Convex optimization algorithms for cooperative RSS-based sensor localization

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Convex optimization algorithms for cooperative RSS-based sensor localization Jian Zheng a , Xiaoping Wu b,∗ a

College of Transport & Communications, Shanghai Maritime University, Shanghai, 201306, PR China

b

School of Information Engineering, Zhejiang A&F University, HangZhou, 311300, PR China

article

info

Article history: Received 4 November 2015 Received in revised form 4 May 2016 Accepted 7 June 2016 Available online xxxx Keywords: Localization Received signal strength (RSS) Convex optimization Semidefinite programming (SDP) Convex second-order cone program (SOCP)

abstract To obtain a global solution for the source location estimates, the cost function of RSS-based sensor localization is relaxed as convex optimization problem which can be solved by interior point method. Weighted squared least square (WSLS) and weighted least square (WLS) based optimization functions are proposed to locate the source nodes. The corresponding semidefinite programming (SDP), second-order cone program (SOCP) and mixed SOC/SDP algorithms are designed by considering the known or unknown transmit powers. The computational complexity of the proposed algorithms is derived by analyzing the number of variables and equality constraints produced in the relaxation. The simulations show that the mixed SOC/SDP runs faster than the SDP, although the algorithms have the approximately equal accuracy performance. Whether the transmit power is known or not, the accuracy performance of the WLS-SDP is better than that of the WSLS-SDP and WSLSSOC/SDP algorithms. However the computational complexity of the WLS-SDP is greatly larger than that of WSLS-SOC/SDP and WSLS-SDP due to a large number of variables. © 2016 Elsevier B.V. All rights reserved.

1. Introduction The capability of accurately positioning sensor nodes enables their innovative applications nowadays in the environmental monitoring and emergency rescue especially for maritime ships, so sensor localization is an indispensable component of wireless sensor networks (WSNs) [1–4]. Global positioning system (GPS) is the most important technology to provide location awareness around the globe through a constellation of at least 24 satellites. However, the effectiveness of GPS is limited at each low-cost and tiny sensor node for its huge volume, energy consumption and hardware cost. Accurately positioning sensor nodes is itself a challenging problem due to extremely limited resources and low cost of sensor nodes [5,6]. In the field of sensor localization, it is often assumed that a few anchor nodes with known positions are used to derive the locations of other source nodes. Besides positions of anchors, the additional information that are also assumed to be known are the distance measurements between neighboring sensors. Most of the accuracy localization techniques are based on the ranging approaches of the radio signal transmitted between the anchors and source nodes. The ranging approaches measure the Euclidean distances among the sensors with certain range techniques and locate the sensors using geometric methods. Many ranging methods use techniques such as, time of arrival (TOA) [7–9], time difference of arrival (TDOA) [10], angle of arrival (AOA) [11,12], received signal strength (RSS) [13–15], or acoustic energy strength [16] measurements to measure distance among anchor and source nodes. WSNs consist of a number of relatively inexpensive and randomly distributed

∗

Corresponding author. E-mail addresses: [email protected] (J. Zheng), [email protected] (X. Wu).

http://dx.doi.org/10.1016/j.pmcj.2016.06.002 1574-1192/© 2016 Elsevier B.V. All rights reserved.

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sensor nodes that can communicate with each other, so RSS-based technique provides inherent tradeoffs between the positioning performance and the implementation complexity due to its low complexity and cost in software and hardware implementations [17]. Some estimation algorithms have been proposed for sensor localization over the past years, such as maximum likelihood (ML) [18,19] estimator, linear estimators [20–22] and convex optimization approaches [23–26]. The ML estimator is an asymptotically unbiased estimator for sensor localization problem. However it is a nonlinear least squares estimator and requires a reasonable initialization close to the true solution. Without a good initial guess, however, local convergence may occur and convergence is not guaranteed. The linear estimators always represent the location estimation as algebraic closedform solutions and avoid the selection of initial solution. However the performance of linear estimator is poor especially when the shadowing is very high. The basic idea of the convex optimization technique is to relax the nonconvex formulation to a convex optimization program which always guarantees a global solution and does not rely on the initialization. Generally speaking the solution of convex optimization is an approximate estimate due to the relaxation of optimization equations. Based on whether the measurements between source nodes are involved or not, sensor localization is generally divided into non-cooperative and cooperative localization [27,28]. Non-cooperative localization only uses the measurements between source node and anchor node (source–anchor measurements) to locate the unique position of source nodes. The received signal strength transmitted by source node can be measured at not only anchor node but also source node. The limited number of accessible anchor nodes have led to the emergence of cooperative localization in which the source–source RSS measurements are involved [29,30]. Both source–anchor and source–source measurements are exploited in the estimation of source location for cooperative localization, so the positioning performance and robustness are improved due to more useful information. Convex optimization of sensor localization has a relatively complicated structure and high computational complexity, so a number of algorithms focus on the relaxation of convex optimization to trade off the computational complexity and positioning accuracy, such as semidefinite programming (SDP) [31–34], convex second-order cone program (SOCP) [35,36] and mixed SOC/SDP [37] algorithms. The SDP algorithm provides a tighter relaxation and hence results in a better localization accuracy compared to SOCP and runs slower due to a number of variables and equality constraints generated by the relaxation of convex optimization. The lower complexity of SOCP is because for a given problem, the number and size of variables and constraints required for solving SOCP are smaller than those required for solving SDP. However the disadvantage is that SOCP position estimates always lie within the convex hull of the anchor positions, so the SOCP algorithm cannot be directly applied to the cooperative localization. The mixed SOC/SDP algorithm trades off the positioning accuracy and computational complexity since the less variables are produced in the convex relaxation. These convex optimization algorithms are also proposed to deal with the RSS-based sensor localization by linearly approximating the exponential relationship between RSS measurement and distance. In [37], the model of RSS-based sensor localization is converted into a weighted least squares (WLS) estimation problem by using the unscented transformation (UT) and approximating the nonlinear function. The proposed WLS estimation problem can be solved with the convex optimization technique efficiently, but the UT leads to the large localization error especially when the geometric conditions are poor. The transmit power will be subject to a large fluctuation because its value is dependent on the height and orientation of the node antenna, as well as antenna gain and its battery which will decrease with time. Considering the optimization function of weighted squared least squares (WSLS), the WSLS-SDP algorithm is put forward in [33] by converting the ML minimization problem into a convex problem which can be solved efficiently. However the computational complexity of the SDP is high due to a large number of variables and equality constraints. To reduce the computation cost, the SOCP is proposed in [35] for the distributed RSS-based sensor localization, since the SOCP requires a lower number of iterations to converge. In this paper we assume that the propagation model, path-loss exponent and shadowing variances are known a priori through a calibration phase. The SOCP, SDP and mixed SOC/SDP algorithms are designed for cooperative RSS-based sensor localization when the transmit powers of source nodes are assumed to be known or unknown. To reduce the computation complexity of the convex optimization, the optimization function of WSLS-SDP proposed in [33] is firstly relaxed into the mixed WSLS-SOC/SDP algorithm. Then a novel optimization function of weighted least squares (WLS) is put forward by using the ML estimator. Compared with the WSLS-based optimization function, the WLS-based optimization function has less approximation error. Therefore we derive the WLS-SOCP and WLS-SDP algorithms for cooperative RSS-based sensor localization. Furthermore, these algorithms are extended to the scenario of unknown transmit power. The corresponding uWSLS-SDP, uWSLS-SOC/SDP and uWLS-SDP algorithms are introduced to estimate the source locations. The computational complexities of the proposed algorithms are also derived to analyze the theoretical performance. This paper mainly presents the convex optimization approaches for RSS-based senor localization by considering different optimization functions and parameter conditions. The rest of this paper is structured as follows. Section 2 presents the problem specification of sensor localization in the network. Sections 3 and 4 describe in detail the algorithm design with known and unknown transmit powers. Section 5 drives the computational complexities of proposed algorithms. Section 6 analyzes the simulation results. The conclusion is represented in Section 7. This paper contains a number of symbols. Following the convention, we represent the matrices as bold case letters. If we denote the matrix as (∗), (∗)−1 represents matrix inverse. ∥ ∗ ∥ denotes ℓ2 norm. diag{∗} represents a diagonal matrix. [A]i,j denotes the element at the ith row and jth column of matrix A. For arbitrary symmetric matrices A, A ≽ 0 means that A is positive semidefinite.

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2. Problem specification We consider that a sensor network of size M + N is deployed in a 2-dimensional geographical area. Note that threedimensional region sensor localization is treated in the same manner. The first M sensor nodes are source nodes whose positions are to be determined and denoted as xi = [xi yi ]T ∈ R2 , i = 1, 2, . . . , M. The remaining N nodes are anchors with precisely known position ak = [xk yk ]T ∈ R2 , k = M + 1, M + 2, . . . , M + N. The received signal strength through a radio channel is attenuated because of three nearly independent factors, namely, path-loss, shadow fading and multipath fading. The path-loss quantifies the attenuation of the transmitted power which decreases as di,j increases. The shadow fading represents as low variation in a RSS measurement due to obstacles in propagation paths. The multipath fading is caused by reception of multiple time delayed copies of a transmitted signal through multipath propagation and can be smoothed out by averaging the RSS measurements over frequency and time. As a result, the average RSS in dB measured at node j, denoted by Pi,j , is Pi,j = Pi,0 − 10β log10 di,j + ni,j

i ∈ S , j ∈ Ei

(1)

where S = {i|i = 1, 2, . . . , M }, Ei = Ai ∪ Bi . S represents the set of indices of the source nodes. Ai or Bi means that source node i is connected to anchor node j or source node j, respectively. L = |Ei , i ∈ S | represents the total link number of source nodes in the sensor network. Pi,0 is the transmit power in dB emitted from ith source node, and β denotes the path-loss factor whose value varies from 2 to 6. In particular, β is equal to 2 in free space. ni,j is shadow fading and modeled as uncorrelated zero-mean Gaussian variables with variances δi2,j . It is assumed that the noise variances δi2,j are known via a calibration of the environment before deploying these nodes. The goal of the localization problem based on RSS measurements is to estimate xi given Pi,j . All the RSS measurements are collected and sent to a central information processor where localization is performed in a centralized manner. This assumes that each RSS measurement received at the jth node can be correctly associated to the ith node. Denote the positions of the source nodes, transmit powers in vector form by x = [xT1

...

xT2

p0 = [P1,0

...

P2,0

2M

xTM ]T P M ,0 ]

(2) T

(3)

M

where x ∈ R , p0 ∈ R . Our aim is to estimate x based on the ranging model depicted as (1) where p0 is known in Section 3 and unknown in Section 4. 3. Known transmit power In the section the source location x is estimated by using the observed RSS measurements when the transmit power p0 is assumed to be available. In this case there are in total 2M unknown parameters that should be estimated when the sensor nodes are deployed in the 2-dimensional geographical area. The well known maximum likelihood (ML) solution to the sensor localization is written as min x

 1 (Pi,j − Pi,0 + 10β log10 di,j )2 . 2 δ i , j i∈S j∈Ei

(4)

The ML optimization expression of (4) is a nonlinear and non-convex problem and rather challenging as a multidimensional search problem whose solution requires an initialization. The shortcoming of the ML estimator leads us to employ suboptimal estimators, such as convex optimization and linear estimation algorithms. In the following, we introduce the convex optimization algorithms which do not require the initialization and overcome the shortcoming of the ML estimator. To obtain the convex form, the measurement equation of (1) is rewritten by using some approximations. Weighting squared least square (WSLS) and weighting least square (WLS) optimization functions are introduced to exploit the problem for cooperative sensor localization. Then we relax these nonconvex optimization functions to convex expressions which can be solved effectively with well known algorithms such as interior point methods. The mixed SOC/SDP algorithm is firstly designed by using the WSLS optimization function. Then based on the WLS optimization function the WLS-SDP algorithm is designed by relaxing the nonconvex problem to convex expression. 3.1. WSLS-based algorithms Converting the nonlinear expression to linear equation, we start with rewriting (1) as di,j = 10

Pi,0 −Pi,j +ni,j 10β

i ∈ S , j ∈ Ei .

(5)

For sufficiently small noise, the right-hand side of (5) can be approximated using the first-order Taylor series expansion as di,j = λi,j +

λi,j ln10 ni,j i ∈ S , j ∈ Ei 10β

(6)

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where λi,j = 10

Pi,0 −Pi,j 10β

,

λi,j ln10 ni , j 10β

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is a zero-mean Gaussian random variable with variance

of (6) and ignoring the second order term, (6) can also be rewritten as d2i,j = λ2i,j +

λ2i,j ln10 5β

Apparently the noise

5β

100β 2

i ∈ S , j ∈ Ei .

ni,j

λ2i,j ln10

λ2i,j (ln10)2 δi2,j

. By squaring both sides

(7)

ni,j is a zero-mean Gaussian random variable with variance

λ4i,j (ln10)2 δi2,j 25β 2

. By using the WSLS-based

optimization function, the ML estimator of the equivalent measurement model of (7) is obtained by the following nonconvex optimization problem



min x

h2i,j (d2i,j − λ2i,j )2

i∈S j∈Ei

s.t. ∥xi − aj ∥ = di,j i ∈ S , j ∈ Ai ∥xi − xj ∥ = di,j i ∈ S , j ∈ Bi

(8)

where the weight hi,j is written as 5β . λ2i,j δi,j ln10

hi,j =

(9)

The ML expression of (8) is also nonlinear and nonconvex, but the optimization expression is easy to be converted into the convex form. A new matrix X is firstly defined X = [x1

...

x2

xM ]

(10)

2 ×M

where X ∈ R . In order to form a semidefinite programming in the later relaxation procedure, we would like to introduce the change of variables





I2 XT

Z=

X XT X

(11)

where Z ∈ R(M +2)×(M +2) is required to be determined, I2 denotes the 2 by 2 identity matrix. It is not difficult to find that

  T    aj aj 2   i ∈ S , j ∈ Ai di,j = −ei Z −ei  T     02 02 2  di,j = Z i ∈ S , j ∈ Bi ei − ej ei − ej

(12)

where ei is an M × 1 column vector with 1 at the ith entry and 0’s elsewhere, 02 is a 2 × 1 zero column vector. So the optimization problem of (8) is also equivalent to



min

Z,α,{di,j }

αi,j

i∈S j∈Ei

s.t. h2i,j (d2i,j − λ2i,j )2 ≤ αi,j d2i,j = d2i,j =



aj −e i



T 

02

Z

aj −ei

T 

ei − ej

Z



i ∈ S , j ∈ Ai

02

ei − ej



i ∈ S , j ∈ Bi

(13)

where α = [αi,j |i ∈ S , j ∈ Ei ]. Unfortunately, the optimization problem of (13) is still non-convex. Let Y = XT X, which is relaxed into Y ≽ XT X. Then (11) is rewritten as

 Z=



I2 XT

X ≽ 0M +2 Y

(14)

where Y ∈ RM ×M , 0M +2 denotes the M + 2 by M + 2 zero matrix. By relaxing Z ≽ 0M +2 , [33] relaxed the nonconvex optimization problem of (13) into convex WSLS-SDP form min



Z,α,{di,j }

s.t.

αi,j

i∈S j∈Ei

αi,j hi,j (d2i,j − λ2i,j )



hi,j (d2i,j − λ2i,j ) ≽ 02 1



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i ∈ S , j ∈ Ei d2i,j =



aj −e i



d2i,j

=

T  Z

02

aj −e i

T  Z

ei − ej



i ∈ S , j ∈ Ai



02

ei − ej

i ∈ S , j ∈ Bi

Z ≽ 0M +2 .

(15)

The proposed WSLS-SDP algorithm provides a convex optimization solution. However the WSLS-SDP algorithm runs slower due to a larger of variables and equality constraints. The set of all possible solutions obtained from SDP relaxation is a subset of all possible solutions that can be obtained by SOCP. In other words, SDP provides a tighter relaxation than SOCP solution since the SOCP solution set includes the SDP solution set. To reduce the computational complexity, here the robust SOCP algorithm is provided for the SDP optimization model. To obtain the SOCP form, a new variable ui,j is defined and written as ui,j = hi,j (d2i,j − λ2i,j ).

(16)

Then the inequality constraint of (13) is transformed as

∥u∥ ≤ υ where u = [ui,j |i ∈ S , j ∈ Ei ], the new variable υ is also given by  αi,j . υ=

(17)

(18)

i∈S j∈Ei

So the SDP optimization problem of (15) can be equivalently written as its epigraph form min

Z,u,{di,j },υ

υ

s.t. ∥u∥ ≤ υ hi,j (d2i,j − λ2i,j ) = ui,j d2i,j = d2i,j



T 

aj

−e i 

=

02

Z

i ∈ S , j ∈ Ei



aj

−e i T 

ei − ej

Z

i ∈ S , j ∈ Ai

02



ei − ej

i ∈ S , j ∈ Bi

Z ≽ 0M +2

(19)

where the SOCP and SDP constraints are mixed by using the WSLS-based optimization function, so the solution to (19) is also called the WSLS-SOC/SDP algorithm for the cooperative sensor localization. For using the same optimization function, the solution of WSLS-SOC/SDP algorithm is very close to that of WSLS-SDP algorithm. However the computation complexity of WSLS-SOC/SDP algorithm is lower than that of WSLS-SDP algorithm. 3.2. WLS-based algorithms Due to the larger approximation errors, the accuracy performance of WSLS optimization function is not very well. In this subsection, a WLS-based optimization function is exploited to obtain the convex solutions for the source location estimates. By applying the approximated expression of (6) and the WLS optimization function, similarly the ML solution can be represented as min x



gi2,j (di,j − λi,j )2

i∈S j∈Ei

s.t. ∥xi − aj ∥ = di,j

i ∈ S , j ∈ Ai ∥xi − xj ∥ = di,j i ∈ S , j ∈ Bi

(20)

where the weight gi,j is also written as gi,j =

10β

λi,j δi,j ln10

.

(21)

Similarly we relax the target function as inequality constraint and write the equivalent expression of (20) in the following form min τ

x,t,{di,j }

s.t. ∥t∥ ≤ τ gi,j (di,j − λi,j ) = ti,j i ∈ S , j ∈ Ei ∥xi − aj ∥ = di,j i ∈ S , j ∈ Ai

∥xi − xj ∥ = di,j i ∈ S , j ∈ Bi

(22)

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where t = [ti,j |i ∈ S , j ∈ Ei ]. By relaxing the distance equality constraint to inequality once again, (22) can also be written as the following SOCP form min τ

x,t,{di,j }

s.t. ∥t∥ ≤ τ gi,j (di,j − λi,j ) = ti,j i ∈ S , j ∈ Ei ∥xi − aj ∥ ≤ di,j i ∈ S , j ∈ Ai

∥xi − xj ∥ ≤ di,j i ∈ S , j ∈ Bi .

(23)

Based on the WLS-based optimization function, we obtain the SOCP form for RSS-based sensor localization, so the solution to (23) is called the WLS-SOCP algorithm. For the relaxation from the distance equality constraint to inequality, SOCP position estimates always lie within the convex hull of the anchor positions. However we cannot ensure that the location of source node is within the convex hull of the anchor positions when the location of source node is unknown. The SOCP algorithm cannot also be applied for the cooperative localization in which the source–source measurements are involved. To overcome the shortcoming of SOCP, we obtain the SDP solution for RSS-based sensor localization considering the random position distribution of anchor nodes. Producing the change of variables

  D=

d  d 1



1

(24)

where d = [di,j |i ∈ S , j ∈ Ei ], D ∈ R(L+1)×(L+1) . It is directly shown that

 [D]p,p = d2i,j [D]L+1,p = di,j

(25)

where p = 1, 2, . . . , L. p represents the pth connection of node pair (i, j). By expanding the optimization function of (20), the target function can be rewritten as gi2,j (di,j − λi,j )2 = gi2,j ([D]p,p − 2λi,j [D]L+1,p + λ2i,j ).

(26)

The expression of (12) is also rewritten as

  T    aj aj   [ D ] = Z i ∈ S , j ∈ Ai  p,p −ei −ei  T     02 02  [D]p,p = Z i ∈ S , j ∈ Bi . ei − ej ei − ej

(27)

Removing the constant of the target function, we construct a semidefinite relaxation of (20) which is given by min Z ,D

L 

gi2,j ([D]p,p − 2λi,j [D]L+1,p )

p=1



aj s.t. [D]p,p = −ei

 [D]p,p =

T 

aj Z −ei

T 

02

Z

ei − ej



i ∈ S , j ∈ Ai



02 ei − ej

i ∈ S , j ∈ Bi

D ≽ 0L+1 , Z ≽ 0M +2

[D]L+1,L+1 = 1 rank(D) = 1.

(28)

Appendix demonstrates that matrix D must have rank one when (D, Z) is an optimal solution of (28). So by dropping the rank constraints in (28), we obtain the following WLS-SDP form min Z,D

L 

gi2,j ([D]p,p − 2λi,j [D]L+1,p )

p=1

s.t. [D]p,p =



aj −ei

T  Z

aj −ei



i ∈ S , j ∈ Ai

J. Zheng, X. Wu / Pervasive and Mobile Computing (



02 ei − ej

[D]p,p = D ≽ 0L+1 ,

T  Z

02 ei − ej



)

–

7

i ∈ S , j ∈ Bi

Z ≽ 0M +2

[D]L+1,L+1 = 1.

(29)

The number of variables in the defined Z is quadratic to the total RSS measurements, so the WLS-SDP algorithm runs slower due to a larger number of variables produced in the convex relaxation. However the convex algorithm of the WLS-based optimization provides more accurate source location estimates than that of the WSLS-based optimization due to the less approximation error. 4. Unknown transmit power Sometimes each source node has a specific transmit power depending on, e.g., its battery and antenna gain. In addition, the transmit power might change with time, e.g., when batteries begin to exhaust. Consequently, each source node has to report its transmit power to anchor nodes constantly during RSS measurements which requires additional hardware and software in both anchor nodes and source nodes making the network more convoluted. In this section the transmit powers are considered as nuisance parameters and assumed to be unknown, so the source transmit powers are estimated jointly with the source locations. In this case, there are in total 2M + M unknown parameters that should be estimated including the source node coordinates and the transmit powers. When the source transmit powers are unknown, the convex optimization relaxation follows the same procedure as described previously for the known transmit power case but with a slightly different relaxation. Similarly WSLS and WLS target optimization functions are discussed to design the convex optimization algorithms. 4.1. WSLS-based algorithms When the transmit power is considered as unknown parameter, we define a new measurement related parameter µi,j and a new variable ρi,0 , which are given by

 −P µ = 10 10iβ,j i,j Pi,0  ρi,0 = 10 10β .

(30)

So (5) can be rewritten as di,j = µi,j ρi,0 +

λi,j ln10 ni,j i ∈ S , j ∈ Ei 10β

(31)

where λi,j = µi,j ρi,0 . Similarly by squaring both sides of (31) and neglecting the second order term, (31) is rewritten as d2i,j = µ2i,j ηi,0 +

λ2i,j ln10 5β

ni,j

i ∈ S , j ∈ Ei

(32)

where ηi,0 is a new defined variable, ηi,0 = ρi2,0 . So the ML estimator for WSLS optimization function can be represented as min x,η0



h2i,j (d2i,j − µ2i,j ηi,0 )2

i∈S j∈Ei

s.t. ∥xi − aj ∥ = di,j

i ∈ S , j ∈ Ai

∥xi − xj ∥ = di,j i ∈ S , j ∈ Bi

(33)

where η0 denotes the unknown parameter vector related with the transmit powers of all source nodes, η0 = [ηi,0 |i ∈ S ]. Using the similar method expressed by (15), we obtain the SDP form min

x,η0 ,α,{di,j }

s.t.



αi,j

i∈S j∈Ei

 αi,j hi,j (d2i,j − µ2i,j ηi,0 ) ≽ 02 1 hi,j (d2i,j − µ2i,j ηi,0 ) i ∈ S , j ∈ Ei  T   aj aj 2 di,j = Z i ∈ S , j ∈ Ai −ei −e i  T   02 02 d2i,j = Z i ∈ S , j ∈ Bi ei − ej ei − ej 

Z ≽ 0M +2 .

(34)

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When assuming that the transmit powers of source nodes are unknown, (34) provides the SDP solution by utilizing the WSLS-based optimization function. So the solution to (34) is also called as uWSLS-SDP algorithm. However the computation complexity of SDP algorithm is higher than SOCP algorithm. To trade off the contradiction between the computation complexity and positioning accuracy, we relax the convex optimization problem as uWSLS-SOCP/SDP form min

Z,u,η0 ,{di,j },v

υ

s.t. ∥u∥ ≤ υ hi,j (d2i,j − µ2i,j ηi,0 ) = ui,j d2i,j = d2i,j



aj

T 

−ei 

=

02

Z



aj

−ei T 

ei − ej

Z

i ∈ S , j ∈ Ei i ∈ S , j ∈ Ai

02



ei − ej

i ∈ S , j ∈ Bi

Z ≽ 0M +2 .

(35)

The solutions to (34) and (35) rely on the estimated λi,j which is determined by the transmit power and not available in the beginning. Preliminarily considering λi,j as identical we obtain the initial estimate λi,j . Then putting the initial estimate into these optimization expressions would produce better solutions for the source location estimates. 4.2. WLS-based algorithms Compared with the WSLS-based optimization function, the WLS optimization function has less approximation error. In the subsection, a novel convex algorithm is designed by using the WLS-based optimization function when the transmit powers are assumed to be unknown. According to the approximation expression of (31), the ML estimator of WLS-based optimization problem can be represented as min x,ρ0



gi2,j (di,j − µi,j ρi,0 )2

i∈S j∈Ei

s.t. ∥xi − aj ∥ = di,j

i ∈ S , j ∈ Ai

∥xi − xj ∥ = di,j i ∈ S , j ∈ Bi

(36)

where ρ0 = [ρ1,0 ρ2,0 . . . ρM ,0 ]T , ρ0 denotes the unknown vector related with the transmit power. Similar with the expression of (23), we obtain the SOCP convex optimization form min

x,t,ρ0 ,{di,j }

τ

s.t. ∥t∥ ≤ τ gi,j (di,j − µi,j ρi,0 ) = ti,j

i ∈ S , j ∈ Ei ∥xi − aj ∥ ≤ di,j i ∈ S , j ∈ Ai

∥xi − xj ∥ ≤ di,j i ∈ S , j ∈ Bi .

(37)

By considering the transmit powers as unknown parameters, the SOCP solution to (37) obtains the estimated positions of source nodes with WLS-based optimization function. Due to the relaxation of distance constraints, the errors of estimated source locations are especially large. The accuracy performance degrades greatly when the SOCP algorithm is applied to RSS-based sensor localization. In the following, we further derive the SDP solutions to the source location estimates by using the WLS optimization function. By stacking all the connects in an ascending order of i and j, the matrix form of (31) is written as IL d + Fρ0 = ε

(38)

where IL denotes the L × L identity matrix, the row element of d is equal to [di,j ], the row vectors of F are equal to

[01×(i−1) − µi,j 01×(M −i) ]. The row elements of ε are equal to [ as

λi,j ln10 ni,j ]. 10β

Let b = [d ρ0 ]T , A = [IL F]. (38) is rewritten

Ab = ε.

(39)

So the cost function of the ML estimator in (36) can be alternatively written as Tr{Σ (Ab)T Ab} = Tr{Σ (ABAT )}

(40)

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where B = bbT , the weight matrix Σ is obtained with





100β 2

Σ = diag

.

λ2i,j δi2,j (ln10)2

(41)

The diagonal elements of the matrix B are denoted as [B]p,p . Apparently it is shown that

  T    aj aj   j ∈ Ai Z [ B ] = p , p  −ei −ei  T     02 02  [B]p,p = j ∈ Bi Z ei − ej ei − ej

(42)

where p is the new order by stacking all links in an ascending order of i and j, p = 1, 2, . . . , L. Then the optimization problem of the ML estimator in (36) is rewritten as min Tr{Σ (ABAT )} Z,b,B

s.t. [B]p,p =

 [B]p,p =



aj −ei

Z

T 

02

Z

ei − ej



T 



02



j ∈ Ai j ∈ Bi

ei − ej



I2 Z= XT

aj −e i

X , XT X

B = bbT .

(43)

It is noted that the cost function of (43) is linear with the variables of B. However the constraints in (43) make the problem nonconvex. To obtain the convex SDP form, B = bbT is relaxed as the following,



B bT



b ≽ 0L+M +1 . 1

(44)

The above relaxations of the constraints degrade the performance the source location estimates, but the convex SDP form is obtained. Using the relaxations, the optimization problem of (44) is rewritten as the SDP form min Tr{Σ (AHAT )} Z,b,B

s.t. [B]p,p =

 [B]p,p =



aj −ei

02 ei − ej

T  Z

T  Z

aj −e i



02



ei − ej

j ∈ Ai j ∈ Bi

Z ≽ 0M +2



B bT



b ≽ 0L+M +1 . 1

(45)

The SDP optimization problem of (45) is convex and can be solved with well known algorithms such as interior point methods which are self initialized and requires no initialization from the user. Extracting from Z and b we can obtain the location estimates X along with the transmit powers of the source nodes. By considering the unknown transmit power, so the SDP solution to (45) is also called as uWLS-SDP algorithm. The weight matrix Σ is determined by the variable λi,j which is unknown in the beginning. We can preliminarily consider λi,j as identical and obtain the initial estimate. Then putting the initial estimate into this optimization problem would produce better estimates. 5. Complexity analysis The number of equality constraints is denoted as m, the number of the SOCP and SDP constraints are denoted as Nsocp socp

sdp

and Nsdp , respectively. The corresponding dimension of the ith SOCP and SDP cone is denoted as ni and ni , respectively. To compare the computational cost for WLS-SOCP, WLS-SDP, WSLS-SDP, WSLS-SOC/SDP, uWLS-SOCP, uWLS-SDP, uWSLSSDP and uWSLS-SOC/SDP algorithms, Table 1 shows the number of variables, equality constraints, SOCP constraints, SDP constraints and dimension of SOCP and SDP cone for these different algorithms. The complexity of these proposed algorithms is calculated as a function of M, the number of source nodes, N, the number  of anchor nodes,  and L, the total number of connections. For a network with full connectivity, we can obtain that L = M N + (M − 1)/2 .

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Table 1 Parameters in computing the computational complexity. socp

sdp

Methods

Variables

m

Nsocp

ni

Nsdp

ni

WLS-SOCP WLS-SDP WSLS-SDP proposed in [33] WSLS-SOC/SDP uWLS-SOCP uWLS-SDP uWSLS-SDP proposed in [33] uWSLS-SOC/SDP

4L + 1 (L + 1)2 + (M + 2)2 (M + 2)2 + 4L (M + 2 )2 + L + 1 4L + M + 1 (L + M + 1)2 + (M + 2)2 (M + 2)2 + 4L + M (M + 2 )2 + L + M + 1

L L+4 2L + 3 L+3 L L+4 2L + 3 L+3

L+1 0 0 1 L+1 0 0 1

L + 1, L of size 3 0 0 L+1 L + 1, L of size 3 0 0 L+1

0 2 L+1 1 0 2 L+1 1

0 L + 1, M + 2 L of size 2, M + 2 M +2 0 L + M + 1, M + 2 L of size 2, M + 2 M +2

(a) Locations in the first scenario.

(b) Locations in the second scenario.

Fig. 1. The RMSE of the proposed algorithms versus the standard deviation of shadowing.

As can be seen from Table 1, the WLS-SDP algorithm has (L + 1)2 + (M + 2)2 variables, while the proposed WLS-SOCP in (23) has only 4L + 1 variables. Moreover, the number of equality constraints for WLS-SOCP algorithm is also smaller than for the proposed WLS-SDP algorithm. When WSLS-based optimization function is proposed to relax into convex problem, the number of variables for the WSLS-SOC/SDP algorithm is (M + 2)2 + L + 1. However the proposed WSLS-SDP algorithm has (M + 2)2 + 4L variables. When there are a large number of connections, it means than L is greatly larger than M. So the number of variables for the WSLS-SDP algorithm is greatly larger than that of the WSLS-SOC/SDP algorithm. The number of equality constraints for WSLS-SOC/SDP algorithm is L + 3, which is also smaller than 2L + 3, the number of equality constraints for WSLS-SDP algorithm. When the transmit powers of source nodes are assumed to be unknown, the number of variables for the corresponding algorithms is increased by M due to the unknown M transmit powers of source nodes. Similarly the number of variables, equality constraints for uWSLS-SOC/SDP algorithm are always smaller than for the proposed uWLS-SDP and uWSLS-SDP algorithms. So the SOC/SDP algorithm has a smaller number of variables and equality constraint metric compared with the SDP algorithm. A convex optimization problem can be solved by iterative optimization techniques, e.g., interior-point

Nsocp socp + i = 1 ni √ √  3 + m ) M log(1/ϵ) , where M log(1/ϵ) is the required least iterations, ϵ is the accuracy

methods. As is known, the worst-case complexity of solving the mixed SOCP and SDP algorithm is O (m2



Nsdp 2

sdp2

Nsdp

sdp3

m + m i=1 ni i = 1 ni socp of the convex optimization solution. The complexity of solving the mixed SOC/SDP is linear with ni , while that of solving sdp

the SDP is quadratic in ni . Using the parameters listed in Table 1, we can further calculate the computational complexity of different algorithms. The detailed calculation of computational complexity has not been derived here, but it is apparently shown that the complexity of mixed SOCP/SDP algorithm is evidently lower than that of SDP algorithm. 6. Evaluation In this section simulations are conducted to evaluate the performance of these proposed algorithms with two different scenarios by assuming the transmit power to be known or unknown. In the first scenario, five anchor nodes were placed regularly on the corners and in the center of a square 100 m × 100 m and ten source nodes were randomly distributed in the region. The geographic locations of the source nodes and anchor nodes are shown as in Fig. 1(a) in the first scenario. In the second scenario, the locations of the source nodes are the same as in Fig. 1(a), but the anchor nodes were deployed

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Table 2 Performance and complexity comparison of different algorithms for the network in Fig. 1(a). Algorithms

Variables

Equality constraints

RMSE

CPU runtime (s)

WLS-SDP WSLS-SDP proposed in [33] WSLS-SOC/SDP uWLS-SDP uWSLS-SDP proposed in [33] uWSLS-SOC/SDP

9 360 524 240 11 169 534 250

99 193 98 99 193 98

8.17 9.08 9.05 8.43 9.63 9.73

0.79 0.17 0.10 1.62 0.43 0.28

irregularly. Fig. 1(b) shows the configuration of the network in the second scenario. The transmit power can be adjusted with different antenna gain and set at −45 dB in the simulations. Full connectivity is initially assumed, meaning that each source node can be connected to all anchor nodes and also to all other source nodes. The path loss exponent β is set to 4, unless otherwise noted. The proposed SOCP, SDP and SOC/SDP are all implemented by the CVX toolbox using SeDuMi as the solver. 6.1. Impacts of shadowing Considering the configuration of the network given in Fig. 1, we perform Monte Carlo simulations with 500 ensemble runs to evaluate the root mean square error (RMSE) of the location estimation. Each standard deviation δi,j of the shadowing is varied from 0.5 to 4 dB. By averaging over all estimated source locations and noise realizations, the RMSEs for the two different scenarios are plotted in Fig. 2(a) and (b) where the CRLB or uCRLB denotes the Cramér–Rao Lower Bound of location estimation with known or unknown transmit power, respectively. The CRLB and uCRLB are calculated according to that of [33]. The proposed WLS-SOCP and uWLS-SOCP are no longer applicable here because in its formulation, SOCP position estimates always lie within the convex hull of the anchor positions. It can be seen from Fig. 2(a) that the RMSE performance degrades as the shadowing increases. The proposed algorithms perform well especially when shadowing is very small. The RMSE of WLS-SDP algorithm is less and more close to the CRLB performance compared with the WSLS-SDP and WSLS-SOC/SDP algorithms. When the standard deviation of the shadowing is set to 4 dB, the RMSEs of the WSLS-SDP and WSLS-SOC/SDP are 9.1 m and 8.9 m, respectively. However the RMSE of the WLS-SDP is only 8.1 m when the standard deviation of the shadowing is also set to 4 dB. The WSLS-SDP and WSLS-SOC/SDP algorithms can achieve almost the same estimation accuracy for using the similar optimization function. When the anchor nodes are placed in the irregular shape shown as in Fig. 1(b), the situation of the three algorithms is similar with the first scenario. Compared with the uWSLS-SDP and uWSLS-SOC/SDP algorithms, the uWLS-SDP also performs better due to the less approximation error when the transmit power is assumed to be unknown. However the irregular deployment of anchor nodes leads that the RMSE of each algorithm is far from the corresponding CRLB compared with the regular deployment. For instance, when the standard deviation of the shadowing is set to 4 dB, the RMSE of WLS-SDP is 8.1 m and very close to 7.3 m of the CRLB in the first scenario. However for the second scenario, the RMSE of WLS-SDP is 9.2 m and far from 5.8 m of the CRLB. 6.2. CPU runtime Another interesting performance is the CPU runtime which reveals the computational complexity of the proposed algorithms. When the network configuration is shown as in Fig. 1(a) by considering the full measurements of all the source nodes and anchor nodes, the number of measurements, variables, equality constraints, RMSE and average CPU runtime are listed in Table 2. It is also shown that the RMSE of uWSLS-SOC/SDP algorithm is also close to that of the uWSLS-SDP algorithm. However the uWSLS-SOC/SDP algorithm runs faster than uWSLS-SDP algorithm due to less variables. In Table 2, we list the average running time of the considered algorithms. The running time is measured by averaging over 500 noise realizations when the standard deviation of the shadowing is set to 4 dB. The average running time of WSLS-SDP algorithm is 0.17 s. However the average running time of WSLS-SOC/SDP algorithm is reduced to 0.10 s. When the transmit powers are unknown, the uWSLS-SOC/SDP should be run twice of the SOC/SDP calculation because of the estimation of the weighting terms. On the other hand, there are more unknown parameters to be estimated for considering the unknown transmit powers. Therefore, the required running time of uWSLS-SOC/SDP is more than twice as that of WSLS-SOC/SDP. Although the RMSE of WLS-SDP algorithm is smaller, the complexity of WLS-SDP algorithm is evidently higher than that of WSLS-SDP and WSLS-SOC/SDP algorithms. 6.3. Weighting term Above proposed algorithms rely on the weighting terms which are determined by the transmit powers. However when the transmit powers of source nodes are unavailable, uWSLS-SDP, uWSLS-SOC/SDP and uWLS-SDP algorithms initially set the value of the weighting terms to one and estimate the transmit powers (here the corresponding algorithms are called as uSLS-SDP, uSLS-SOC/SDP and uLS-SDP, respectively). Then using the approximate estimates of transmit powers to relocate

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(a) The first scenario.

)

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(b) The second scenario. Fig. 2. RMSE of estimated source location versus the standard deviation of shadowing.

Fig. 3. Evaluation of the weighting terms.

the source nodes we obtain more accurate source locations by employing the weighting terms. Due to the estimation of the weighting terms, the uWSLS-SDP, uWSLS SOC/SDP and uWLS-SDP algorithms run twice of the corresponding uSLS-SDP, uSLS-SOC/SDP and uLS-SDP as for the same optimization model. Similarly when each standard deviation of the shadowing is varied from 0.5 to 4 dB, Fig. 3 plots the RMSE performance of the proposed algorithms for the first and the second scenario. Considering the weighting terms to one, the uSLS-SOC/SDP algorithm performs worse compared with the uWSLS-SOC/SDP. For instance, the RMSE of uSLS-SOC/SDP is 18.5 m in the first scenario when the standard deviation of the shadowing is 4 dB. However the RMSE of uWSLS-SOC/SDP is only 11.1 m when the standard deviation of the shadowing is also 4 dB. Compared with the uLS-SDP, the RMSE of the uWLS-SDP is less due to the right weighting terms. When the standard deviation of the shadowing is set to 4 dB, the RMSEs of uWLS-SDP and uLS-SDP are 12.1 m and 21.0 m in the second scenario, respectively. 6.4. Estimated transmit power When the transmit power is estimated along with the locations of the source nodes the transmit power is assumed to be known. To test the RMSE performance of the estimated transmit power, the standard deviation of the shadowing is also varied from 0.5 to 4 dB. When the anchor nodes are placed regularly in the first scenario, Fig. 4 plots the RMSE of the estimated transmit power with the uWSLS-SDP, uWSLS-SOC/SDP and uWLS-SDP algorithms. As can be seen, the RMSE performance also becomes worse as the standard deviation of the shadowing increases. For instance, the RMSE of the uWLSSDP is 0.2 dB when the standard deviation of the shadowing is set to 0.5 dB. However when the standard deviation of the

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Fig. 4. RMSE of estimated transmit power versus the standard deviation of shadowing.

Fig. 5. RMSE of estimated source location versus communication radius.

shadowing is increased to 4 dB, the RMSE of the uWLS-SDP is also increased to 1.7 dB. Compared with the uWSLS-SDP and uWSLS-SOC/SDP, the uWLS-SDP provides better accuracy performance for the estimate of transmit power. 6.5. Communication radius In the previous simulations, we assumed that the network has full connectivity. However, this assumption is not valid in all practical cases. Only if the range measurement of nodes is less than communication radius, there is a connection. When each standard deviation of the shadowing is set to 1 dB and the communication radius is varied from 50 to 90 m, Fig. 5 shows the RMSE of the proposed algorithms versus the communication radius for network configuration in the first scenario. With the increasing of communication radius, there are more connections in the network configuration. So the RMSE performance performs well when the communication radius is large. Fig. 5 shows that by increasing the communication radius, the estimation accuracy improves, as expected. When the transmit power is assumed to be unknown, the uncertainty of estimated parameters is increased. So the transmit power of the uCRLB has more effect than that of the CRLB when the communication radius is small. 6.6. Path loss exponent In this subsection, we investigate the effect of path loss exponent (PLE) on the performance of the proposed algorithms. Similarly the network configuration is set as Fig. 1(a). The standard deviation of the shadowing is also set to 1 dB. When PLE β is varied from 2 to 6, Fig. 6 plots the RMSE performance versus PLE β . As can be seen, the RMSE performance of

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Fig. 6. RMSE of estimated source location versus PLE β .

all the algorithms degrades, especially at small values of PLE β . The RMSE performance of WLS-SDP always approaches the Cramér–Rao Lower Bound which provides the optimal location accuracy. The RMSE of WSLS-SCOP/SDP or uWSLS-SOCP/SDP is greatly larger than that of WLS-SDP especially at small values of PLE β . However the WLS-SDP runs slower than WSLSSCOP/SDP or uWSLS-SOCP/SDP. In Table 2, due to a larger number of variables the average running time of WLS-SDP is 0.79 s, which is greatly bigger than that of WSLS-SCOP/SDP or uWSLS-SOCP/SDP. 7. Conclusion In this paper by applying the WLS and WSLS optimization functions, the WSLS-SOC/SDP and WLS-SDP algorithms are proposed for cooperative RSS-based sensor localization. Then the algorithms are extended to scenario of unknown transmit power, the uWSLS-SOC/SDP and uWLS-SDP algorithms are designed by relaxing the nonconvex optimization model into convex problem. Compared with the existed WSLS-SDP algorithm, the computation complexity of the WSLSSOC/SDP algorithm is lower due to less variables and equality constraints. However the WSLS-SDP algorithm achieves similar accuracy performance with the WSLS-SDP algorithm for the same optimization function. The WLS optimization function has less approximation error, so the RMSE performance of WLS-SDP is better than that of the WSLS-SOC/SDP and WSLSSDP algorithms. However the WLS-SDP runs slower than WSLS-SOC/SDP and WSLS-SDP due to a large number of variables produced in the convex optimization relaxation. Acknowledgments This study is supported by Zhejiang provincial Natural Science Foundation LY16F020036, NSF China Major Program 61190114 and ZAFU Scientific Research Development Foundation Project 2013FR086. Appendix. Derive for rank of matrix D Every principal submatrix of positive semidefinite matrix D is itself a positive semidefinite matrix. Then we can obtain that

[D]p,p [D]L+1,L+1 ≥ [D]2L+1,p p = 1, 2, . . . , L.

(A.1)

Since [D]L+1,L+1 = 1, (A.1) can be rewritten as

[D]L+1,p ≤

 [D]p,p p = 1, 2, . . . , L.

(A.2)

The target function of (28) can be further represented as L  p=1

gi2,j ([D]p,p − 2λi,j [D]L+1,p ) ≥

L  p=1

gi2,j ([D]p,p − 2λi,j

 [D]p,p )

(A.3)

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[D]p,p , p = 1, 2, . . . , L. So we conclude that when [D]L+1,p where the condition of equality is [D]L+1,p = (28) achieves minimum value. Similarly a principal submatrix of positive semidefinite matrix D corresponding to rows and columns p, q and L + 1 is a positive semidefinite matrix. This implies that

 = [D]p,p ,





[D]p,p  [D]p,q  [D]p,p

[D]p,q [D]q,q [D]q,q

  [D]p,p [D]q,q  ≽ 03

(A.4)

1

where p < q, p, q = 1, 2, . . . , L. Then (A.4) can be rewritten as



0 [D]p,q − [D]p,p [D]q,q

[D]p,q −



[D]p,p [D]q,q



0

The latter condition is satisfied if and only if [D]p,q = when (28) is optimal.



≽ 02 .

(A.5)

[D]p,p [D]q,q . So we further conclude that the matrix D has rank one

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