculty of Science and Technology / Department of Electromechanical Engineering

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Tensors can be viewed as multilinear arrays or generalizations of the notion of matrices. Tensor decompositions have applications in various fields such as psychometrics, signal processing, numerical linear algebra and data mining. When the data are nonnegative, the nonnegative tensor factorization (NTF) better reflects the underlying structure. With NTF it is possible to extract information from a given dataset and construct lower-dimensional bases that capture the main features of the set and concisely describe the original data. Nonnegative tensor factorizations are commonly computed as the solution of a nonlinear bound-constrained optimization problem. Some inherent difficulties must be taken into consideration in order to achieve good solutions. Many existing methods for computing NTF optimize over each of the factors separately; the resulting algorithms are often slow to converge and difficult to control. We propose an all-at-once approach to computing NTF. Our method optimizes over all factors simultaneously and combines two regularization strategies to ensure that the factors in the decomposition remain bounded and equilibrated in norm. We present numerical experiments that illustrate the effectiveness of our approach. We also give an example of digital-inpainting, where our algorithm is employed to construct a basis that can be used to restore digital images. / Science, Faculty of / Computer Science, Department of / Graduate

This thesis deals with Attack and Defence Models involving four different pay-off functions In each model, by means of standard min-max and convexity arguments, the optimal attack strategy, optimal defence strategy, and value are calculated. / Science, Faculty of / Mathematics, Department of / Graduate

Existence problems (proving that a set is nonempty) abound in mathematics, so we look for generally applicable solutions (such as optimization techniques). To test and improve these techniques, we apply them to the Hadamard Conjecture (proving that Hadamard matrices exist in dimensions divisible by 4), which is a good example to study since Hadamard matrices have interesting applications (communication theory, quantum information theory, experiment design, etc.), are challenging to find, are easily distinguished from other matrices, are known to exist for many dimensions, etc.. In this thesis we study optimization algorithms (Exhaustive search, Hill Climbing, Metropolis, Gradient methods, generalizations thereof, etc.), improve their performance (when using a Graphical Processing Unit), and use them to attempt to find Hadamard matrices (real, and complex). Finally, we give an algorithm to prove non-trivial lower bounds on the Hamming distance between any given matrix with elements in {+1,-1} and the set of Hadamard matrices, then we use this algorithm to study matrices with similar properties to Hadamard matrices, but which are far away (with respect to the Hamming distance) from them.

The Ottawa Police Service (OPS) deploys its resources based on the needs of predefined zones. However, the current zoning approach has been acknowledged as inefficient due to negative impacts on costs, proficiency, quality of services and time management. The zoning approach has also been acknowledged as inefficient due to its static nature, its inflexibility and its inability to adjust systematically according to the number of currently available police vehicles. It also cannot assist in addressing demand changes throughout the day in order to reduce call responses in neighbouring zones. Therefore, the demand variation could lead to a significant decrease in police efficiency, since those officers who have been allocated to other zones are not able to participate in events outside their zones without permission. It may cause a high volume of waiting calls and increased response time depending on the time of day, shifts, seasons, etc. Hence, the OPS needs to find a new model for resource deployment that can provide the same coverage but with better service quality. Resource allocation has always been a challenge for emergency services like police, fire emergency, and ambulance services since it has a direct impact on the efficiency and effectiveness of the service activities. The ambulance and fire emergency services have received research attention while the optimization of police resources remains largely ignored. While there are many similarities between ambulance and police deployment there are also significant differences that mean the direct transfer of ambulance models to police deployment is not feasible. This research addresses the lack of an effective tool for the deployment of police resources. We develop a simulation model that analyzes potential deployment plans in order to determine their effect on response times. The model has been developed in partnership with the Ottawa Police Service (OPS) and will address the obstacles, disadvantages, and geographical constraints of the existing allocation model. The OPS needs to align deployment with the service demand and their operational goals (response times, visibility, workload, compliance, etc.). Repositioning police vehicles in real time, helps in responding to future calls more effectively without adding more officers.

Prestressed girder bridges are a very common type of bridges constructed all over the world. The girder bridges are ideal as short to medium spans (15 m to 60 m) structures, due to their moderate self-weight, structural efficiency, ease of fabrication, fast construction, low initial cost, long life expectancy, low maintenance, simple deck removal, and replacement process. Thus, the vast applicability of prestressed girder bridges provides the motivation to develop optimization methodologies, techniques, and models to optimize the design of these widely-used types of bridges, in order to achieve cost-effective design solutions. Most real-world structural engineering problems involve several elements of uncertainty (e.g. uncertainty in loading conditions, in material characteristics, in analysis/simulation model accuracy, in geometric properties, in manufacturing precision, etc). Such uncertainties need to be taken into consideration in the design process in order to achieve uniform levels of safety and consistent reliability in the structural systems. Consideration of uncertainties and variation of design parameters is made through probabilistic calibration of the design codes and specifications. For all current bridge design codes (e.g. AASHTO LRFD, CHBDC, or European code) no calibration is yet made to the Serviceability Limit State or Fatigue Limit State. Eventually, to date only Strength I limit state has been formally calibrated with reliability basis. Optimum designs developed without consideration of uncertainty associated with the design parameters can lead to non-robust designs, ones for which even slight changes in design variables and uncertain parameters can result in substantial performance degradation and localized damages. The accumulated damage may result in serviceability limitations or even collapse, although the structural design meets all code requirements for ultimate flexural and shear capacity. In order to search for the best optimization solution between cost reduction and satisfactory safety levels, probabilistic approaches of design optimization were applied to control the structural uncertainties throughout the design process, which cannot be achieved by deterministic optimization. To perform probabilistic design optimization, the basic design parameters were treated as random variables. For each random variable, the statistical distribution type was properly defined and the statistical parameters were accurately derived. After characterizing the random variables, in the current research, all the limit state functions were formulated and a comprehensive reliability analysis has been conducted to evaluate the bridge’s safely level (reliability index) with respect to every design limit state. For that purpose, a computer-aided model has been developed using Visual Basic Application (VBA). The probabilities of failure and corresponding reliability indexes determined by using the newly developed model, with respect to limit state functions considered, were obtained by the First-Order Reliability Method (FORM) and/or by Monte Carlo Simulation MCS technique. For the overall structural system reliability, a comprehensive Failure Mode Analysis (FMA) has been conducted to determine the failure probability with respect to each possible mode of failure. The Improved Reliability Bounds (IRB) method was applied to obtain the upper and lower bounds of the system reliability. The proposed model also provides two methods of probabilistic design optimization. In the first method, a reliability-based design optimization of prestressed girder bridges has been formulated and developed, in which the calculated failure probabilities and corresponding reliability indexes have been treated as probabilistic constraints. The second method provides a quality-controlled optimization approach applied to the design of prestressed girder bridges where the Six Sigma quality concept has been utilized. For both methods, the proposed model conducts simulation-based optimization technique. The simulation engine performs Monte Carlo Simulation while the optimization engine performs metaheuristic scatter search with neural network accelerator. The feasibility of any bridge design is very sensitive to the bridge superstructure type. Failing to choose the most suitable bridge type will never help achieving cost-effective design alternatives. In addition to the span length, many other factors (e.g. client’s requirements, design requirements, project’s conditions, etc.) affect the selection of bridge type. The current research focusses on prestressed girder bridge type. However, in order to verify whether selecting the prestressed girder bridge type, in a specific project, is the right choice, a tool for selecting the optimum bridge type was needed. Hence, the current research provides a new model for selecting the most suitable bridge type, by integrating the experts’ decision analysis, decision tree analysis and sensitivity analysis. Experts’ opinions and decisions form essential tool in developing decision-making models. However the uncertainties associated with expert’s decisions need to be properly incorporated and statistically modelled. This was uniquely addressed in the current study.