Semi-Global Exponential Stability of Augmented Primal-Dual Gradient Dynamics for Constrained Convex Optimization

Primal-dual gradient dynamics that find saddle points of a Lagrangian have been widely employed for handling constrained optimization problems. Building on existing methods, we extend the augmented primal-dual gradient dynamics (Aug-PDGD) to incorporate general convex and nonlinear inequality constraints, and we establish its semi-global exponential stability when the objective function is strongly convex. We also provide an example of a strongly convex quadratic program of which the Aug-PDGD fails to achieve global exponential stability. Numerical simulation also suggests that the exponential convergence rate could depend on the initial distance to the KKT point

On the Performance Bound of Sparse Estimation with Sensing Matrix Perturbation

This paper focusses on the sparse estimation in the situation where both the the sensing matrix and the measurement vector are corrupted by additive Gaussian noises. The performance bound of sparse estimation is analyzed and discussed in depth. Two types of lower bounds, the constrained Cram\'{e}r-Rao bound (CCRB) and the Hammersley-Chapman-Robbins bound (HCRB), are discussed. It is shown that the situation with sensing matrix perturbation is more complex than the one with only measurement noise. For the CCRB, its closed-form expression is deduced. It demonstrates a gap between the maximal and nonmaximal support cases. It is also revealed that a gap lies between the CCRB and the MSE of the oracle pseudoinverse estimator, but it approaches zero asymptotically when the problem dimensions tend to infinity. For a tighter bound, the HCRB, despite of the difficulty in obtaining a simple expression for general sensing matrix, a closed-form expression in the unit sensing matrix case is derived for a qualitative study of the performance bound. It is shown that the gap between the maximal and nonmaximal cases is eliminated for the HCRB. Numerical simulations are performed to verify the theoretical results in this paper.Comment: 32 pages, 8 Figures, 1 Tabl

Improve Single-Point Zeroth-Order Optimization Using High-Pass and Low-Pass Filters

Single-point zeroth-order optimization (SZO) is useful in solving online black-box optimization and control problems in time-varying environments, as it queries the function value only once at each time step. However, the vanilla SZO method is known to suffer from a large estimation variance and slow convergence, which seriously limits its practical application. In this work, we borrow the idea of high-pass and low-pass filters from extremum seeking control (continuous-time version of SZO) and develop a novel SZO method called HLF-SZO by integrating these filters. It turns out that the high-pass filter coincides with the residual feedback method, and the low-pass filter can be interpreted as the momentum method. As a result, the proposed HLF-SZO achieves a much smaller variance and much faster convergence than the vanilla SZO method and empirically outperforms the residual-feedback SZO method, which is verified via extensive numerical experiments

Cooperative Spectrum Sharing in Cognitive Radio Networking

Driven by the massive growth in communications data traffic as well as flourishing users' demands, we need to fully utilize the existing scarce spectrum resource. However, there have been several studies and reports over the years showing that a large portion of licensed spectrum is actually underutilized in both temporal and spatial domains. Moreover, aiming at facing the dilemma among the fixed spectrum allocation, the ever enormous increasing traffic demand and the limited spectrum resource, cognitive radio (CR) was proposed by Mitola to alleviate the under usage of spectrum. Thus, cognitive radio networking (CRN) has emerged as a promising paradigm to improve the spectrum efficiency and utilization by allowing secondary users (SUs) to utilize the spectrum hole of primary users (PUs). By using spectrum sensing, SUs can opportunistically access spectrum holes for secondary transmission without interfering the transmissions of the PUs and efficient spectrum utilization by multiple PUs and SUs requires reliable detection of PUs. Nevertheless, sensing errors such as false alarm and misdetection are inevitable in practical networks. Hence, the assumption that SUs always obtain the exact channel availability information is unreasonable. In addition, spectrum sensing must be carried out continuously and the SU must terminate its transmission as soon as it senses the re-occupancy by a PU. As a better alternative of spectrum sensing, cooperation has been leveraged in CRN, which is referred as cooperative cognitive radio networking (CCRN). In CCRN, in order to obtain the transmission opportunities, SUs negotiate with the PUs for accessing the spectrum by providing tangible service for PUs. In this thesis, we study cluster based spectrum sharing mechanism for CCRN and investigate on exploiting the cooperative technique in heterogeneous network. First, we develop cooperation protocols for CRN. Simultaneous transmission can be realized through quadrature signalling method in our proposed cooperation protocol. The optimal power allocation has been analyzed and closed-form solution has been derived for amplify and forward mode. Second, we study a cluster based spectrum sharing mechanism. The spectrum sharing is formulated as a combinatorial non-linear optimization problem which is NP-hard. Afterwards, we solve this problem by decomposing it into cluster allocation and time assignment, and we show that the result is close to the optimal solution. Third, we propose a macrocell-femtocell network cooperation scheme for heterogeneous networks under closed access mode. The cooperation between the femtocell network and macrocell network is investigated. By implementing the cooperation, not only the macrocell users' (MUEs') and femtocell users' (FUEs') utility can be improved compared with the non-cooperation case, but also the energy consumption as well as the interference from the femtocell network to the macrocell network can be reduced

Feeder Reconfiguration in Distribution Networks Based on Convex Relaxation of OPF

The feeder reconfiguration problem chooses the on/off status of the switches in a distribution network in order to minimize a certain cost such as power loss. It is a mixed-integer nonlinear program and, hence, hard to solve. In this paper, we propose a heuristic algorithm that is based on the recently developed convex relaxation of the ac optimal power flow problem. The algorithm is computationally efficient and scales linearly with the number of redundant lines. It requires neither parameter tuning nor initialization for different networks. It successfully computes an optimal configuration on all four networks we have tested. Moreover, we have proved that the algorithm solves the feeder reconfiguration problem optimally under certain conditions for the case where only a single redundant line needs to be opened. We also propose a more computationally efficient algorithm and show that it incurs a loss in optimality of less than 3% on the four test networks

Time-Varying Optimization and Its Application to Power System Operation

The main topic of this thesis is time-varying optimization, which studies algorithms that can track optimal trajectories of optimization problems that evolve with time. A typical time-varying optimization algorithm is implemented in a running fashion in the sense that the underlying optimization problem is updated during the iterations of the algorithm, and is especially suitable for optimizing large-scale fast varying systems. Motivated by applications in power system operation, we propose and analyze first-order and second-order running algorithms for time-varying nonconvex optimization problems. The first-order algorithm we propose is the regularized proximal primal-dual gradient algorithm, and we develop a comprehensive theory on its tracking performance. Specifically, we provide analytical results in terms of tracking a KKT point, and derive bounds for the tracking error defined as the distance between the algorithmic iterates and a KKT trajectory. We then provide sufficient conditions under which there exists a set of algorithmic parameters that guarantee that the tracking error bound holds. Qualitatively, the sufficient conditions for the existence of feasible parameters suggest that the problem should be "sufficiently convex" around a KKT trajectory to overcome the nonlinearity of the nonconvex constraints. The study of feasible algorithmic parameters motivates us to analyze the continuous-time limit of the discrete-time algorithm, which we formulate as a system of differential inclusions; results on its tracking performance as well as feasible and optimal algorithmic parameters are also derived. Finally, we derive conditions under which the KKT points for a given time instant will always be isolated so that bifurcations or merging of KKT trajectories do not happen. The second-order algorithms we develop are approximate Newton methods that incorporate second-order information. We first propose the approximate Newton method for a special case where there are no explicit inequality or equality constraints. It is shown that good estimation of second-order information is important for achieving satisfactory tracking performance. We also propose a specific version of the approximate Newton method based on L-BFGS-B that handles box constraints. Then, we propose two variants of the approximate Newton method that handle explicit inequality and equality constraints. The first variant employs penalty functions to obtain a modified version of the original problem, so that the approximate Newton method for the special case can be applied. The second variant can be viewed as an extension of the sequential quadratic program in the time-varying setting. Finally, we discuss application of the proposed algorithms to power system operation. We formulate the time-varying optimal power flow problem, and introduce partition of the decision variables that enables us to model the power system by an implicit power flow map. The implicit power flow map allows us to incorporate real-time feedback measurements naturally in the algorithm. The use of real-time feedback measurement is a central idea in real-time optimal power flow algorithms, as it helps reduce the computation burden and potentially improve robustness against model mismatch. We then present in detail two real-time optimal power flow algorithms, one based on the regularized proximal primal-dual gradient algorithm, and the other based on the approximate Newton method with the penalty approach

Distributed Information-based Source Seeking

In this paper, we design an information-based multi-robot source seeking algorithm where a group of mobile sensors localizes and moves close to a single source using only local range-based measurements. In the algorithm, the mobile sensors perform source identification/localization to estimate the source location; meanwhile, they move to new locations to maximize the Fisher information about the source contained in the sensor measurements. In doing so, they improve the source location estimate and move closer to the source. Our algorithm is superior in convergence speed compared with traditional field climbing algorithms, is flexible in the measurement model and the choice of information metric, and is robust to measurement model errors. Moreover, we provide a fully distributed version of our algorithm, where each sensor decides its own actions and only shares information with its neighbors through a sparse communication network. We perform intensive simulation experiments to test our algorithms on large-scale systems and physical experiments on small ground vehicles with light sensors, demonstrating success in seeking a light source