Trust region subproblems arise within a class of unconstrained methods called trust region methods. The subproblems consist of minimizing a quadratic function subject to a norm constraint. This thesis is a survey of different methods developed to find an approximate solution to the subproblem. We study the well-known method of More and Sorensen and two recent methods for large sparse subproblems: the so-called Lanczos method of Gould et al. and the Rendland Wolkowicz algorithm. The common ground to explore these methods will be semidefinite programming. This approach has been used by Rendl and Wolkowicz to explain their method and the More and Sorensen algorithm; we extend this work to the Lanczos method. The last chapter of this thesis is dedicated to some improvements done to the Rendl and Wolkowicz algorithm and to comparisons between the Lanczos method and the Rendl and Wolkowicz algorithm. In particular, we show some weakness of the Lanczos method and show that the Rendl and Wolkowicz algorithm is more robust.

In the first part of the thesis we present several interior point algorithms for solving certain positive definite programming problems. One of the algorithms is adapted for finding out whether there exists or not a positive definite matrix which is a real linear combination of some given symmetric matrices A1,A2, . . . ,Am. In the second part of the thesis we discuss stability of nonlinear dynamical systems. We search using algorithms described in the first part, for Lyapunov functions of a few forms. A suitable Lyapunov function implies the existence of a hyperellipsoidal attraction region for the dynamical system, thus guaranteeing stability.

This thesis addresses the problem of developing a quantum counter-part of the well established classical theory of control. We dwell on the fundamental fact that quantum states are generally not perfectly distinguishable, and quantum measurements typically introduce noise in the system being measured. Because of these, it is generally not clear whether the central concept of the classical control theory --- that of observing the system and then applying feedback --- is always useful in the quantum setting. We center our investigations around the problem of transforming the state of a quantum system into a given target state, when the system can be prepared in different ways, and the target state depends on the choice of preparation. We call this the "quantum tracking problem" and show how it can be formulated as an optimization problem that can be approached both numerically and analytically. This problem provides a simple route to the characterization of the quantum trade-off between information gain and disturbance, and is seen to have several applications in quantum information. In order to characterize the optimality of our tracking procedures, some figure-of-merit has to be specified. Naturally, distance measures for quantum states are the ideal candidates for this purpose. We investigated several possibilities, and found that there is usually a compromise between physically motivated and mathematically tractable measures. We also introduce an alternative to the Uhlmann-Jozsa fidelity for mixed quantum states, which besides reproducing a number of properties of the standard fidelity, is especially attractive because it is simpler to compute. We employ some ideas of convex analysis to construct optimal control schemes analytically. In particular, we obtain analytic forms of optimal controllers for stabilizing and tracking any pair of states of a single-qubit. In the case of stabilization, we find that feedback control is always useful, but because of the trade-off between information gain and disturbance, somewhat different from the type of feedback performed in classical systems. In the case of tracking, we find that feedback is not always useful, meaning that depending on the choice of states one wants to achieve, it may be better not to introduce any noise by the application of quantum measurements. We also demonstrate that our optimal controllers are immediately applicable in several quantum information applications such as state-dependent cloning, purification, stabilization, and discrimination. In all of these cases, we were able to recover and extend previously known optimal strategies and performances. Finally we show how optimal single-step control schemes can be concatenated to provide multi-step strategies that usually over-perform optimal control protocols based on a single interaction between the controller and the system.

This thesis addresses the problem of developing a quantum counter-part of the well established classical theory of control. We dwell on the fundamental fact that quantum states are generally not perfectly distinguishable, and quantum measurements typically introduce noise in the system being measured. Because of these, it is generally not clear whether the central concept of the classical control theory --- that of observing the system and then applying feedback --- is always useful in the quantum setting. We center our investigations around the problem of transforming the state of a quantum system into a given target state, when the system can be prepared in different ways, and the target state depends on the choice of preparation. We call this the "quantum tracking problem" and show how it can be formulated as an optimization problem that can be approached both numerically and analytically. This problem provides a simple route to the characterization of the quantum trade-off between information gain and disturbance, and is seen to have several applications in quantum information. In order to characterize the optimality of our tracking procedures, some figure-of-merit has to be specified. Naturally, distance measures for quantum states are the ideal candidates for this purpose. We investigated several possibilities, and found that there is usually a compromise between physically motivated and mathematically tractable measures. We also introduce an alternative to the Uhlmann-Jozsa fidelity for mixed quantum states, which besides reproducing a number of properties of the standard fidelity, is especially attractive because it is simpler to compute. We employ some ideas of convex analysis to construct optimal control schemes analytically. In particular, we obtain analytic forms of optimal controllers for stabilizing and tracking any pair of states of a single-qubit. In the case of stabilization, we find that feedback control is always useful, but because of the trade-off between information gain and disturbance, somewhat different from the type of feedback performed in classical systems. In the case of tracking, we find that feedback is not always useful, meaning that depending on the choice of states one wants to achieve, it may be better not to introduce any noise by the application of quantum measurements. We also demonstrate that our optimal controllers are immediately applicable in several quantum information applications such as state-dependent cloning, purification, stabilization, and discrimination. In all of these cases, we were able to recover and extend previously known optimal strategies and performances. Finally we show how optimal single-step control schemes can be concatenated to provide multi-step strategies that usually over-perform optimal control protocols based on a single interaction between the controller and the system.

This thesis addresses the problem of developing a quantum counter-part of the well established classical theory of control. We dwell on the fundamental fact that quantum states are generally not perfectly distinguishable, and quantum measurements typically introduce noise in the system being measured. Because of these, it is generally not clear whether the central concept of the classical control theory --- that of observing the system and then applying feedback --- is always useful in the quantum setting. We center our investigations around the problem of transforming the state of a quantum system into a given target state, when the system can be prepared in different ways, and the target state depends on the choice of preparation. We call this the "quantum tracking problem" and show how it can be formulated as an optimization problem that can be approached both numerically and analytically. This problem provides a simple route to the characterization of the quantum trade-off between information gain and disturbance, and is seen to have several applications in quantum information. In order to characterize the optimality of our tracking procedures, some figure-of-merit has to be specified. Naturally, distance measures for quantum states are the ideal candidates for this purpose. We investigated several possibilities, and found that there is usually a compromise between physically motivated and mathematically tractable measures. We also introduce an alternative to the Uhlmann-Jozsa fidelity for mixed quantum states, which besides reproducing a number of properties of the standard fidelity, is especially attractive because it is simpler to compute. We employ some ideas of convex analysis to construct optimal control schemes analytically. In particular, we obtain analytic forms of optimal controllers for stabilizing and tracking any pair of states of a single-qubit. In the case of stabilization, we find that feedback control is always useful, but because of the trade-off between information gain and disturbance, somewhat different from the type of feedback performed in classical systems. In the case of tracking, we find that feedback is not always useful, meaning that depending on the choice of states one wants to achieve, it may be better not to introduce any noise by the application of quantum measurements. We also demonstrate that our optimal controllers are immediately applicable in several quantum information applications such as state-dependent cloning, purification, stabilization, and discrimination. In all of these cases, we were able to recover and extend previously known optimal strategies and performances. Finally we show how optimal single-step control schemes can be concatenated to provide multi-step strategies that usually over-perform optimal control protocols based on a single interaction between the controller and the system.

Multi-robot systems have recently witnessed a swell in interest in the past few years because of their various applications such as agricultural autonomy, medical robotics, industrial and commercial automation and, search and rescue. In this thesis, we particularly investigate the behavior of multi-robot systems with respect to stable topology control in asymmetric interaction settings. From theoretical perspective, we first classify stable topologies, and identify the conditions under which we can determine whether a topology is stable or not. Then, we design a limited fields-of-view (FOV) controller for robots that use sensors like cameras for coordination which induce asymmetric robot to robot interactions. Finally, we conduct a rigorous theoretical analysis to qualitatively determine which interactions are suitable for stable directed topology control of multi-robot systems with asymmetric interactions. In this regard, we solve an optimal topology selection problem to determine the topology with the best interactions based on a suitable metric that represents the quality of interaction. Further, we solve this optimal problem distributively and validate the distributed optimization formulation with extensive simulations.  For experimental purposes, we developed a portable multi-robot testbed which enables us to conduct multi-robot topology control experiments in both indoor and outdoor settings and validate our theoretical findings. Therefore, the contribution of this thesis is two fold: i) We provide rigorous theoretical analysis of  stable coordination of multi-robot systems with directed graphs, demonstrating the graph structures that induce stability for a broad class of coordination objectives; ii) We develop a testbed that enables validating multi-robot topology control in both indoor and outdoor settings. / Doctor of Philosophy / In this thesis, we address the problem of collaborative tasks in a multi-robot system where we investigate how interactions within members of the multi-robot system can induce instability. We conduct rigorous theoretical analysis and identify when the system will be unstable and hence classify interactions that will lead to stable multi-robot coordination. Our theoretical analysis tries to emulate realistic interactions in a multi-robot system such as limited interactions (blind spots) that exist when on-board cameras are used to detect and track other robots in the vicinity. So we study how these limited interactions induce instability in the multi-robot system. To verify our theoretical analysis experimentally,  we developed a portable multi-robot testbed that enables us to test our theory on stable coordination of multi-robot system with a team of Unmanned Aerial Vehicles (UAVs) in both indoor and outdoor settings. With this feature of the testbed we are able to investigate the difference in the multi-robot system behavior when tested in controlled indoor environments versus an uncontrolled outdoor environment. Ultimately, the motivation behind this thesis is to emulate realistic conditions for multi-robot cooperation and investigate suitable conditions for them to work in a stable and safe manner. Therefore, our contribution is twofold ; i) We provide rigorous theoretical analysis that enables stable coordination of multi-robot systems with limited interactions induced by sensor capabilities such as cameras; ii) We developed a testbed that enables testing of our theoretical contribution with a team of real robots in realistic environmental conditions.

Nowadays, electricity markets are becoming more deregulated, especially development of smart grid and introduction of renewable energy promote regulations of energy markets. On the other hand, the uncertainties of new energy sources and market participants’ bidding bring more challenges to power system operation and transmission system planning. These problems motivate us to study spot price (also called locational marginal pricing) of electricity markets, the strategic bidding of wind power producer as an independent power producer into power market, transmission expansion planning considering wind power investment, and analysis of the maximum loadability of a power grid. The work on probabilistic spot pricing for a utility grid includes renewable wind power generation in a deregulated environment, taking into account both the uncertainty of load forecasting and the randomness of wind speed. Based on the forecasted normal-distributed load and Weibull-distributed wind speed, probabilistic optimal power flow is formulated by including spinning reserve cost associated with wind power plants and emission cost in addition to conventional thermal power plant cost model. Simulations show that the integration of wind power can effectively decrease spot price, also increase the risk of overvoltage. Based on the concept of loacational marginal pricing which is determined by a marketclearing algorithm, further research is conducted on optimal offering strategies for wind power producers participating in a day-ahead market employing a stochastic market-clearing algoivrithm. The proposed procedure to drive strategic offers relies on a stochastic bilevel model: the upper level problem represents the profit maximization of the strategic wind power producer, while the lower level one represents the marketing clearing and the corresponding price formulation aiming to co-optimize both energy and reserve. Thirdly, to improve wind power integration, we propose a bilevel problem incorporating two-stage stochastic programming for transmission expansion planning to accommodate large-scale wind power investments in electricity markets. The model integrates cooptimizations of energy and reserve to deal with uncertainties of wind power production. In the upper level problem, the objective of independent system operator (ISO) modelling transmission investments under uncertain environments is to minimize the transmission and wind power investment cost, and the expected load shedding cost. The lower level problem is composed of a two stage stochastic programming problem for energy schedule and reserve dispatch simultaneously. Case studies are carried out for illustrating the effectiveness of the proposed model. The above market-clearing or power system operation is based on direct current optimal power flow (DC-OPF) model which is a linear problem without reactive power constraints. Power system maximum loadability is a crucial index to determine voltage stability. The fourth work in this thesis proposes a Lagrange semi-definite programming (SDP) method to solve the non-linear and non-convex optimization alternating current (AC) problem of the maximum loadability of security constrained power system. Simulation results from the IEEE three-bus system and IEEE 24-bus Reliability Test System (RTS) show that the proposed method is able to obtain the global optimal solution for the maximum loadability problem. Lastly, we summarize the conclusions from studies on the above mentioned optimization problems of electric power market under modern grid, as well as the influence of wind power integration on power system reliability, and transmission expansion planning, as well as the operations of electricity markets. Meanwhile, we also present some open questions on the related research, such as non-convex constraints in the lower-level problem of a bilevel problem, and integrating N-1 security criterion of transmission planning. / Graduate / lei.ming296@gmail.com

O número de contato do Rn (em inglês, kissing number) é o maior número de esferas de raio unitário e interiores dois-a-dois disjuntos que podem tocar simultaneamente uma esfera de raio unitário central. Nesta dissertação estudamos métodos que limitam o tamanho de tais configurações através de técnicas de otimização, como dualidade e programação semidefinida. O principal resultado obtido foi o cálculo de melhores limitantes para o número de contato nas dimensões 9 a 23; o que foi possível graças à exploração de simetrias dos polinômios presentes no limitante proposto por Bachoc e Vallentin (2008), levando à consideração de programas semidefinidos menores. Por fim, o limitante estudado é estendido para uma classe mais geral de problemas. / The kissing number of Rn is the maximum number of pairwise-nonoverlapping unit spheres that can simultaneously touch a central unit sphere. In this thesis we study methods to bound from above the size of such configurations using optimization techniques, like duality and semidefinite programming. The main result achieved is the computation of better bounds for the kissing number in dimensions 9 to 23; a result possible due to the exploitation of symmetries in the polynomials present in the bound proposed by Bachoc and Vallentin (2008), leading to the consideration of smaller semidefinite programs. Finally, the studied bound is extended to a bigger class of problems.

This dissertation is motivated by the pressing need for solving real-world large-scale optimization problems with the main objective of developing scalable algorithms that are capable of solving such problems efficiently. Large-scale optimization problems naturally appear in complex systems such as power networks and distributed control systems, which are the main systems of interest in this work. This dissertation aims to address four problems with regards to the theory and application of large-scale optimization problems, which are explained below: Chapter 2: In this chapter, a fast and parallelizable algorithm is developed for an arbitrary decomposable semidefinite program (SDP). Based on the alternating direction method of multipliers, we design a numerical algorithm that has a guaranteed convergence under very mild assumptions. We show that each iteration of this algorithm has a simple closed-form solution, consisting of matrix multiplications and eigenvalue decompositions performed by individual agents as well as information exchanges between neighboring agents. The cheap iterations of the proposed algorithm enable solving a wide spectrum of real-world large-scale conic optimization problems that could be reformulated as SDP. Chapter 3: Motivated by the application of sparse SDPs to power networks, the objective of this chapter is to design a fast and parallelizable algorithm for solving the SDP relaxation of a large-scale optimal power flow (OPF) problem. OPF is fundamental problem used for the operation and planning of power networks, which is non-convex and NP-hard in the worst case. The proposed algorithm would enable a real-time power network management and improve the system's reliability. In particular, this algorithm helps with the realization of Smart Grid by allowing to make optimal decisions very fast in response to the stochastic nature of renewable energy. The proposed algorithm is evaluated on IEEE benchmark systems. Chapter 4: The design of an optimal distributed controller using an efficient computational method is one of the most fundamental problems in the area of control systems, which remains as an open problem due to its NP-hardness in the worst case. In this chapter, we first study the infinite-horizon optimal distributed control (ODC) problem (for deterministic systems) and then generalize the results to a stochastic ODC problem (for stochastic systems). Our approach rests on formulating each of these problems as a rank-constrained optimization from which an SDP relaxation can be derived. We show that both problems admit sparse SDP relaxations with solutions of rank at most~3. Since a rank-1 SDP matrix can be mapped back into a globally-optimal controller, the rank-3 solution may be deployed to retrieve a near-global controller. We also propose computationally cheap SDP relaxation for each problem and then develop effective heuristic methods to recover a near-optimal controller from the low-rank SDP solution. The design of several near-optimal structured controllers with global optimality degrees above 99\% will be demonstrated. Chapter 5: The frequency control problem in power networks aims to control the global frequency of the system within a tight range by adjusting the output of generators in response to the uncertain and stochastic demand. The intermittent nature of distributed power generation in smart grid makes the traditional decentralized frequency controllers less efficient and demands distributed controllers that are able to deal with the uncertainty in the system introduced by non-dispatchable supplies (such as renewable energy), fluctuating loads, and measurement noise. Motivated by this need, we study the frequency control problem using the results developed in Chapter 4. In particular, we formulate the problem and then conduct a case study on the IEEE 39-Bus New England system. The objective is to design a near-global optimal distributed frequency controller for the New England test system by optimally adjusting the mechanical power input to each generator based on the real-time measurement received from neighboring generators through a user-defined communication topology.

In practice, the presence of uncertain parameters in optimization problems introduces new challenges in modeling and solvability to operations research. There are three main paradigms proposed for optimization problems under uncertainty. These include stochastic programming, robust optimization, and sensitivity analysis. In this thesis, we examine, improve, and combine the latter two paradigms in several relevant models and applications. In the second chapter, we study a two-stage adjustable robust linear optimization problem in which the right-hand sides are uncertain and belong to a compact, convex, and tractable uncertainty set. Under standard and simple assumptions, we reformulate the two-stage problem as a copositive optimization program, which in turns leads to a class of tractable semidefinite-based approximations that are at least as strong as the affine policy, which is a well studied tractable approximation in the literature. We examine our approach over several examples from the literature and the results demonstrate that our tractable approximations significantly improve the affine policy. In particular, our approach recovers the optimal values of a class of instances of increasing size for which the affine policy admits an arbitrary large gap. In the third chapter, we leverage the concept of robust optimization to conduct sensitivity analysis of the optimal value of linear programming (LP). In particular, we propose a framework for sensitivity analysis of LP problems, allowing for simultaneous perturbations in the objective coefficients and right-hand sides, where the perturbations are modeled in a compact, convex, and tractable uncertainty set. This framework unifies and extends multiple approaches for LP sensitivity analysis in the literature and has close ties to worst-case LP and two-stage adjustable linear programming. We define the best-case and worst-case LP optimal values over the uncertainty set. As the concept aligns well with the general spirit of robust optimization, we denote our approach as robust sensitivity analysis. While the best-case and worst-case optimal values are difficult to compute in general, we prove that they equal the optimal values of two separate, but related, copositive programs. We then develop tight, tractable conic relaxations to provide bounds on the best-case and worst case optimal values, respectively. We also develop techniques to assess the quality of the bounds, and we validate our approach computationally on several examples from—and inspired by—the literature. We find that the bounds are very strong in practice and, in particular, are at least as strong as known results for specific cases from the literature. In the fourth chapter of this thesis, we study the expected optimal value of a mixed 0-1 programming problem with uncertain objective coefficients following a joint distribution. We assume that the true distribution is not known exactly, but a set of independent samples can be observed. Using the Wasserstein metric, we construct an ambiguity set centered at the empirical distribution from the observed samples and containing all distributions that could have generated the observed samples with a high confidence. The problem of interest is to investigate the bound on the expected optimal value over the Wasserstein ambiguity set. Under standard assumptions, we reformulate the problem into a copositive programming problem, which naturally leads to a tractable semidefinite-based approximation. We compare our approach with a moment-based approach from the literature for two applications. The numerical results illustrate the effectiveness of our approach. Finally, we conclude the thesis with remarks on some interesting open questions in the field of optimization under uncertainty. In particular, we point out that some interesting topics that can be potentially studied by copositive programming techniques.