Convex optimization

About: Convex optimization is a research topic. Over the lifetime, 24906 publications have been published within this topic receiving 908795 citations. The topic is also known as: convex optimisation.

Algorithms for Leader Selection in Stochastically Forced Consensus Networks

Abstract: We are interested in assigning a pre-specified number of nodes as leaders in order to minimize the mean-square deviation from consensus in stochastically forced networks. This problem arises in several applications including control of vehicular formations and localization in sensor networks. For networks with leaders subject to noise, we show that the Boolean constraints (which indicate whether a node is a leader) are the only source of nonconvexity. By relaxing these constraints to their convex hull we obtain a lower bound on the global optimal value. We also use a simple but efficient greedy algorithm to identify leaders and to compute an upper bound. For networks with leaders that perfectly follow their desired trajectories, we identify an additional source of nonconvexity in the form of a rank constraint. Removal of the rank constraint and relaxation of the Boolean constraints yields a semidefinite program for which we develop a customized algorithm well-suited for large networks. Several examples ranging from regular lattices to random graphs are provided to illustrate the effectiveness of the developed algorithms.

Rate maximization in multi-antenna broadcast channels with linear preprocessing

Abstract: The sum rate capacity of the multi-antenna broadcast channel has recently been computed. However, the search for efficient practical schemes that achieve it is still ongoing. In this paper, we focus on schemes with linear preprocessing of the transmitted data. We propose two criteria for the preceding matrix design: one maximizing the sum rate and the other maximizing the minimum rate among all users. The latter problem is shown to be quasiconvex and is solved exactly via a bisection method. In addition to preceding, we employ a signal scaling scheme that minimizes the average bit-error-rate (BER). The signal scaling scheme is posed as a convex optimization problem, and thus can be solved exactly via efficient interior-point methods. In terms of the achievable sum rate, the proposed technique significantly outperforms traditional channel inversion methods, while having comparable (in fact, often superior) BER performance

A System-Level Approach to Controller Synthesis

Abstract: Biological and advanced cyber-physical control systems often have limited, sparse, uncertain, and distributed communication and computing in addition to sensing and actuation. Fortunately, the corresponding plants and performance requirements are also sparse and structured, and this must be exploited to make constrained controller design feasible and tractable. We introduce a new “system level” (SL) approach involving three complementary SL elements. SL parameterizations (SLPs) provide an alternative to the Youla parameterization of all stabilizing controllers and the responses they achieve, and combine with SL constraints (SLCs) to parameterize the largest known class of constrained stabilizing controllers that admit a convex characterization, generalizing quadratic invariance. SLPs also lead to a generalization of detectability and stabilizability, suggesting the existence of a rich separation structure, that when combined with SLCs is naturally applicable to structurally constrained controllers and systems. We further provide a catalog of useful SLCs, most importantly including sparsity, delay, and locality constraints on both communication and computing internal to the controller, and external system performance. Finally, we formulate SL synthesis problems, which define the broadest known class of constrained optimal control problems that can be solved using convex programming.

Robust Downlink Beamforming in Multiuser MISO Cognitive Radio Networks With Imperfect Channel-State Information

Abstract: This paper studies the problem of robust downlink beamforming design in a multiuser multiple-input-single-output (MISO) cognitive radio network (CR-Net) in which multiple secondary users (SUs) coexist with multiple primary users (PUs) of a single-cell primary radio network (PR-Net). It is assumed that the channel-state information (CSI) for all relevant channels is imperfectly known, and the imperfectness of the CSI is modeled using a Euclidean ball-shaped uncertainty set. Our design objective is to minimize the transmit power of the SU-Transmitter (SU-Tx) while simultaneously achieving a lower bound on the received signal-to-interference-plus-noise ratio (SINR) for the SUs and imposing an upper limit on the interference power (IP) at the PUs. The design parameters at the SU-Tx are the beamforming weights, and the proposed methodology to solve the problem is based on the worst-case design scenario through which the performance metrics of the design are immune to variations in the channels. The original problem is a separable homogeneous quadratically constrained quadratic problem (QCQP), which is an NP-hard problem, even for uncertain CSI. We reformulate our original design problem to a relaxed semidefinite program (SDP) and then investigate three different approaches based on convex programming. Finally, simulation results are provided to validate the robustness of the proposed methods.

Optimal Strategies and Minimax Lower Bounds for Online Convex Games

Abstract: A number of learning problems can be cast as an Online Convex Game: on each round, a learner makes a prediction x from a convex set, the environment plays a loss function f, and the learner’s long-term goal is to minimize regret. Algorithms have been proposed by Zinkevich, when f is assumed to be convex, and Hazan et al., when f is assumed to be strongly convex, that have provably low regret. We consider these two settings and analyze such games from a minimax perspective, proving minimax strategies and lower bounds in each case. These results prove that the existing algorithms are essentially optimal.