Semidefinite Programming Relaxation Technique Research Articles

A Sum of Squares-Based Approach to Robust Nonlinear Mixed H2/H∞ Output-Feedback Control for a Class of Uncertain Polynomial Systems

This paper considers the robust nonlinear mixed H2/H∞ output-feedback control problem for a class of uncertain polynomial systems. Generally, the solvable conditions of such nonlinear output-feedback control problems are nonconvex, whose computations are challenging. By using the semidefinite programming relaxation technique based on sum of squares (SOS) decomposition, the solvable conditions of above control problem are formulated in terms of state-dependent linear matrix equality (LMI), which can be effectively solved. This conversion effectively overcomes the computational difficulties caused by the non-convexity of output-feedback H∞ control design for nonlinear systems. Furthermore, the state-observer and the controller can be designed simultaneously through a single-step SOS condition and constructed in a simple analytical form, thus reducing the computational complexity in a certain degree. In the simulation, the nonlinear mass-spring-damper system is considered to illustrate the effectiveness and feasibility of the proposed approach. The results show that the proposed method not only guarantees the stability of the system, but also has good transient performance and robust performance.

Convex relaxation for optimal fixture layout design

This article proposes a general fixture layout design framework that directly integrates the system equation with the convex relaxation method. Note that the optimal fixture design problem is a large-scale combinatorial optimization problem; we relax it to a convex Semi-Definite Programming (SDP) problem by adopting sparse learning and SDP relaxation techniques. It can be solved efficiently by existing convex optimization algorithms and thus generates a near-optimal fixture layout. A real case study in the half-to-half fuselage assembly process indicates the superiority of our proposed algorithm compared to the current industry practice and state-of-art methods.

Pay-As-You-Go: A Wireless Power Transfer-Enabled Beamforming for Cooperative Communication Systems

This letter considers a wireless power transfer enabled cooperative communication system where the end-users (EUs) make use of WPT to send energy to intermediate RUs to receive information. In traditional cooperative communications systems, service provider provides the energy for cooperation and the networks are optimized from the service providers' perspectives. In the this system, EUs compensate for the energy requirements of RUs, and EUs should get a fair service back from RUs. To administrate EUs WPT and ensure the fair service policy, this proposes a novel beamforming technique named Pay-As-You-Go beamforming. This adapts data rates proportional to the amount of energy received from each EUs while optimizing the energy per bit cost of the overall system. Then formulated as an non convex optimization problem and solved with the semi-definite programming relaxation techniques.

Robust adaptive beamforming against large steering vector mismatch based on minimum dispersion

Many efforts have been recently devoted to adaptive beamformers based on the minimum dispersion (MD) criterion for non-Gaussian signals, which results in improved signal-to-noise ratio performance. However, large steering vector mismatch and inappropriate choice of uncertainty set will cause performance degradation to the existing MD-based methods. To address these issues, we develop a robust MD-based beamformer against large steering vector mismatch, which utilizes multiple small uncertainty sets to cover the whole uncertainty region. The ℓp−norm (p≥1) minimization problem turns to be nonlinear and nonconvex because of these multi-uncertainty-set constraints. To solve the problem, a projected gradient algorithm is utilized to transform the original problem into a nonconvex quadratically constrained quadratic programming (QCQP) problem. The semidefinite programming relaxation technique is then employed in each iteration to deal with the nonconvex QCQP problem. Numerical results demonstrate that the proposed beamformer offers a significant performance improvement in case of large steering vector mismatch for non-Gaussian signals compared with the conventional schemes.

Frequency-Hopping Code Design for Target Detection via Optimization Theory

We present a signaling scheme for information embedding into the illumination of radar using frequency-hopping pulses. A frequency-hopping-based joint radar-communication system enables implementing a primary radar operation and a secondary communication function simultaneously. Then, we consider the problems of radar codes optimization under a peak-to-average-power ratio and an energy constraint. These radar codes design problems can be converted into non-convex quadratic programs with a finite or an infinite number of quadratic constraints. All problems are proved to be NP-hard optimization problems. Therefore, we develop optimization approaches, resorting to semi-definite programming relaxation technique along with to the idea of trigonometric polynomials, offering expected approximate solutions with a polynomial time calculation burden. We assess the capability of the proposed schemes, considering both the detection probability and the robustness in correspondence of Doppler shifts offered by the Neyman–Pearson detector. Simulation results show an improvement in detection performance as the average signal-to-noise ratio value increases, while still maintaining low symbol error rates between the proposed system nodes.

Chordal Conversion Based Convex Iteration Algorithm for Three-Phase Optimal Power Flow Problems

The three-phase optimal power flow (OPF) problem has recently attracted a lot of research interests due to the need to coordinate the operations of large-scale and heterogeneous distributed energy resources in unbalanced electric power distribution systems. The nonconvexity of the three-phase OPF problem is much stronger than that of the single-phase OPF problem. Instead of applying the semidefinite programming relaxation technique, this paper advocates a convex iteration algorithm to solve the nonconvex three-phase OPF problem. To make the convex iteration algorithm computationally efficient for large-scale distribution networks, the chordal conversion based technique is embedded in the convex iteration framework. By synergistically combining the convex iteration method and the chordal based conversion technique, the proposed three-phase OPF algorithm is not only computationally efficient but also guarantees global optimality when the trace of the regularization term becomes zero. At last, to further improve the computational performance, a greedy grid partitioning algorithm is proposed to decompose a single large matrix representing a distribution network to many smaller matrices. The simulation results using standard IEEE test feeders show that the proposed algorithm is computationally efficient, scalable, and yields global optimal solutions while resolving the rank conundrum.

3-D Target Localization in Wireless Sensor Networks Using RSS and AoA Measurements

This paper addresses target localization problems in both noncooperative and cooperative 3-D wireless sensor networks (WSNs), for both cases of known and unknown sensor transmit power, i.e., $P_{T}$ . We employ a hybrid system that fuses distance and angle measurements, extracted from the received signal strength and angle-of-arrival information, respectively. Based on range and angle measurement models, we derive a novel nonconvex estimator based on the least squares criterion. The derived nonconvex estimator tightly approximates the maximum-likelihood estimator for small noise. We then show that the developed estimator can be transformed into a generalized trust region subproblem framework, by following the squared range approach, for noncooperative WSNs. For cooperative WSNs, we show that the estimator can be transformed into a convex problem by applying appropriate semidefinite programming relaxation techniques. Moreover, we show that the generalization of the proposed estimators for known $P_{T}$ is straightforward to the case where $P_{T}$ is not known. Our simulation results show that the new estimators have excellent performance and are robust to not knowing $P_{T}$ . The new estimators for noncooperative localization significantly outperform the existing estimators, and our estimators for cooperative localization show exceptional performance in all considered settings.

MIMO Radar Transmit Beampattern Design for DOA Estimation with Sidelobe Suppression

We address the problem of transmit beamspace design for multiple-input multiple-output (MIMO) radar with colocated antennas in direction-of-arrival (DOA) estimation application. Three transmit beampattern sidelobe suppression strategies for designing the transmit beamspace matrix are introduced. The design of transmit beamspace matrix is based on minimizing the difference between a desired transmit beampattern and the actual one while keeping the sidelobe levels under control. Uniform elemental power distribution across the transmit antenna is guaranteed; at the same time, signal rotational invariance property is considered, which enables search-free based DOA estimation algorithms to be utilized at the receiver. The proposed optimization problems are nonconvex and are solved by using semidefinite programming relaxation technique. Moreover, the DOA estimation Cramer-Rao bound with transmit beamspace matrix is discussed. Simulation results show the superiority of the proposed techniques over the existing methods.

Nearly optimal minimax estimator for high-dimensional sparse linear regression

We present estimators for a well studied statistical estimation problem: the estimation for the linear regression model with soft sparsity constraints ($\ell_q$ constraint with $0<q\leq1$) in the high-dimensional setting. We first present a family of estimators, called the projected nearest neighbor estimator and show, by using results from Convex Geometry, that such estimator is within a logarithmic factor of the optimal for any design matrix. Then by utilizing a semi-definite programming relaxation technique developed in [SIAM J. Comput. 36 (2007) 1764–1776], we obtain an approximation algorithm for computing the minimax risk for any such estimation task and also a polynomial time nearly optimal estimator for the important case of $\ell_1$ sparsity constraint. Such results were only known before for special cases, despite decades of studies on this problem. We also extend the method to the adaptive case when the parameter radius is unknown.

Identification Aspects of SDP Based Polynomial Optimization Relaxations

The goal of parametric system identification is to provide estimates for parameters of a certain model structure based on given measurement data. This problem can always be presented as an optimization problem with an appropriate choice of cost and constraint functions. Apart from the simplest cases the resulting optimization problems are nonconvex with multiple local minima. Due to the existence of these, usually there are no guaranties that the model resulting from a given identification method is a global minimizer. This paper applies semidefinite programming (SDP) relaxation techniques to the optimization problem arising in time domain identification. From the solution of the defined sequence of SDPs a sequence of system models can be extracted that converges to the globally optimal system model. We give a short overview of the SDP relaxation technique for polynomial optimization problems (POP), then this technique is applied to the identification problem. The properties of the resulting POP are examined in detail. The solutions of the SDP sequence usually converge to the optimizer only in the limit. Model structures where finite convergence occurs and issues regarding detecting finite convergence are also considered.

Minimax Design of IIR Digital Filters Using SDP Relaxation Technique

This paper presents a new algorithm using semidefinite programming (SDP) relaxation to design infinite impulse response digital filters in the minimax sense. Unlike traditional design algorithms that try to directly minimize the error limit, the proposed algorithm employs a bisection searching procedure to locate the minimum error limit of the approximation error. Given a fixed error limit at each iteration, the SDP relaxation technique is adopted to formulate the design problem in a convex form. In practice, the true minimax design cannot be always obtained. Thus, a regularized feasibility problem is adopted in the bisection searching procedure. The stability of the designed filters can also be guaranteed by adjusting the regularization coefficient. Unlike other sequential design methods, the proposed algorithm tries to find a feasible solution at each iteration of the sequential design procedure within a feasible set defined by the relaxed constraints. This feasible set is not restricted within the neighborhood of a given point obtained from the previous iteration. Thus, the proposed method can avoid being trapped in the locally minimum point. Four examples are presented in this paper to demonstrate the effectiveness of the proposed method.