A method for nonsmooth equation system
6.2. Sensor localization problem
6.2.1. Problem formulation
A wireless sensor network (WSN) consist of spatially distributed autonomous sensor to monitor physical or environmental conditions, such as temperature, sound, pressure, etc.
and to cooperatively pass their data through the network to main location. The develop-ment of wireless sensor network was motivated by military application such as battlefield surveillance, now such networks are used in many industrial applications, such as industrial process, monitoring and control, machine health monitoring.
The wireless sensor network is built of “nodes” from a few to several hundred or even thousands, where each node is connected to one or several sensors. Wireless sensor network presents novel tradeoffs in system design. On the one hand, the low cost of nodes facilitates
Table 6.1.: Numerical results for Problem 6.4.
Number of atoms Best known value Best obtained value
2 -1.000000 -1.000000
3 -3.000000 -3.000000
4 -6.000000 -6.000000
5 -9.103852 -9.103852
6 -12.712062 -12.712062
7 -16.505384 -16.505384
8 -19.821489 -19.821489
9 -24.113360 -24.113360
10 -28.422532 -28.422531
11 -32.765970 -32.765970
12 -37.967600 -37.967600
13 -44.326801 -44.326801
14 -47.845157 -47.845156
15 -52.322627 -52.322627
16 -56.815742 -56.815741
17 -61.317995 -61.317994
18 -65.842309 -66.284568
19 -72.659782 -72.659782
20 -77.177043 -77.177042
21 -81.684571 -81.684571
22 -86.573675 -86.809782
23 -92.844461 -92.844461
massive scale and highly parallel computation. On the other hand, each node is likely to have limited power, limited reliability and only local communication with a modest number of neighbors. These application contexts and potential massive scale make it unrealistic to rely on careful placement or uniform arrangement of sensors. It is still impossible to localize each sensor by GPS because of the expensive cost and the limited power and memory of sensors.
This leads to the area of sensor localization problem which intend to localize position of each sensor in a network by giving measured distances between the connected pairs of sensors.
The distance information can be obtained by strategies like time of arrival (TOA), time-difference of arrival (TDoA) and received signal strength (RSS). Generally, there is some
degree of error in the distance information because of the inaccuracy of measurement and power (or memory) constraints. If without any sensor’s position beforehand, one can just estimate the relative position information of sensor network by giving measured distance of connected pairs of sensors. But for some networks, this is not good enough, such as networking for tracking, network for monitoring environmental information (temperature, sound levels, light, etc.). Thus, in order to obtain the absolute position of sensors, it should be assumed that we already know the position of a few anchor nodes. Those anchor nodes are called beacon.
For simplicity, let the sensors be placed on a plane. Suppose that we have m known points (anchors) which belong to A ={ak∈ R2|k = 1, 2, · · · , m} and n unknown points (sensors) which belong to S ={xi ∈ R2|i = 1, 2, · · · , n}. A collection of point-pair Neis defined as
Ne={{ak, xj}|ak ∈ A, xj ∈ S} ,
and a collection of point-pair Nu is defined as
Nu ={{xi, xj}|xi ∈ S, xj ∈ S} .
For a pair of points in Ne, we have a Euclidean distance measure dkj. For a pair of points in Nu, we have a Euclidean distance measure dij. Then, the sensor localization problem is to find position of sensors (xis), such that
∥ ak− xj ∥2= d2kj for any{ak, xj} ∈ Ne,
∥ xi− xj ∥2= d2ij for any{xi, xj} ∈ Nu.
(6.5)
For a small number of sensors, it might be possible to compute sensor locations by solving the system of equations (6.5). Sturmfels [111] proposed a method for solving polynomial equations. However, solving polynomial system can be very expensive when there are a lot
of sensors. Furthermore, this polynomial system may be inconsistent if the distances dkj or dij have errors, which often occur in practice.
The polynomial system of equations (6.5) can also be convert into the following equivalent global optimization problem, (6.6) or problem (6.7) is zero. Thus, the sensor localization problem is equivalent to find the global minimizer of problem (6.6) or problem (6.7). Note that the problem (6.6) is nonsmooth problem, while the problem (6.7) is quadratic smooth problem, but both of them are nonconvex.
6.2.2. Numerical results
In this subsection, we solve some test sensor localization problems. The test problem gen-erator is from SFSDP [59, 58, 60]. SFSDP is a MATLAB package for sensor localization problem using semidefinite programming (SDP). In SFSDP, there are four m-files, SFSDP-plus.m, SFSDP.m, generateProblem.m and test SFSDP.m. Among them, generateProblem.m is used to generate test problems. It is called by
[xM atrix0, distanceM atrix0] = generateProblem(sDim, noisyF ac, radiorange,...
noOf Sensors, anchorT ype, noOf Anchors, randSeed).
The explanation of the input and output parameters can be found in the code. Figure 6.2.2 illustrates an example of sensor network generate by m-file “generateProblem.m” with 50
sensors and 4 anchors.
Figure 6.2.: An example of sensor network with 50 sensors and 4 anchors.
In order to evaluate the numerical performance of different solvers on sensor localization problems, Kim [59] developed the root mean square distance,
RMSD =
where ˆxiis the true sensor location (for test problems, the true sensor location is known), x∗i is the approximate sensor location obtained by optimization solvers. The root mean square distance reflects the average error between true sensor locations and obtained sensor loca-tions. In order to evaluate the numerical performance of the hybrid quasisecant method for solving the molecular conformation problem, we apply measures such as average error and standard deviation of errors to investigate the statistical properties of the numerical results.
Following the notation of the root mean square distance, we consider ˆxi(i = 1, 2,· · · , n) as the true sensor locations, x∗i(i = 1, 2,· · · , n) as the approximation sensor locations ob-tained by optimization solver, and
di =∥ ˆxi− x∗i ∥, i = 1, 2, · · · , n
as the error between the corresponding sensors. Then the following notations are used in evaluation of the algorithm.
• dave: the average error of corresponding sensors;
• dstd: the standard deviation of errors of corresponding sensors.
In the following, we solve some test problems of sensor localization by the hybrid qua-sisecant method proposed in the Chapter 4. In order to see the whole picture, we test sensor networks with the number of nodes 20-50 (with increment 10), 100-500 (with increment 100), 1000 and 2000 , respectively.
Table 6.2.: Numerical results for sensor localization problem.
No. f∗ RMSD dave dstd
20 3.8751710× 10−9 1.1409744× 10−11 3.0889847× 10−5 1.4022229× 10−5 30 2.0706732× 10−9 9.5128657× 10−11 3.0889847× 10−5 1.4022229× 10−5 40 2.3013408× 10−9 4.8699112× 10−10 1.7453671× 10−5 1.3676125× 10−5 50 1.8709465× 10−9 1.1781585× 10−09 1.7453671× 10−5 1.3676125× 10−5 100 5.8333877× 10−9 8.2989634× 10−10 1.7453671× 10−5 1.3676125× 10−5 200 2.4626913× 10−9 2.6716611× 10−10 1.7453671× 10−5 1.3676125× 10−5 300 1.3537058× 10−8 1.0683495× 10−10 1.7453671× 10−5 1.3676125× 10−5 400 1.3448554× 10−8 1.1369356× 10−10 9.8926555× 10−6 3.9835417× 10−6 500 4.4267586× 10−8 1.2075006× 10−09 3.4372143× 10−5 5.1096592× 10−6 1000 1.5312145× 10−6 2.2130698× 10−08 1.4392109× 10−4 3.7667449× 10−5 2000 1.3000758× 10−6 9.6764080× 10−09 9.6162666× 10−5 2.0721106× 10−5
The results are presented in Table 6.2. From the table, we can see that the hybrid qua-sisecant method solves all problems with high accuracy, however as the number of nodes increase the accuracy becomes slightly lower than that for small number of nodes. The root mean square distance of each problem are still extremely small, which implies that the hy-brid quasisecant method is promising in solving the sensor localization problem with large number of nodes. The average and standard deviation of distance between obtained sensor localization and true sensor localization are very small, which yields the robustness of the hybrid quasisecant method.
Figures 6.3 and 6.4 demonstrate the results of sensor localization problems solved by the hybrid quasisecant method with 50 and 400 sensors, respectively. From Figures 6.3(b) and 6.4(b), we can see that all the true sensor locations (represented by “*”) are well estimated by obtained sensor locations (represented by “o”).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(b) Approximation of obtained sensor localization with the true sensor localization
Figure 6.3.: Results of sensor network with 50 sensors and 4 anchors.
Figure 6.5 depicts the decline of the objective function value when number of sensors equals 1000 and 2000. From Figure 6.5(a), the optimal solution is obtained after 20 iterations by the global search. From Figure 6.5(b), the optimal solution is obtained after 194 iterations by the global search. These two figures show that the global search strategy used in the hybrid quasisecant method is able to find good starting points (lower basins) for local search.
6.3. Conclusion
In this Chapter, we apply the hybrid quasisecant method proposed in Chapter 4 to solve some practical problems, more specifically, molecular conformation and sensor localization
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(b) Approximation of obtained sensor localization with the true sensor localization
Figure 6.4.: Results of sensor network with 400 sensors and 4 anchors.
problems. For the molecular conformation problem, the hybrid quasisecant method success-fully solved all the test problems with number of atoms from 2 to 23. Better solutions were obtained for problems with number of atoms 18 and 22. For the sensor localization problem, the hybrid quasisecant method successfully solved all problems. In these sensor localization problems, the number of sensors is up to 2000, i.e., the number of variables is up to 4000.
From the numerical performance of the hybrid quasisecant method for solving molecular conformation and sensor localization problems, we can conclude that the hybrid quasisecant method is efficient and robust for practical applications.
0 5 10 15 20 25 0
1 2 3 4 5 6 7
Index of global search
Function value
(a) Number os sensors is 1000
0 20 40 60 80 100 120 140 160 180 200
0 5 10 15 20 25 30 35 40
Index of global search
Function value
(b) Number of sensors is 2000
Figure 6.5.: Decrease of the objective function value