Energy-Based Acoustic Source Localization Methods: A Survey <s> Semidefinite Programming Method <s> Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics. <s> BIB001 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Semidefinite Programming Method <s> This paper addresses the energy-based localization problem in wireless sensor networks. The maximum likelihood (ML) location estimation problem is a difficult optimization problem due to the non-convexity of the objective function, and finding an exact solution is difficult. In this work, an approximate solution to the ML localization is presented, by relaxing the minimization problem into semidefinite programming form. Simulation results show that the proposed algorithm outperforms the existing solutions. <s> BIB002 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Semidefinite Programming Method <s> In this paper, the energy-based localization problem in wireless sensor networks is addressed. The maximum likelihood (ML) location estimation problem is a difficult optimization problem due to the non-convexity of the objective function, and an exact solution is difficult to obtain. An approximate solution to the ML problem is proposed, by relaxing the minimization problem into semidefinite programming form. Simulation results show that the proposed algorithm outperforms the existing solutions. <s> BIB003 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Semidefinite Programming Method <s> Source localization in wireless sensor networks (WSNs) aims to determine the position of a source in a network, given inaccurate position-bearing measurements. This paper addresses the problem of locating a single source from noisy acoustic energy measurements in WSNs. Under the assumption of Gaussian energy measurement errors, the maximum likelihood (ML) estimator requires the minimization of a nonlinear and nonconvex cost function which may have multiple local optima, thus making the search for the globally optimal solution hard. In this work, an approximate solution to the ML location estimation problem is presented by relaxing the minimization problem to a convex optimization problem, namely second-order cone programming. Simulation results demonstrate the superior performance of the convex relaxation approach. More precisely, the new approach shows an improvement of 20 % in terms of localization accuracy when compared to the existing approaches at moderate to high noise levels. Simulation results further show comparable performance of the new approach and the state-of-art approaches at low noise levels. <s> BIB004
For the centralized methods, in BIB002 BIB004 BIB003 , the author formulated the energy-based source localization problem in different ways. For example, the problem was solved by using the semidefinite programming method BIB002 , second-order cone programming relaxation (SOCP) BIB004 . Denoting Θ = θθ T , and relaxing it into Θ θθ T , then the source localization problem can be formulated using following SDP: subject to: As we can see from Equation (18), the condition for the optimization function in Equation (17) (the initial maximum-likelihood (ML) problem) is an approximated one compared to the fact that Θ = θθ T . Hence, we can claim that the SDP is an approximated solution to ML estimation of the source localization problem. Motivated by the idea of complexity reduction while keeping the performance at an acceptable level, the nonconvex ML-based localization problem is approximated by a convex second-order cone programming (SOCP) optimization problem, which can be solved very efficiently by interior point methods BIB001 . The basic idea of SOCP is similar to the SDP formulation. Both SOCP and SDP can be considered as approximated ML solutions. The SDP method perform well in terms of localization accuracy. However, its main disadvantage is that it can only be used for centralized implementations.