Edge-Based Semidefinite Programming Relaxation of ... - CiteSeerX

Edge-Based Semidefinite Programming Relaxation of Sensor Network Localization with Lower Bound Constraints February 10, 2010

Ting Kei Pong1 Abstract The sensor network localization problem has been much studied. Based on the edge-based semidefinite programming relaxation (ESDP) recently proposed by Wang, Zheng, Boyd, and Ye [28], we consider strengthening the relaxation by adding lower bound constraints. We study some key properties of lower-bound-constrained ESDP relaxation (ESDPlb ), showing that, when distances are exact, zero individual trace is necessary and sufficient for a sensor to be correctly positioned by an interior solution. To extend this characterization of accurately positioned sensors to the noisy case, we propose a noise-aware version of ESDPlb (ρ-ESDPlb ) for which small individual trace is necessary and sufficient for a sensor to be accurately positioned by a certain analytic center solution, assuming the noise level is sufficiently small. We then propose a postprocessing heuristic based on ρ-ESDPlb . In simulations, when applied to a solution obtained by solving ρ-ESDP proposed by Pong and Tseng [21], it usually improves the solution accuracy by at least 10%; and it provides a certificate for identifying accurately positioned sensors in the refined solution, which is not common for existing refinement heuristics. Moreover, the method can distribute its computation over the sensors via local communication, making it practical for positioning and tracking in real time. Key words. Sensor network localization, semidefinite programming relaxation, error bound, log-barrier, coordinate gradient descent.

1

Introduction

A problem that has received considerable attention is that of ad hoc wireless sensor network localization [2, 4, 6–18, 20–23, 25, 26]. In the basic version of this problem, we have n distinct points in Rd (d ≥ 1). We are given the Cartesian coordinates of the last n − m points (called “anchors”) xm+1 , . . . , xn , and an estimate dij > 0 of the Euclidean distance between “neighboring” points i and j for all (i, j) ∈ A, where A ⊆ ({1, . . . , m}×{1, . . . , n}) ∪ ({1, . . . , n}×{1, . . . , m}).2 We wish to estimate the Cartesian coordinates of the first m points (called “sensors”). This problem is closely related to distance geometry problems arising in the determination of protein structure [5, 19] and to graph rigidity [1, 26]. The sensor network localization problem is NP-hard in general [19, 24]. This can be proved for d = 1 by reduction from the set partition problem, and the proof readily extends for d > 1; also see [2, 22] for related studies. Thus, efforts have been directed at solving this problem approximately. These include heuristics based on Euclidean geometry, shortest path, and local improvement; see for example [22, 23] and references therein. A different approach involves solving a convex relaxation, and then refining the resulting solution through local improvement. This has been effective in simulation and, under appropriate assumptions, the solution is provably exact/accurate. In the edge-based semidefinite programming (ESDP) relaxation approach of Wang et al. [28], the sensor network localization problem is formulated as the following nonconvex minimization problem: X min |kxi − xj k2 − d2ij |, (1) x1 ,...,xm (i,j)∈A

1 Department of Mathematics, University of Washington, Seattle, Washington ([email protected]) 2 The set A is undirected in the sense that (i, j) = (j, i) and d = d ij ji for all (i, j) ∈ A.

1

98195,

U.S.A.

¡ where k · k denotes the Euclidean norm. Letting X := x1 · · · matrix, the ESDP relaxation is X ¯ ¯ ¯`ij (Z) − d2ij ¯ min Z

(i,j)∈A

µ

s.t. Z =

Y X

XT Id

¶ ,

 yii yij xTi yij yjj xTj  º 0 xi xj Id ¶ µ yii xTi º0 xi Id 

¢ xm and Id denote the d × d identity

∀(i, j) ∈ As ,

(2)

∀i ≤ m,

¡ ¢ where Y = yij 1≤i,j≤m , “s.t.” is short for “subject to”, `ij (Z) :=

( yii − 2yij + yjj yii −

2xTi xj

if i, j ≤ m; 2

+ kxj k

if i ≤ m < j,

(3)

and As := {(i, j) ∈ A | i, j ≤ m}. Properties of its solutions are studied in [21, 28].3 In particular, the notion of individual traces, tri (Z) := yii − kxi k2 , i = 1, . . . , m, first introduced by Biswas and Ye [6, Section 4] for the SDP relaxation, is adapted to the ESDP relaxation to test for solution accuracy. In practice, measured distances may be inexact, i.e., true

d2ij = kxi

true

− xj k2 + δij

∀(i, j) ∈ A, true

(4)

where δ = (δij )(i,j)∈A ∈ R|A| denotes the measurement noise, and xi denotes the true position of the true ith point (so that xi = xi for i > m); see [11, Eq. (2)], [12, Eq. (3a)–(3f)], [18, Section 2]. Methods for sensor network localization can be highly sensitive to such noises. A noise-aware variant of the ESDP relaxation (ρ-ESDP) was introduced in [21]. Exploiting the partial separable structure of the formulation, a distributed algorithm has been developed to solve ρ-ESDP. For other convex relaxations, see for example [7, 14, 20, 27]. Various local refinement heuristics have been proposed to refine the solutions obtained from solving the convex relaxations. In [3, 7], the SDP relaxation is solved and the solution is used as a starting point for a local gradient descent algorithm to refine the solution. In [14], the authors also solved the SDP relaxation but they used the matlab function nsqnonlin to refine the solution, while in [21], the ρ-ESDP is considered and a local distributed gradient descent algorithm is used to refine the solution. These heuristics work well in simulation, but there is no certificate to tell which sensor position is improved after refinement. The convex relaxations can be strengthened by adding lower bound constraints; see [6, 8, 9, 15]. In this approach, the basic problem (1) becomes: X ¯ ¯ ¯kxi − xj k2 − d2ij ¯ min x1 ,...,xm

s.t. 3 Throughout,

(i,j)∈A

¯ kxi − xj k ≥ rij ∀(i, j) ∈ A¯1 ∪ A¯2 =: A.

“solution” of an optimization problem means a globally optimal solution.

2

where A¯1 ⊆ {(i, j) : i, j ≤ m, (i, j) ∈ / A}, A¯2 ⊆ {(i, j) : i ≤ m < j, (i, j) ∈ / A}, r = (rij )(i,j)∈A¯ is an estimated lower bound. A lower-bound-constrained ESDP relaxation (ESDPlb ) is as follows: X ¯ ¯ ¯`ij (Z) − d2ij ¯ min Z

(i,j)∈A

µ

¶ Y XT X Id 2 `ij (Z) ≥ rij   yii yij xTi yij yjj xTj  º 0

s.t. Z =

µ

xi yii xi

xj Id ¶ xTi º0 Id

∀(i, j) ∈ A¯

(5)

∀(i, j) ∈ As ∪ A¯1 ∀i ≤ m,

where As is defined as in the remark after (3). Notice that the objective function and the constraints in (5) do not depend on yij , (i, j) 6∈ As ∪ A¯1 . The following example shows that addition of suitable lower bound constraints will lead to an improvement in localization accuracy. This justifies our consideration of the lower bound constraints. true

Example 1.√Consider n = 4, m = 1. Let x1 = (0, 1)T , x2 = (−1, 0)T , x3 = (1, 0)T , x4 = (0, −1)T , d12 = d13 = 2, d14 is not known and r14 = 2. Then the problem becomes min |y11 − 2xT2 x1 + kx2 k2 − 2| + |y11 − 2xT3 x1 + kx3 k2 − 2| Z µ ¶ y11 xT1 s.t. Z = º0 x1 I2 y11 − 2xT4 x1 + kx4 k2 ≥ 4.

(6)

  1 0 1 We claim that Z0 := 0 1 0 is the unique solution. To this end, note that Z0 is clearly feasible with 1 0 1 objective value zero. This shows that Z0 is an optimal solution of (6) and the optimal value is zero. Since the optimal value is zero, from |y11 − 2xT2 x1 + kx2 k2 − 2| + |y11 − 2xT3 x1 + kx3 k2 − 2| = 0, we see that for any optimal solution, y11 = 1 and x1 = (0, t)T for some t ∈ [−1, 1]. Next, we have from y11 − 2xT4 x1 + kx4 k2 ≥ 4 that 1 + 2t + 1 ≥ 4 ⇒ t ≥ 1. Hence, t = 1, showing that Z0 is the unique solution to (6). The true position of x1 is recovered. On the other hand, if no lower bound constraint is imposed, it is easy to see from the above derivation that the solution set is     1 0 t  0 1 0 : t ∈ [−1, 1] .   t 0 1 Thus, the true position of x1 cannot be recovered without the lower bound constraint. Despite Example 1, solving (5) with all available lower bound constraints takes much more time than solving (2). A strategy in [8] adapted to our context suggests that one first solves (2), and then solves (5) with only the violated lower bound constraints4 . But this still takes at least double the time to solve 4 We

say that a lower bound constraint corresponding to (i, j) ∈ A¯ is violated by Z if kxi − xj k < rij .

3

a problem. It is advantageous to be able to incorporate lower bound constraints and yet not significantly increase solution time. Our aims are three-fold. First, we study approximation accuracy of (5) as measured by the individual traces of interior solutions5 . We show that, in the noiseless case, zero individual trace of some interior solution is a necessary and sufficient condition for xi to be invariant over the solutions of (5), provided that the lower bound estimates are realistic (see Definition 1); see Theorem 1. This is a generalization of [21, Theorem 1]. Second, we propose a noise-aware variant of ESDPlb (ρ-ESDPlb ). We show that xi is accurately positioned if the ith individual trace of some analytic center solution is small, when noise is small; and that the localization error is bounded above in the order of the square root of the trace of this solution; see Theorem 3. We then propose an algorithm to compute this analytic center solution in Section 5. Our algorithm is distributable, and hence potentially implementable in real time. Third, we propose in Section 6.3 a new refinement heuristic based on ρ-ESDPlb for solutions of ρ-ESDP. This involves solving ρ-ESDPlb on the reduced network obtained by fixing sensors accurately positioned by ρ-ESDP as new anchors. The trace test can be used to determine which sensors are accurately positioned after refinement: this provides an accuracy certificate that does not commonly exist for refinement heuristics. Simulation shows that when the number of accurately positioned sensors identified is large enough (> 0.7m), there is usually a 10 to 20% improvement in solution accuracy, while the solution time does not increase significantly; see Section 6.3. Throughout, S n denotes the space of n × n real symmetric matrices, and T denotes transpose. For a vector x ∈ Rp , kxk and kxk∞ denote the Euclidean norm of x and the ∞-norm of x, respectively. For A ∈ Rp×q , aij denotes the (i, j)th entry of A, and kAkF denotes the Fr¨obenius norm of A. For A, B ∈ S p , ¡ ¢ A º B means A − B is positive semidefinite. For A ∈ S p and I ⊆ {1, . . . , p}, AI = aij i,j∈I denotes the principal submatrix of A comprising the rows and columns of A indexed by I. We will abbreviate “m + 1, m + 2, . . . , m + d” as “m+ .” Thus, Z{i,j,m+ } and Z{i,m+ } are, respectively, the (2 + d) × (2 + d) and (1 + d) × (1 + d) principal submatrices of Z appearing in the second and third constraint of (2). For any finite set J , |J | denotes the cardinality of J . For any I ⊆ {1, . . . , m}, we shall make use of the following sets:

2

N (I) :=

{j ∈ / I | (i, j) ∈ A for some i ∈ I},

A(I) := A¯1 (I) :=

{(i, j) ∈ A | i ∈ I, j ∈ / I}, {(i, j) ∈ A¯1 | i ∈ I, j ∈ / I}.

Trace test for uniquely positioned sensors by ESDP

δ,r Let Sesdp denote the solution set of (5) with yij set arbitrarily to zero for all (i, j) 6∈ As ∪ A¯1 (see the remark following (5)), where dij is given by (4) and r is the lower bound estimates in the second constraint δ,r δ,r of (5). Note that Sesdp is a closed convex set and hence its relative interior ri(Sesdp ) is well defined. As in [26, Proposition 1] and [27, page 162], we make the the following assumption:

Assumption 1. Each connected component of the graph G := ({1, . . . , n}, A) contains an anchor index. In what follows, we define ¯ µ ½ ¯ Y δ,r ¯ Iesdp := i ∈ {1, . . . , m} ¯ xi is invariant over all Z = X

XT Id

¶

¾ ∈ Sesdp

Definition 1. The lower bound estimate r = (rij )(i,j)∈A¯ is said to be realistic if Z 5 Throughout,

we abbreviate relative interior solution as interior solution.

4

δ,r

true

. 0,r ∈ Sesdp .

δ,r In the case when δ = 0 and r is realistic, those sensors indexed by Iesdp are correctly positioned by solving (5). Thus it is of interest to identify these index sets. The following result can be proved similarly as [27, Proposition 4.1]; see also [28, Theorem 2]. ¯

δ,r Proposition 1. For any δ ∈ R|A| , r ∈ R|A| , Z ∈ ri(Sesdp ) and i ∈ {1, . . . , m}, if tri (Z) = 0, then δ,r i ∈ Iesdp . 0 0 The converse implication holds true when δ = 0 and A¯ = ∅. Let Sesdp and Iesdp denote the solution set of (2) and the set of indices of invariant sensors for (2) in the noiseless case respectively. 0 ) and i ∈ {1, . . . , m}, it Proposition 2 ( [21, Theorem 1]). Suppose that A¯ = ∅. For any Z ∈ ri(Sesdp 0 holds that tri (Z) = 0 if and only if i ∈ Iesdp . r denote the feasible set of (5) with yij set to zero for all (i, j) ∈ / As ∪ A¯1 ; here r represents the Let Fesdp r lower bound estimates in the second constraint. For any Z ∈ Fesdp , we have from the fourth constraint in (5) that tri (Z) ≥ 0, i = 1, . . . , m. r r For any Z, Z 0 ∈ Fesdp and any α ∈ [0, 1], we have Z α := αZ + (1 − α)Z 0 ∈ Fesdp and

tri (Z α ) = αtri (Z) + (1 − α)tri (Z 0 ) + α(1 − α)kxi − x0i k2 ,

i = 1, . . . , m.

r Thus each individual trace is a concave function on Fesdp . The following result follows from the concavity r and nonnegativity of the individual trace on Fesdp . ¯

δ,r ), then tri (Z) = 0 for all Lemma 1. For any δ ∈ R|A| , r ∈ R|A| , if tri (Z) = 0 for some Z ∈ ri(Sesdp δ,r Z ∈ Sesdp .

3

ESDPlb : realistic and noiseless case

Suppose there is no distance measurement noise and that the lower bound estimates are realistic. Then a result similar to Proposition 2 holds for ESDPlb . We first quote the following lemmas, which were proved for the case A¯ = ∅. The proofs extend readily to the case when r is realistic. For notational convenience, we denote à true ! ( true ´ ³ true 0 if (i, j) 6∈ As ∪ A¯1 ; true true true Y (X )T true Z := . X := x1 · · · xm , yij := true true true X Id else, (xi )T xj true

0,r r Since r is realistic, we have Z ∈ Sesdp ⊆ Fesdp . Lemma 2 and Lemma 3 follow readily from [21, Lemma 2] and [21, Lemma 3], respectively. r , we have Lemma 2. (a) For any Z ∈ Fesdp

µ

yii − kxi k2 yij − xTi xj

yij − xTi xj yjj − kxj k2

¶ º 0.

0,r (b) For any Z ∈ Sesdp and (i, j) ∈ As , if kxi − xj k = dij , then tri (Z) = trj (Z).

Lemma 3. Suppose δ = 0 and r is realistic. 0,r 0,r , we have tri (Z) = trj (Z) for all Z ∈ Sesdp . (a) For any (i, j) ∈ A with i, j ∈ Iesdp

5

0,r 0,r (b) For any (i, j) ∈ A with i ∈ Iesdp and j > m, we have tri (Z) = 0 for all Z ∈ Sesdp .

¯ A ∈ Rd×k , B, ¯ B ∈ Rk×k , and α ∈ [0, 1], we have upon letting Lemma 4 ( [21, Lemma 4]). For any A, ¯ + (1 − α)B that X α = αA¯ + (1 − α)A and Y α = αB Y α − (X α )T X α

¡ ¢ ¡ ¢ ¯ − A¯T A) ¯ + (1 − α)(B − AT A) + α(1 − α) A¯ − A T A¯ − A . = α(B

In the lower-bound-constrained case, the following lemma is crucial for the proof of our main theorem. Lemma 5. Suppose that δ = 0 and r is realistic. 2 0,r 0,r ) such that `ij (Z 0 ) = rij . Then tri (Z) = 0 and Z 0 ∈ ri(Sesdp (a) Suppose there exists (i, j) ∈ A¯2 , i ∈ Iesdp 0,r for all Z ∈ Sesdp . 0,r 0,r 2 (b) Suppose there exists (i, j) ∈ A¯1 , i, j ∈ Iesdp and Z 0 ∈ ri(Sesdp ) such that `ij (Z 0 ) = rij . Then 0,r tri (Z) = trj (Z) for all Z ∈ Sesdp . 2 0,r 0,r . Fix ), it follows from the assumption that `ij (Z) = rij for all Z ∈ Sesdp Proof. (a) Since Z 0 ∈ ri(Sesdp 0,r . Since Z any Z ∈ Sesdp

true

true

0,r , we have kxi ∈ Sesdp

true

0,r i ∈ Iesdp , it follows that kxi

true

2 − xj k2 = rij . On the other hand, since j > m and

true

− xj k = kxi − xj k. Hence, true

kxi k2 − 2xTi xj + kxj k2 = kxi − xj k2 = kxi

true

2 − xj k2 = rij = `ij (Z) = yii − 2xTi xj + kxj k2 ,

and thus tri (Z) = yii − kxi k2 = 0. This proves (a). 2 0,r 0,r (b) Similarly, we have from the assumption that `ij (Z) = rij for all Z ∈ Sesdp . Fix any Z ∈ Sesdp . Since Z true

kxi

true

true

0,r ∈ Sesdp , we have kxi

true

true

2 0,r . On the other hand, since i, j ∈ Iesdp , it follows that − xj k2 = rij

− xj k = kxi − xj k. Hence, true

kxi − xj k2 = kxi

true

2 − xj k2 = rij = `ij (Z).

As in the proof of [21, Lemma 2(b)], this implies tri (Z) = trj (Z). This proves (b). We are now ready to state our main theorem in this section: the converse of Proposition 1 in the noiseless case. The proof is given in Appendix B. 0,r 0,r Theorem 1. For any i ∈ Iesdp , we have tri (Z) = 0 for all Z ∈ Sesdp .

Theorem 1, together with Proposition 1, states that, in the noiseless case, xi is accurately positioned if and only if the ith trace is zero, provided that r is realistic. This completely characterizes when a sensor can be accurately positioned by solving ESDPlb in the realistic and noiseless case. How about the noisy case? Recall in [21, Section 4] that the trace test for accuracy can be uninformative when there is noise in the distance measurements for ESDP relaxation. Hence, in the next section, we develop a noise-aware variant of ESDPlb so that the trace test provably works under some assumptions on the noise.

4

A noise-aware ESDPlb

As in [21, Section 5], we consider a noise-aware variant of ESDPlb (ρ-ESDPlb ), while assuming r to be ρ,δ,r realistic. Let Sresdp denote the set of Z satisfying r Z ∈ Fesdp

and |`ij (Z) − d2ij | ≤ ρij

6

∀(i, j) ∈ A.

with ρ = (ρij )(i,j)∈A ≥ 0. This reduces to the ρ-ESDP proposed in [21] when A¯ = ∅. It is routine to ρ,δ,r show that Sresdp is uniformly bounded on {|δ| ≤ ρ ≤ e}, where e is the vector of all ones. Moreover, if ρ ≥ |δ| (i.e., ρij ≥ |δij | for all (i, j) ∈ A), then ρ,δ,r

0,r Sesdp ⊆ Sresdp .

For each |δ| < ρ, let Z min

ρ,δ,r Z∈S resdp

ρ,δ,r

be the unique solution of the following log-barrier problem: X

B(Z) := −

ln det(Z{i,j,m+ } ) −

¯1 (i,j)∈As ∪A

m X

X

ln tri (Z) − γ

2 ln(`ij (Z) − rij ),

(7)

¯ (i,j)∈A

i=1

where γ > 0 is a weight parameter. The parameter γ adjusts how much weight we put on satisfying the ρ,δ,r lower bound constraints. Since |δ| < ρ, there exists a Z ∈ Sresdp satisfying B(Z) < ∞ (e.g., take any 0,r Z ∈ Sesdp and increase yii , i = 1, . . . , m, by a sufficiently small amount). Moreover, the objective function ρ,δ,r

of (7) is strictly convex and Sresdp is compact. Hence Z ρ,δ,r

ρ,δ,r

ρ,δ,r

is well defined, unique, and B(Z

ρ,δ,r

) < ∞.

true

0 The following result shows that tri (Z ) ≈ 0 and xi ≈ xi whenever |δ| < ρ ≈ 0, for all i ∈ Iesdp . The proof follows a similar line of arguments as the proof of [21, Theorem 3]. Hence we only provide a sketch of the proof.

Theorem 2. (a) Every cluster point of {Z

ρ,δ,r

0,r }, as |δ| < ρ → 0, belongs to ri(Sesdp ).

0,r (b) For each i ∈ Iesdp ,

lim

|δ| rij ∀(i, j) ∈ A∗1 ∪ A∗2 ,

in place of (26) in [21]. We then define B a (Z) to be X X B a (Z) := − ln λi,j ln tri (Z) − k (Z) − (i,j,k)∈J a

0,r i∈I / esdp

X

2 ln(`ij (Z) − rij ),

∗ (i,j)∈A∗ 1 ∪A2

a ¯ with λi,j k (Z) and J defined as in [21]. We also use the following estimate for each (i, j) ∈ A, Ã ρ,δ,r ! µ ¶ ρ,δ,r ρ,δ,r Z + Z¯ 1 − ln `ij ≤ − ln `ij (Z ) = − ln(`ij (Z )) + ln 2, 2 2

(8)

0,r in addition to (28) and (29) in [21], where Z¯ ∈ ri(Sesdp ). The first inequality in (8) is a consequence of operator monotonicity of `ij (·). (b) The proof is the same as [21, Theorem 3(b)], except that we use Theorem 1 in place of [21, Theorem 1]. ρ,δ,r

0,r The following result shows that Iesdp is identified by those i with tri (Z ) ≈ 0 for any |δ| < ρ ≈ 0. ´ ³p ρ,δ,r true It also shows that the distance from xi to its true position xi is O tri (Z ρ,δ,r ) . The proof follows a similar line of arguments as [21, Theorem 4]. Hence we only provide a sketch of the proof.

7

Theorem 3. (a) There exists η¯ > 0, ρ¯ > 0 such that tri (Z tri (Z

ρ,δ,r

ρ,δ,r

) < η¯ for some |δ| < ρ ≤ ρ¯e

=⇒

0,r i ∈ Iesdp ,

) ≥ 0.1¯ η for some |δ| < ρ ≤ ρ¯e

=⇒

0,r i∈ / Iesdp ,

where e := (1, ..., 1)T ∈ R|A| . (b) There exists a K > 0 such that for i ∈ {1, . . . , m}, ρ,δ,r

kxi

true

− xi

³ ´ 12 ρ,δ,r k ≤ K tri (Z )

∀|δ| < ρ.

Proof. (a) Follow the same proof as [21, Theorem 4(a)] except that we use Theorem 2 in place of [21, Theorem 3]. ρ,δ,r (b) Use the same idea as [21, Theorem 4(b)], making use of the fact that Z solves max G(Z) :=

ρ,δ,r Z∈S resdp

5

Y

det Z{i,j,m+ }

¯1 (i,j)∈As ∪A

(r)

An LPCGD

m Y

tri (Z)

i=1

Y ¡

2 `ij (Z) − rij

¢γ

.

¯ (i,j)∈A

method for solving ρ-ESDPlb

The results of Section 4 suggest solving (7), with ρ > 0 small but above the noise level, and then checking the individual traces of the solution to determine which sensors are accurately positioned. We shall adapt the LPCGD method proposed in [21, Section 6] to solve (7). For completeness, we briefly describe the method below. We assume that r is realistic. For any scalar a > 0, let ha (t) :=

1 1 1 max{0, t − a}2 + max{0, −t − a}2 = max{0, |t| − a}2 . 2 2 2

For any ρ = (ρij )(i,j)∈A > 0, define the smooth convex penalty function fρ (Z) :=

X

hρij (`ij (Z) − d2ij ).

(9)

(i,j)∈A ρ,δ,r

r Then when ρ ≥ |δ|, Z ∈ Sresdp if and only if Z ∈ Fesdp and fρ (Z) = 0. We augment fρ by a scalar µ > 0 multiple of the log-barrier function B from (7):

fρµ (Z) := fρ (Z) + µB(Z). Then fρµ is convex, differentiable on domB, partially separable (i.e., a sum of functions, each of few variables), and fρµ (Z) → ∞ as Z approaches any boundary point of domB. A standard argument shows that, when ρ > |δ|, arg minZ fρµ (Z) → Z ρ,δ,r as µ → 0. If ρ 6> |δ|, then it is still true that every cluster point of arg minZ fρµ (Z) as µ → 0 is a solution of min fρ (Z).

Z∈F r

esdp

8

(10)

We denote by Zi the subvector of variables xi , yii , {yij | (i, j) ∈ As ∪ A¯1 } and by ∇Zi fρµ the gradient of fρµ with respect to Zi , i = 1, . . . , m. B is twice differentiable on domB, and we denote its Hessian with respect to Zi by ∇2Zi B. Although the quadratic penalty function ha is not twice differentiable, the generalized Hessian ∂ 2 ha is well defined and given by   if |t| > a;  1 ∂ 2 ha (t) =

[0, 1] if |t| = a;   0 else.

We make the (somewhat arbitrary) selection of 1 if |t| > a and 0 else. This yields, via (9) and the chain rule, a selection of ∂Z2 i fρ (Z), which we denote by Hi,ρ (Z). The corresponding selection of ∂Z2 i fρµ (Z) is µ Hi,ρ (Z) := Hi,ρ (Z) + µ∇2Zi B(Z). µ µ Since Hi,ρ (Z) º 0 and ∇2Zi B(Z) Â 0, we have Hi,ρ (Z) Â 0 for Z ∈ domB. Moreover, Hi,ρ (Z) has an “arrow” sparsity structure, so its Cholesky factorization can be efficiently computed in linear time. The algorithm is described below: final

0. Choose initial µ > 0 and Z ∈ domB with Z{m+ } = Id . Choose µ > 0 and a continuous function ψ : (0, ∞) → (0, ∞) such that limµ↓0 ψ(µ) = 0. Choose stepsize parameters 0 < β < 1 and 0 < ω < 12 . Go to step 1. 1. If there exists i ∈ {1, . . . , m} such that k∇Zi fρµ (Z)k > ψ(µ), then construct the block-coordinate generalized Newton direction: µ Di = −(Hi,ρ (Z))−1 ∇Zi fρµ (Z),

and repeat step 1 with Z

new

= Z[α],

where Z[α] is obtained from Z by replacing Zi with Zi + αDi and α is the largest element of {1, β, β 2 , · · · } satisfying fρµ (Z[α]) ≤ fρµ (Z) + αωDiT ∇Zi fρµ (Z). Otherwise, go to step 2. final

2. If µ ≤ µ

, then stop. Otherwise, decrease µ and return to step 1. (r)

We call this adaptation the LPCGD method. (r) The LPCGD method is highly parallelizable since, for any i, j ∈ {1, . . . , m} that share no neighbor and have no lower bound estimates, Zi and Zj share no variable and can be updated simultaneously. Moreover, the computation distributes over the sensors since each sensor i needs to communicate only with its neighbors, or, in the presence of lower bound constraints, with sensors 2 or 3 hops away in order to update Zi . This is an important practical consideration, especially when tracking the position of moving sensors in real time, since the coordination of communication/computation over all sensors is expensive and the graph topology may change; see [13, 18, 23]. Only the changing of µ needs centralized coordination among all sensors, but this needs to be done only infrequently. (r) ρ,0,r In the noiseless case, by setting ρ to be sufficiently small, the LPCGD method computes a Z , within a desired accuracy, that approximates closely an interior solution of (5); see Theorem 2.

9

6 6.1

Numerical simulations Problem generation true

true

To facilitate comparison with existing work, we follow [6, 7, 21, 27, 28] and generate x1 , · · · , xn independently according to a uniform distribution on the unit square [−0.5, 0.5]2 , and set m = 0.9n; true true we denote this type of network by unif. We also generate x1 , · · · , xn on the unit square with m ≈ 0.95n such that the anchors form a square grid; we denote this type of network by grid. We true true then set A = {(i, j) : kxi − xj k < rr}, and true

dij = kxi

true

− xj k · |1 + ²ij · σ|

∀(i, j) ∈ A,

where ²ij is a random variable, rr ∈ (0, 1) is the radio range, and σ ∈ [0, 1] is the noisy factor. As in [6, 7, 14, 21, 27, 28], each ²ij is normally distributed with mean 0 and variance 1. We use the parameter values of σ = 0, 0.01 and rr = 0.06 for n = 1000, rr = 0.035 for n = 4000 and rr = 0.02 for n = 10000; see Table 1. Finally, we take all i, j so that (i, j) ∈ / A but there exists k such that (i, k), (j, k) ∈ A. This essentially means that every sensor has access to lower bound estimates between itself and any sensors/anchors within 2-hops from it. Then we set A¯ to be the collection of all such edges, and set rij = rr

¯ ∀(i, j) ∈ A.

Then r is realistic. P 1 2 3 4 5 6 7 8 9 10 11 12

n 1000 1000 1000 1000 4000 4000 4000 4000 10000 10000 10000 10000

m 900 900 951 951 3600 3600 3804 3804 9000 9000 9516 9516

type unif unif grid grid unif unif grid grid unif unif grid grid

rr 0.06 0.06 0.06 0.06 0.035 0.035 0.035 0.035 0.02 0.02 0.02 0.02

σ 0 0.01 0 0.01 0 0.01 0 0.01 0 0.01 0 0.01

|A| 5423 5273 5143 5172 29998 29051 29704 29521 60941 61491 61255 61254

¯ |A| 9233 8727 8521 9245 58897 57980 57580 58621 114247 115550 115121 115067

Table 1: Input parameters for the test problems.

6.2

(r)

Implementation of LPCGD (r)

We coded in Fortran-77 the LPCGD final

µ

= 10−14 ,

method of Section 5, with initial µ = 0.1 and ( µ if µ > 10−7 ; β = 0.5, ω = 0.1. ψ(µ) = final 10−7 if µ ≤ µ ≤ 10−7 ,

(11)

µ We choose i in Step 1 in a cyclic order, compute Di using a Cholesky factorization of Hi,ρ (Z), and decrease µ by a factor of 10 in Step 2. These choices were made with little experimentation and can true conceivably be improved. We initialize xi = xi + ∆i , with the components of ∆i randomly generated from the square [−.2, .2]2 . We then set yii = kxi k2 + 1 and yij = xTi xj .

10

Since the Gaussian distribution has an unbounded support, the condition ρ > |δ| for ρ-ESDPlb is not guaranteed to hold for a fixed ρ > 0. As in [21, Section 7], we set ¶ µ 1 ρij = d2ij − 1 ∀(i, j) ∈ A, (12) (1 − 2ˆ σ )2 where 0 ≤ σ ˆ < 12 is our estimate of σ. If σ ˆ ≥ σ > 0, then ρij > |δij | for over 95% of the edges on average. (r) For each Z found by our LPCGD code, we judge a sensor i to be accurately positioned if 2 tri (Z) ≤ (a0 + a1 σ ˆ ) d¯i ,

(13)

P where d¯i := |N1(i)| j∈N (i) dij and a0 , a1 are positive constants. The test (13) is justified by Proposition 1, Theorems 1 and 3. Specifically, when δ = 0 and we set σ ˆ ≈ 0, we have Z approximately equal to 0,r 0 some Z0 ∈ ri(Sesdp ) and, by Proposition 1 and Theorem 1, i ∈ Iesdp if and only if tri (Z0 ) = 0, implying tri (Z) ≈ 0. When δ 6= 0 is sufficiently small and we set σ ˆ such that |δ| < ρ and ρ is sufficiently small, ρ,δ,r 0 we have from Theorem 3(a) that i ∈ Iesdp if and only if tri (Z ) is sufficiently small, implying tri (Z) is sufficiently small. We use the same constants a0 = 0.01 and a1 = 30 as in [21, Section 7]. We denote by map the number of sensors that are judged to be accurately positioned. We check the accuracy of these computed positions by computing the maximum error between them and the true positions: true errap = max kxi − xi k. i accurately positioned

For comparison, we also compute the maximum error and the root-mean-square deviation (RMSD) between computed positions and true positions of all sensors: err =

true

max kxi − xi

i=1,...,m

à RMSD

6.3

=

m

k,

true 1 X kxi − xi k2 m i=1

! 21 .

Simulation results: using ρ-ESDPlb as refinement heuristics

We derive from ρ-ESDPlb a refinement heuristics. ρ-ESDPlb is conceivably good for postprocessing solutions obtained from solving ρ-ESDP proposed in [21, Section 5] since it is stronger than ρ-ESDP and the trace test can serve as solution accuracy certificate, when noise is small; see Theorem 3. We first solve ρ-ESDP using the LPCGD algorithm as described in [21, Section 7], with the same preprocessing so that the number of neighboring sensors is below 10 for each i = 1, ..., m, for faster solution time. Then we postprocess the solution. More precisely, we consider the following 3 strategies: N Solve ρ-ESDP using the LPCGD algorithm, with number of neighboring sensors kept below 10 for each i = 1, ..., m. No refinement is done. F Solve ρ-ESDP using the LPCGD algorithm, with number of neighboring sensors kept below 10 for each i = 1, ..., m. To the solution obtained, fix sensors with individual traces satisfying (13) as new anchors. Then solve ρ-ESDP on the new network by the LPCGD algorithm, keeping number of neighboring sensors below 10. FL Solve ρ-ESDP using the LPCGD algorithm, with number of neighboring sensors kept below 10 for each i = 1, ..., m. To the solution obtained, fix sensors with individual traces satisfying (13) as new anchors. Take p% of the lower bound constraints that are most violated, include them as long as 11

the number of lower bound constraints added per sensor/anchor is below 10, and that the total number of lower bound constraints added is below m − map . Then solve ρ-ESDPlb on the new (r) network by the LPCGD algorithm, keeping the number of neighboring sensors below 10. Here, N stands for “No refinements”, F stands for “Fixing sensors” and FL stands for “Fixing sensors and adding Lower bound constraints”. We set p = 20 in FL for our simulation. Intuitively, a larger p means more lower bound constraints are incorporated (see vioused in Table 2). This implies that the refinement takes more time, but on the other hand, one should expect a more accurate solution. Lower bound constraints have also been considered in [8] for the SDP relaxation. We consider the following strategy VL adapted from strategy I1 there: VL Solve ρ-ESDP using the LPCGD algorithm, with number of neighboring sensors kept below 10 for each i = 1, ..., m. Take p% of the lower bound constraints that are most violated, include them as long as the number of lower bound constraints added is less than 10 per sensor/anchor, and that the (r) total number of lower bound constraints added is below m − map . Then solve (7) by the LPCGD algorithm, keeping the number of neighboring sensors below 10. Note that new anchors are NOT introduced in the second stage in VL. VL stands for “Variant on adding Lower bound constraints”. (r) Like LPCGD , the LPCGD algorithm is a Fortran implementation that solves ρ-ESDP. The Fortran codes were compiled by Gnu F-77 compiler (Version 3.4.6). All codes were run on a Dell POWEREDGE 1950, under Debian 4.0 Kernel Linux. In the simulation below, we also use initial µ = .1, (11), (12) and (13) for the LPCGD algorithm. In all of our tests below, we set σ ˆ = max{σ, 10−6 } in (12) and (13), hence assuming the best knowledge of noise. In strategy F, we fix sensors with small traces as new anchors and re-solve ρ-ESDP in the refinement stage. In theory, it should not bring about improvement in solution quality. However, in practice, we only obtain an approximate solution from the LPCGD algorithm. By adding new anchors and re-solving, we minimize a different objective function that comprises fewer log barrier functions in the refinement stage. In the noiseless case, intuitively, since σ ˆ ≈ 0, we are getting close to some interior solution of (2) by minimizing a sum of log barrier functions. By switching to a sum of fewer log barrier functions, we get close to the solution more quickly; see Table 2.6 In this case, strategy F can be thought of as a heuristic to solve ρ-ESDP more quickly and accurately. On the other hand, for the noisy case, we are approaching a different center solution in the refinement stage. We do not have a good explanation for the observed improvement in RMSD in the noisy cases. In strategy FL, we consider only violated lower bound constraints to exclude redundant lower bound estimates. The sum in (7) then comprises fewer log barrier functions, and this results in less solution time. On the other hand, distance measurements alone are likely insufficient to position those less accurately positioned sensors. Hence, putting more weight on satisfying the lower bound constraints in the refinement stage should decrease the RMSD. In simulation, it appears that with a larger γ in (7), the improvement in RMSD is larger, yet a longer solution time is required. We settled on γ = 5 after some experimentation. In contrast to strategy F, strategy FL is a heuristic to improve ρ-ESDP solution by solving a stronger relaxation, namely the ρ-ESDPlb . The simulation results are shown in Table 2. We see that, in general, the strategies F and FL lead to an improvement in the solution quality in terms of larger map , smaller err and smaller RMSD, with at least about 10% improvement in RMSD. Also, FL takes around 10% to 20% more time than F due to the 6 A supporting evidence of this is that, a solution with comparable RMSD and m ap to that obtained by F can be obtained by applying N with much smaller parameters than 10−7 and 10−14 (say 10−10 and 10−18 for Problem 1) in (11). With these parameters, we actually obtain a smaller errap . However, this approach takes significantly more time (around 15 seconds) and number of iterations to converge. Thus F is preferred.

12

P 1 2 3 4 5 6 7 8 9 10 11 12

N iter*/cpu/map /errap /err/RMSD 62/4/678/4.1e-3/.17/2.3e-2 26/2/698/3.7e-2/.17/2.2e-2 81/5/699/3.0e-3/.11/1.0e-2 43/3/753/2.3e-2/.06/9.7e-3 133/9/3337/3.9e-4/.12/6.0e-3 102/7/3276/1.1e-2/.06/6.0e-3 163/11/3684/9.3e-4/.02/8.7e-4 115/8/3703/1.6e-2/.03/2.2e-3 348/23/7359/7.6e-4/.05/2.3e-3 266/18/8182/1.0e-2/.12/3.5e-3 401/27/7783/2.2e-3/.02/8.5e-4 306/21/8788/8.7e-3/.20/2.8e-3

F iter*/cpu/map /errap /err/RMSD 73/4/747/4.1e-3/.17/2.0e-2 30/2/827/3.7e-2/.16/1.9e-2 89/6/836/3.0e-3/.09/7.2e-3 45/3/921/3.4e-2/.06/7.1e-3 141/9/3490/7.4e-4/.11/4.9e-3 110/7/3539/1.8e-2/.05/3.3e-3 164/11/3776/9.3e-4/.02/5.4e-4 116/8/3796/1.6e-2/.03/1.7e-3 383/25/8487/1.5e-3/.05/1.9e-3 280/18/8871/1.4e-2/.12/2.5e-3 424/28/9172/2.2e-3/.02/5.6e-4 312/21/9466/2.1e-2/.20/2.4e-3

FL iter*/cpu/map /errap /err/RMSD/vio/vioused 78/5/747/4.1e-3/.16/1.8e-2*/456/90 32/2/830/3.7e-2/.13/1.6e-2*/644/129 91/6/836/3.0e-3/.07/5.2e-3*/223/45 45/3/930/3.4e-2/.05/6.2e-3*/318/64 147/10/3492/6.9e-4/.10/4.3e-3*/449/90 114/8/3549/1.8e-2/.03/2.7e-3*/1067/213 164/11/3778/9.3e-4/.02/4.6e-4*/66/13 116/8/3800/1.6e-2/.02/1.6e-3*/227/45 390/26/8491/1.5e-3/.04/1.5e-3*/1184/237 284/19/8908/1.6e-2/.12/2.3e-3*/1859/372 424/28/9178/2.2e-3/.01/2.7e-4*/518/102 313/21/9490/8.7e-3/.20/2.3e-3*/1225/245

Table 2: A table comparing the strategies N, F and FL as applied to different test problems described in Section 6.1. cpu times are in seconds. iter* represents iterations in ten thousands. map , errap , err and RMSD are as described in Section 6.2. vio is the number of lower bound constraints violated by the solution obtained from strategy N, while vioused is the actual number of lower bound constraints incorporated. “*” indicates the lowest RMSDamong the strategies.

additional lower bound constraints, but it usually leads to an RMSD that is at least around 10% smaller. This is also illustrated in Figures 1 and 2. In all cases, we observe that map is large (> 0.7m). This is essential for F and FL to be efficient, since the network in the refinement stage has fewer sensors but many more anchors. To compare our strategies N, F and FL with strategy VL adapted from [8], we report in Table 3 simulation results on the first 4 problems from Table 2 using strategy VL. As expected, the time taken by VL is significantly longer than N, F and FL, and the RMSDs are not as low as those achieved by FL. Also, VL seems to give the smallest map among all. The main difference between FL and VL is that, instead of including violated lower bound constraints into the original problem, FL takes advantage of the individual trace test and fix sensors with small trace as new anchors. This results in a network with fewer sensors, thus a faster solution time.

P 1 2 3 4

VL iter*/cpu/map /errap /err/RMSD/vio/vioused 133/8/675/4.0e-3/.16/2.1e-2/460/91 62/4/697/3.7e-2/.15/2.0e-2/819/164 164/10/695/3.1e-3/.09/8.2e-3/225/45 89/6/754/1.9e-2/.06/8.3e-3/466/93

Table 3: A table illustrating the performance of VL to the first 4 problems in Section 6.1. cpu times are in seconds. iter* represents iterations in ten thousands. map , errap , err and RMSD are as described in Section 6.2. vio is the number of lower bound constraints violated by the solution obtained from strategy N, while vioused is the actual number of lower bound constraints incorporated. The advantages of using F or FL is that we still have the individual trace test as an accuracy certificate for the solution obtained, which is not common for refinement heuristics.

13

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4

−0.5 −0.5

0

−0.5 −0.5

0.5

0

0.5

Figure 1: The left figure shows the anchor (“◦”) and the solution found using strategy N for problem 2 in Table 1. Each sensor position (“·”) found is joined to its true position (“∗”) by a line. The right figure shows the same information for the solution found using strategy FL.

Next, as in [3, 7, 14, 21], we explore the possibility of improving solution quality by gradient descent heuristics, after using strategies F or FL. In [3, 21], a steepest descent method is applied to locally minimize the error function X fˆ(X) := (kxi − xj k − dij )2 . (i,j)∈A

As in [21], to maintain the distributed nature of our method, we apply a block-coordinate steepest descent method to locally minimize fˆ. In each iteration, the method chooses an i ∈ {1, . . . , m} with k∇xi fˆ(X)k > 10−3 and updates xi by xi ← xi − α∇xi fˆ(X), (r)

and the stepsize α is chosen by an Armijo rule analogously as in Step 1 of LPCGD . We do refinements as described above after applying strategies N, F and FL. The results are recorded under Nrefine , Frefine and FLrefine in Table 4. For comparison purpose, we also record the results obtained from solving the ESDP relaxation without dropping edges using LPCGD method, and then refining using gradient descent as described above. These results are recorded under Orefine . As observed in Table 4, in most cases, FLrefine yields the smallest RMSD among all strategies, with an RMSD being at least 20% smaller than Orefine , and at least 15% smaller than Nrefine . Also, FLrefine takes at most 2 seconds more than Nrefine does. On the other hand, Frefine usually yields the second smallest RMSD, at least 10% smaller than Nrefine . This is also illustrated in Figures 3 and 4. Hence, if 2-hop communication is expensive, Frefine should be used. Otherwise, FLrefine should be used. In the tests above, we have chosen σ ˆ = max{σ, 10−6 }. If a larger σ ˆ is used, say σ ˆ = 2 max{σ, 10−6 }, it is still true that FL leads to the lowest RMSD, followed by F. However, the RMSD of the solutions obtained ρ,δ,r would become larger. Intuitively, a larger σ ˆ corresponds to a larger ρ, and thus a larger Sesdp . Hence, the Z

ρ,δ,r

, which is some kind of analytic center, is further away from the true solution. 14

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4

−0.5 −0.5

0

−0.5 −0.5

0.5

0

0.5

Figure 2: The left figure shows the anchor (“◦”) and the solution found using strategy N for problem 4 in Table 1. Each sensor position (“·”) found is joined to its true position (“∗”) by a line. The right figure shows the same information for the solution found using strategy FL.

The performance of strategies F and FL depends heavily on the quality of the solution obtained from the first stage. In examples where the localization error of the solution obtained from ρ-ESDP is large (e.g., when localizing the network type bd3 used in [14, Section 5]), F or FL does not lead to reasonable improvement, i.e., the improvement in RMSD is very small while the extra time taken to obtain the refined solution is very large. A reasonable indicator of solution quality is map . Hence, we recommend using these refinement strategies when map is sufficiently large, say, larger than 0.7m, for the solution obtained from solving ρ-ESDP.

7

Conclusion and open questions

We have proposed to refine solution obtained from solving ρ-ESDP by solving ρ-ESDPlb on a reduced network, which is obtained by fixing sensors with small traces as new anchors. Simulation shows that strategies F and FL usually yield at least 10% improvement in the RMSD when map is large. We recommend using FL when 2-hop communication is not too expensive. Otherwise, F should be used. Both methods are distributable and trace test can be used to identify accurately positioned sensors in the refined solution. The refined solution can be further improved by local gradient descent, yielding usually a lower RMSD. The improvement in the RMSD by strategies F and FL depends heavily on the parameters a0 and a1 in the trace test (13). The larger the parameters, the greater the reduction in the number of sensors of the network in the refinement stage, and hence the cpu time is smaller. However, as more and more (possibly not so accurately positioned) sensors are fixed as anchors, the localization error increases. What are the optimal choices of a0 , a1 to minimize the localization error? How do they change with network topology/density? The proposed heuristics are also applicable to other convex relaxations. How much will lower bound

15

P 1 2 3 4 5 6 7 8 9 10 11 12

Orefine cpu/RMSD 5/2.5e-2 2/2.1e-2 6/9.9e-3 4/4.1e-3 18/5.9e-3 13/3.9e-3 21/5.6e-4 16/3.7e-4* 35/2.3e-3 28/2.9e-3 43/2.9e-4 35/1.3e-3

Nrefine cpu/RMSD 4/1.9e-2 2/2.2e-2 5/1.1e-2 3/4.1e-3 9/6.1e-3 7/4.6e-3 11/4.9e-4* 8/4.0e-4 23/2.4e-3 18/3.0e-3 27/5.3e-4 21/1.3e-3

Frefine cpu/RMSD 4/1.8e-2 2/1.9e-2 6/4.1e-3 3/2.4e-3 9/5.5e-3 7/1.3e-3 11/4.9e-4* 8/4.1e-4 25/2.0e-3 18/2.4e-3 28/1.8e-4 21/1.0e-3

FLrefine cpu/RMSD 5/1.7e-2* 2/1.2e-2* 6/2.3e-3* 3/2.1e-3* 10/4.3e-3* 8/9.3e-4* 11/4.9e-4* 8/4.0e-4 26/1.7e-3* 19/2.0e-3* 28/1.1e-4* 21/6.2e-4*

Table 4: Comparing the time and solution RMSD for different refinement strategies on the problems from Table 1. “*” indicates the lowest RMSD among the refinement strategies. constraints help improve the RMSD of the solution in those cases? We only considered adding lower bound constraints within 2-hops in simulation, how about adding lower bound constraints between sensors within 3 or 4 hops? What will be a good lower bound estimate in those cases? Acknowledgement. The author is indebted to Paul Tseng for suggesting this topic, his suggestion to use ρ-ESDPlb as a postprocessing refinement heuristic, providing a possible explanation for the improvement in localization error by using strategy F and many other fruitful discussions. This paper is prepared as part of the PhD dissertation of the author, under supervision of Paul Tseng. The author would also like to thank Maryam Fazel, Anthony Man-Cho So and Rekha Thomas for reading and commenting on an early version of the manuscript.

A

Pathological behaviors under unrealistic lower bound estimates

We have assumed that our lower bound estimates are realistic. What happens if this is not true? The following example shows that Theorem 1 can fail if r is not realistic, even when there is no noise in distance measurement. true

Example 2. Consider n √ = 4, m = 1, x1 = (0, 1)T , x2 = (1, 0)T , x3 =√(−1, 0)T , x4 = (0, −1)T , T x5 = (0, 1) , d12 = d13 = 2, d14 and d15 are not known, and r14 = r15 = 2. Then the problem (5) becomes min |y11 − 2xT2 x1 + kx2 k2 − 2| + |y11 − 2xT3 x1 + kx3 k2 − 2| Z µ ¶ y11 xT1 s.t. Z = º0 (14) x1 I2 T 2 y11 − 2x4 x1 + kx4 k ≥ 2 y11 − 2xT5 x1 + kx5 k2 ≥ 2. µ ¶ 1 0 We claim that Z0 := is the unique solution. To this end, note that Z0 is clearly feasible with 0 I2 16

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4

−0.5 −0.5

0

−0.5 −0.5

0.5

0

0.5

Figure 3: The left figure shows the anchor (“◦”) and the solution found using strategy Nrefine for problem 2 in Table 1. Each sensor position (“·”) found is joined to its true position (“∗”) by a line. The right figure shows the same information for the solution found using strategy FLrefine .

objective value zero. This shows that Z0 is an optimal solution of (14) and the optimal value is zero. Since the optimal value is zero, by |y11 − 2xT2 x1 + kx2 k2 − 2| + |y11 − 2xT3 x1 + kx3 k2 − 2| = 0, we see that for any optimal solution, y11 = 1 and x1 = (0, t)T for some t ∈ [−1, 1]. Next, we have from y11 − 2xT4 x1 + kx4 k2 ≥ 2 that 1 + 2t + 1 ≥ 2 ⇒ t ≥ 0. Similarly, y11 − 2xT5 x1 + kx5 k2 ≥ 2 implies t ≤ 0. Hence, x1 = (0, 0). This shows that Z0 is the unique solution to (14). However, tr1 (Z0 ) = 1 > 0. Note that the lower bound estimate is not realistic since   1 0 1 true 0,r Z = 0 1 0 ∈ / Sesdp . 1 0 1 Even worse, in unrealistic cases, even when there is no noise in the distance measurements, zero ith individual trace for some relative interior solution does not imply that sensor i is accurately positioned; though it is true that xi is invariant. true

Example 3.√ Consider n = 3, m = 1, x1 = (0, 1)T , x2 = (1, 0)T , x3 = (−1, 0)T , x4 = (0, 2)T , d12 = d13 = 2, d14 is not known and r14 = 3. Then the problem (5) becomes min |y11 − 2xT2 x1 + kx2 k2 − 2| + |y11 − 2xT3 x1 + kx3 k2 − 2| Z µ ¶ y11 xT1 (15) s.t. Z = º0 x1 I2 y11 − 2xT4 x1 + kx4 k2 ≥ 9.   1 0 −1 We claim that Z0 :=  0 1 0  is the unique solution. To this end, note that Z0 is clearly feasible −1 0 1 with objective value zero. This shows that Z0 is an optimal solution of (15) and the optimal value is zero. 17

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4

−0.5 −0.5

0

−0.5 −0.5

0.5

0

0.5

Figure 4: The left figure shows the anchor (“◦”) and the solution found using strategy Nrefine for problem 4 in Table 1. Each sensor position (“·”) found is joined to its true position (“∗”) by a line. The right figure shows the same information for the solution found using strategy FLrefine .

Since the optimal value is zero, by |y11 − 2xT2 x1 + kx2 k2 − 2| + |y11 − 2xT3 x1 + kx3 k2 − 2| = 0, we see that for any optimal solution, y11 = 1 and x1 = (0, t)T for some t ∈ [−1, 1]. Next, we have from y11 − 2xT4 x1 + kx4 k2 ≥ 9 that 1 − 4t + 4 ≥ 9 ⇒ −1 ≥ t. Hence, x1 = (0, −1). This shows that Z0 is the unique solution to (14). Moreover, tr1 (Z0 ) = 0 but   1 0 1 true true 0,r x1 6= x1 . Note that the lower bound estimate is not realistic since Z = 0 1 0 ∈ / Sesdp . 1 0 1 Hence, in applications, it is crucial, in view of the above examples, to ensure that the lower bound estimates are realistic. Otherwise, the trace test is not informative of the true position of sensors. One heuristics would be, assuming that all sensors have the same radio range, to take rij = radio range; this is what we have done in Section 6.1. Then r is realistic. A way of choosing tighter realistic lower bound estimates will conceivably give more accurate solutions.

B

Proof of theorem 1

The proof of Theorem 1 highly parallels the proof of [21, Theorem 1], except that we have to make use ¯ of Lemma 5 to deal with the lower bound constraints in (5) corresponding to A. 0,r 0,r ¯ ¯ Fix any ¯i ∈ Iesdp and Z ∈ ri(Sesdp ). Suppose to the contrary that tr¯i (Z) > 0. Let I¯ be the collection 0,r of all nodes in Iesdp such that it is connected to ¯i by a path with arcs in the following set: 0,r 0,r 0,r {(i, j) ∈ As : i, j ∈ Iesdp } ∪ {(i, j) ∈ A¯1 : tri (Z) > 0, trj (Z) > 0 for some Z ∈ ri(Sesdp ), i, j ∈ Iesdp }.

¯ 6= ∅. It follows from Lemma 3(a) and definition that tri (Z) ¯ > 0 for all i ∈ I. ¯ By Assumption 1, N (I)

18

Hence, Lemma 3(b) implies that ¯ ⊆ {1, . . . , m} \ I 0,r . N (I) esdp

(16)

We will arrive at a contradiction below. true 0,r 0,r 0,r / Iesdp , and hence, by By definition of Iesdp , there exists a Z ∈ ri(Sesdp ) such that xj 6= xj for all j ∈ true 0,r 0,r ¯ (16), for all j ∈ N (I). Since Z ∈S and S is convex, we have esdp

α

Z := αZ

true

esdp

0,r + (1 − α)Z ∈ ri(Sesdp ) ∀ 0 ≤ α < 1.

¯ with i ∈ I. ¯ By (16), j ≤ m so that (i, j) ∈ As . Applying Lemma 4 with Fix any (i, j) ∈ ´A(I) ³ true ¡ ¢ true ¯ = Z true , B = Z true and using xi = xtrue (since i ∈ I 0,r ) yield A¯ = xi , A = xi xj , B xj i {i,j} {i,j} esdp à õ !! ¶ 0 0 tri (Z) yij − xi T xj α αT α +α (Y − X X ){i,j} = (1 − α) . true trj (Z) yij − xi T xj 0 kxj − xj k2 ¯ > 0, Lemma 1 implies tri (Z) > 0. Since xj 6= x Since tri (Z) and the first matrix on the right-hand side j α is positive semidefinite (since Z{i,j,m+ } º 0), the right-hand side is nonsingular or, equivalently, Z{i,j,m +} is nonsingular for all 0 < α < 1 sufficiently small. 0,r ¯ with j ∈ On the other hand, let (i, j) ∈ A¯1 (I) / Iesdp . Then, by arguing as in the previous paragraph, α we have that Z{i,j,m+ } is nonsingular for all 0 < α < 1 sufficiently small. α Choose a 0 < α < 1 such that Z{i,j,m + } is nonsingular (and hence positive definite) for all (i, j) ∈ 0,r ¯ ¯ ¯ A(I) ∪ A1 (I) with j ∈ / Iesdp . We now construct a feasible perturbation of Z α . By translating all n points d×d ¯ by a common factor if necessary, we can assume that xα i 6= 0 for all i ∈ I. For each θ > 0, let Uθ ∈ R be an orthogonal matrix satisfying 0 < kUθ − Id kF = O(θ). Then, for θ > 0 sufficiently small, we have  α α T yii yij (Uθ xα i ) 0,r T α α ¯ ∪ A¯1 (I), ¯ j∈  yij  º 0 ∀(i, j) ∈ A(I) / Iesdp . (17) yjj xα j α α Uθ xi xj Id true

We now show that θ > 0 can be further chosen such that the lower bound constraints are satisfied. 0,r ¯ with j ∈ For any (i, j) ∈ A¯1 (I) / Iesdp , the corresponding positive semidefinite constraint in (5) is α satisfied. Hence, with yij unchanged, we see that the corresponding lower bound constraint is also satisfied. This places no further restrictions on the choice of θ. On the other hand, consider any (i, j) ∈ ¯ with j ∈ I 0,r \I. ¯ We shall alter y α in such a way that both the lower bound constraint and the A¯1 (I) ij esdp positive semidefinite constraint are satisfied. To this end, note that trj (Z α ) = 0 by definition of I¯ and that true tri (Z α ) ≥ αtri (Z ) + (1 − α)tri (Z) > 0. (18) 2 These imply `ij (Z α ) > rij by Lemma 5(b). Moreover, by the third constraint in (5) and trj (Z α ) = 0, we α α T α have yij = (xi ) xj . Hence, for sufficiently small θ > 0, we have that α T α α α α α T α 2 0,r ¯ ¯ ¯ yii − 2(Uθ xα i ) xj + yjj = `ij (Z ) + 2(yij − (Uθ xi ) xj ) > rij ∀(i, j) ∈ A1 (I), j ∈ Iesdp \I,

and by (18) that µ α yii α T α (Uθ xi ) xj

T α (Uθ xα i ) xj α yjj

¶ −

¡

Uθ xα i

¢T xα j

¡

Uθ xα i

xα j

¢

The relationship (20) is the same as 

α yii α  yij Uθ xα i

α yij α yjj xα j

T (Uθ xα i ) T  º 0. xα j Id

19

µ =

α 2 yii − kxα i k 0

0 0

(19)

¶ º 0.

(20)

¯ Since Z α ∈ ri(S 0,r ) and tri (Z α ) > 0, it follows from Next, consider any (i, j) ∈ A¯2 with i ∈ I. esdp 2 Lemma 5(a) that `ij (Z α ) > rij . Hence, for θ > 0 sufficiently small, we have that ¯ j > m. ∀(i, j) ∈ A¯2 , i ∈ I,

α 2 2 yii − 2xTj Uθ xα i + kxj k > rij

(21)

¯ we have from Fix any θ > 0 such that (17), (19) and (21) hold. For each (i, j) ∈ As ∪ A¯1 with i, j ∈ I, α r Z ∈ Fesdp that µ

α yii α yij

α yij α yjj

¶ −

¡

(Uθ xα i

xα j

¢

T

)

¡

Uθ xα i

xα j

¢

µ =

α yii α yij

α yij α yjj

¶

¡ − xα i

xα j

¢T ¡

xα i

¢ xα j º 0,

from which it follows that 

α yii α  yij Uθ xα i

α yij α yjj Uθ xα j

T (Uθ xα i ) T  º 0. (Uθ xα j) Id

α α α T α ¯ ¯ ¯ Thus, replacing xα by Uθ xα i in Z i for all i ∈ I, and yij by (Uθ xi ) xj for all (i, j) ∈ A1 (I) with 0,r ¯ α α ˜ ˜ j ∈ Iesdp \I yields a Z that is feasible for (5). Moreover, Z is optimal for (5) (with δ = 0) since, by (3) ¯ ⊆ {1, . . . , m} (see (16)), the objective function of (5) does not depend on xi for i ∈ I, ¯ nor and I¯ ∪ N (I) α 0,r α α ¯ ˜ on yij for (i, j) ∈ A1 . Thus Z ∈ Sesdp but its Uθ xi component differs from the xi component of Z α for ¯ contradicting the definition of I. ¯ all i ∈ I,

References [1] Alfakih, A. Y., Graph rigidity via Euclidean distance matrices, Linear Algebra Appl., 310 (2000), pp. 149–165. [2] Aspnes, J., Goldenberg, D. and Yang, Y. R., On the computational complexity of sensor network localization, in ALGOSENSORS 2004, Turku, Finland, Lecture Notes in Comput. Sci. 3121, Springer-Verlag, New York, 2004, pp. 32–44. [3] Biswas, P., Liang, T.-C., Toh, K.-C., Wang, T.-C. and Ye, Y., Semidefinite programming approaches for sensor network localization with noisy distance measurements, IEEE Trans. Auto. Sci. Eng., 3 (2006), pp. 360–371. [4] Biswas, P., Liang, T.-C., Wang, T.-C. and Ye, Y., Semidefinite programming based algorithms for sensor network localization, ACM Trans. Sensor Networks, 2 (2006), pp. 188–220. [5] Biswas, P., Toh, K.-C. and Ye, Y., A distributed SDP approach for large-scale noisy anchor-free graph realization with applications to molecular conformation, SIAM J. Sci. Comput., 30 (2008), pp. 1251–1277. [6] Biswas, P. and Ye, Y., Semidefinite programming for ad hoc wireless sensor network localization, in Proc. 3rd IPSN, Berkeley, CA, 2004, pp. 46–54. [7] Biswas, P. and Ye, Y., A distributed method for solving semidefinite programs arising from ad hoc wireless sensor network localization, in Mutiscale Optimization Methods and Applications, Nonconvex Optim. Appl. 82, Springer-Verlag, New York, 2006, pp. 69–84. [8] Carter, W., Jin, H. H., Saunders, M. A. and Ye, Y., SpaseLoc: An Adaptive Subproblem Algorithm for Scalable Wireless Sensor Network Localization, SIAM J. Optim., 17 (2006), pp. 1102–1128. 20

[9] Ding, Y., Krislock, N., Qian, J. and Wolkowicz, H., Sensor network localization, Euclidean distance matrix completions, and graph realization, Report, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, November 2008. [10] Doherty, L., Pister, K. S. J. and El Ghaoui, L., Convex position estimation in wireless sensor networks, in Proc. 20th INFOCOM, Vol. 3, Los Alamitos, CA, 2001, pp. 1655–1663. [11] Fari˜ na, N., Miguez, J. and Bugallo, M. F., Novel decision-fusion algorithms for target tracking using ad hoc networks, in Proc. 61st Vehicular Technology Conference, Vol. 4, 2005, pp. 2556– 2559. [12] Gustafsson, F., Gunnarsson, F., Bergman, N., Forssell, U., Jansson, J., Karlsson, R. and Nordlund, P., Particle filters for positioning, navigation, and tracking, IEEE Trans. Signal Proc., 50 (2002), pp. 425–437. [13] Hightower, J. and Borriello, G., Location systems for ubiquitous computing, Computer, 34 (2001), pp. 57–66. [14] Kim, S., Kojima, M. and Waki, H., Exploiting sparsity in SDP relaxation for sensor network localization, SIAM J. Optim., 17 (2009), pp. 192–215. [15] Krislock, N. and Wolkowicz, H., Explicit Sensor Network Localization using Semidefinite Representations and Facial Reductions, Report, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, April 2009. [16] Krislock, N., Piccialli, V. and Wolkowicz, H., Robust semidefinite programming approaches for sensor network localization with anchors, Report, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, May 2006. [17] Liang, T.-C., Wang, T.-C. and Ye, Y., A gradient search method to round the semidefinite programming relaxation solution for ad hoc wireless sensor network localization, Report, Electrical Engineering, Stanford University, Stanford, October 2004. http://serv1.ist.psu.edu:8080/viewdoc/summary?doi=10.1.1.81.7689 [18] Liu, J., Zhang, Y. and Zhao, F., Robust distributed node localization with error management, in Proc. 7th ACM International Symposium on Mobile Ad Hoc Networking and Computing, Florence, Italy, 2006, pp. 250–261. [19] Mor´e, J. J. and Wu, Z., Global continuation for distance geometry problems, SIAM J. Optim., 7 (1997), pp. 814–836. [20] Nie, J., Sum of squares method for sensor network localization, Comput. Optim. Appl., 43 (2009), pp. 151–179. [21] Pong, T. K. and Tseng, P., (Robust) Edge-Based Semidefinite Programming Relaxation of Sensor Network Localization, Math. Program., (2010), DOI: 10.1007/s10107-009-0338-x. [22] Rao, A., Ratnasamy, S., Papadimitriou, C., Shenker, S. and Stoica, I., Geographic routing without location information, in Proc. 9th Annual International Conference on Mobile Computing and Networking (MobiCom ’03), San Diego, CA, 2003, pp. 96–108. [23] Savarese, C., Rabaey, J. M. and Langendoen, K., Robust positioning algorithms for distributed ad-hoc wireless sensor networks, in Proc. USENIX Annual Technical Conference, Monterey, CA, 2002, pp. 317–327. 21

[24] Saxe, J. B., Embeddability of weighted graphs in k-space is strongly NP-hard, in Proc. 17th Allerton Conference in Communications, Control, and Computing, Monticello, IL, 1979, pp. 480–489. [25] Simi´c, S. N. and Sastry, S., Distributed localization in wireless ad hoc networks, Report, Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, 2002; First ACM International Workshop on Wireless Sensor Networks and Applications, Atlanta, GA, 2002, submitted. [26] So, A. M.-C. and Ye, Y., Theory of semidefinite programming for sensor network localization, Math. Program., 109 (2007), pp. 367–384. [27] Tseng, P. Second-order cone programming relaxation of sensor network localizations, SIAM J. Optim., 18 (2007), pp. 156–185. [28] Wang, Z., Zheng, S., Ye, Y. and Boyd, S., Further relaxations of the semidefinite programming approach to sensor network localization, SIAM J. Optim., 19 (2008), pp. 655–673.

22