Explicit Sensor Network Localization using Semidefinite Programming and Facial Reduction

Transcription

1 Explicit Sensor Network Localization using Semidefinite Programming and Facial Reduction Nathan Krislock and Henry Wolkowicz Dept. of Combinatorics and Optimization University of Waterloo INFORMS, San Diego Oct , 2009

2 Outline 1 Preliminaries SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones 2 Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations 3 Clique Unions and Node Absorptions Results (low CPU time; high accuracy) 4

3 Outline 1 Preliminaries SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones 2 Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations 3 Clique Unions and Node Absorptions Results (low CPU time; high accuracy) 4

4 Outline 1 Preliminaries SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones 2 Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations 3 Clique Unions and Node Absorptions Results (low CPU time; high accuracy) 4

5 Outline 1 Preliminaries SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones 2 Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations 3 Clique Unions and Node Absorptions Results (low CPU time; high accuracy) 4

6 SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Sensor Network Localization, SNL, Problem SNL - a Fundamental Problem of Distance Geometry; easy to describe - dates back to Grasssmann 1886 n ad hoc wireless sensors (nodes) to locate in R r, (r is embedding dimension; sensors p i R r, i V := 1,...,n) m of the sensors are anchors, p i, i = n m + 1,...,n) (positions known, using e.g. GPS) pairwise distances D ij = p i p j 2, ij E, are known within radio range R > 0 P = p T 1. p T n = 3 [ ] X R n r A

7 Applications SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Horst Stormer (Nobel Prize, Physics, 1949), 21 Ideas for the 21st Century, Business Week. 8/23-30, 1999 Untethered micro sensors will go anywhere and measure anything - traffic flow, water level, number of people walking by, temperature. This is developing into something like a nervous system for the earth, a skin for the earth. The world will evolve this way. Tracking Humans/Animals/Equipment/Weather (smart dust) geographic routing; data aggregation; topological control; soil humidity; earthquakes and volcanos; weather and ocean currents. military; tracking of goods; vehicle positions; surveillance; random deployment in inaccessible terrains.

8 5 Preliminaries Conferences/Journals/Research Groups/Books/Theses/Codes SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Conference, MELT 2008 International Journal of Sensor Networks Research groups include: CENS at UCLA, Berkeley WEBS, recent related theses and books include: [10, 16, 8, 7, 11, 12, 6, 14, 17] recent algorithms specific for SNL: [1, 2, 3, 4, 5, 9, 15, 18, 13]

9 SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Underlying Graph Realization/Partial EDM Graph G = (V, E,ω) node set V = {1,...,n} edge set (i, j) E ; ω ij = p i p j 2 known approximately The anchors form a clique (complete subgraph) Realization of G in R r : a mapping of node v i p i R r with squared distances given by ω. Corresponding Partial Euclidean Distance Matrix, EDM { d 2 D ij = ij if (i, j) E 0 otherwise (unknown distance), d 2 ij = ω ij are known squared Euclidean distances between sensors p i, p j ; anchors correspond to a clique. 6 NP-Hard

10 SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Underlying Graph Realization/Partial EDM Graph G = (V, E,ω) node set V = {1,...,n} edge set (i, j) E ; ω ij = p i p j 2 known approximately The anchors form a clique (complete subgraph) Realization of G in R r : a mapping of node v i p i R r with squared distances given by ω. Corresponding Partial Euclidean Distance Matrix, EDM { d 2 D ij = ij if (i, j) E 0 otherwise (unknown distance), d 2 ij = ω ij are known squared Euclidean distances between sensors p i, p j ; anchors correspond to a clique. 6 NP-Hard

11 SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Sensor Localization Problem/Partial EDM Sensors and Anchors n = 100, m = 9, R = 2

12 8 Preliminaries SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Connections to Semidefinite Programming (SDP) S n +, Cone of (symmetric) SDP matrices in S n ; x T Ax 0 inner product A, B = trace AB Löwner (psd) partial order A B, A B D = K (B) E n, B = K (D) S n S C (centered Be = 0) P T = [ ] p 1 p 2... p n M r n ; B := PP T S+ n rank B = r; D E ; n be corresponding EDM. (to D E n ) D = ( p i p j 2 n ( 2) i,j=1 ) n = pi T p i + pj T p j 2pi T p j i,j=1 = diag (B) e T + e diag (B) T 2B =: D e (B) 2B =: K (B) (from B S+ n ).

13 8 Preliminaries SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Connections to Semidefinite Programming (SDP) S n +, Cone of (symmetric) SDP matrices in S n ; x T Ax 0 inner product A, B = trace AB Löwner (psd) partial order A B, A B D = K (B) E n, B = K (D) S n S C (centered Be = 0) P T = [ ] p 1 p 2... p n M r n ; B := PP T S+ n rank B = r; D E ; n be corresponding EDM. (to D E n ) D = ( p i p j 2 n ( 2) i,j=1 ) n = pi T p i + pj T p j 2pi T p j i,j=1 = diag (B) e T + e diag (B) T 2B =: D e (B) 2B =: K (B) (from B S+ n ).

14 SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Current Techniques; SDP Relax.; Highly Degen. Nearest, Weighted, SDP Approx. (relax rank B) min B 0,B Ω H (K (B) D) ; rank B = r; typical weights: H ij = 1/ D ij, if ij E. with rank constraint: a non-convex, NP-hard program SDP relaxation is convex, BUT: expensive/low accuracy/implicitly highly degenerate (cliques restrict ranks of feasible Bs) Instead: (Shall) Take Advantage of Degeneracy! clique α, α = k (corresp. D[α]) with embed. dim. = t r < k = rank K (D[α]) = t r = rank B[α] rank K (D[α]) + 1 = rank B = rank K (D) n (k t 1) = Slater s CQ (strict feasibility) fails 9

15 SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Current Techniques; SDP Relax.; Highly Degen. Nearest, Weighted, SDP Approx. (relax rank B) min B 0,B Ω H (K (B) D) ; rank B = r; typical weights: H ij = 1/ D ij, if ij E. with rank constraint: a non-convex, NP-hard program SDP relaxation is convex, BUT: expensive/low accuracy/implicitly highly degenerate (cliques restrict ranks of feasible Bs) Instead: (Shall) Take Advantage of Degeneracy! clique α, α = k (corresp. D[α]) with embed. dim. = t r < k = rank K (D[α]) = t r = rank B[α] rank K (D[α]) + 1 = rank B = rank K (D) n (k t 1) = Slater s CQ (strict feasibility) fails 9

16 10 Preliminaries SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones (S n :) K : S n + S C E n S n S H : T (:E n ) Linear Transformations: D v (B), K (B), T (D) allow: D v (B) := diag(b) v T + v diag(b) T ; D v (y) := yv T + vy T adjoint K (D) = 2(Diag(De) D). K is 1 1, onto between centered & hollow subspaces : S C := {B S n : Be = 0}; S H := {D S n : diag (D) = 0} = R(offDiag ) J := I 1 n eet (orthogonal projection onto M := {e} ); T (D) := 1 2JoffDiag (D)J (= K (D))

17 Properties of Linear Transformations SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones K, T, Diag, D e R(K ) = S H ; N (K ) = R (D e ); R(K ) = R(T ) = S C ; N (K ) = N (T ) = R(Diag ); S n = S H R(Diag ) = S C R (D e ). T (E n ) = S n + S C and K (S n + S C ) = E n.

18 Semidefinite Cone, Faces SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Faces of cone K F K is a face of K, denoted F K, if ( x, y K, 1 2 (x + y) F) = (cone {x, y} F). F K, if F K, F K ; F is proper face if {0} F K. F K is exposed if: intersection of K with a hyperplane. face(s) denotes smallest face of K that contains set S. S n + is a Facially Exposed Cone All faces are exposed.

19 Semidefinite Cone, Faces SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Faces of cone K F K is a face of K, denoted F K, if ( x, y K, 1 2 (x + y) F) = (cone {x, y} F). F K, if F K, F K ; F is proper face if {0} F K. F K is exposed if: intersection of K with a hyperplane. face(s) denotes smallest face of K that contains set S. S n + is a Facially Exposed Cone All faces are exposed.

20 13 Preliminaries SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Facial Structure of SDP Cone; Equivalent SUBSPACES Face F S n + Equivalence to R(U) Subspace of R n F S+ n determined by range of any S relint F, i.e. let S = UΓU T be compact spectral decomposition; Γ S++ t is diagonal matrix of pos. eigenvalues; F = US+U t T (F associated with R (U)) dim F = t(t + 1)/2. face F representation by subspace L (subspace) L = R(T), T is n t full column, then: F := T S t +T T S n +

21 13 Preliminaries SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Facial Structure of SDP Cone; Equivalent SUBSPACES Face F S n + Equivalence to R(U) Subspace of R n F S+ n determined by range of any S relint F, i.e. let S = UΓU T be compact spectral decomposition; Γ S++ t is diagonal matrix of pos. eigenvalues; F = US+U t T (F associated with R (U)) dim F = t(t + 1)/2. face F representation by subspace L (subspace) L = R(T), T is n t full column, then: F := T S t +T T S n +

22 Further Notation SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Matrix with Fixed Principal Submatrix For Y S n, α {1,..., n}: Y[α] denotes principal submatrix formed from rows & cols with indices α. Sets with Fixed Principal Submatrices If α = k and Ȳ Sk, then: S n (α, Ȳ) := { Y S n : Y[α] = Ȳ}, S+ n (α, Ȳ) := { Y S+ n : Y[α] = Ȳ} i.e. the subset of matrices Y S n (Y S+) n with principal submatrix Y[α] fixed to Ȳ.

23 Further Notation SNL < > GR < > EDM < > SDP Further Notation/Preliminaries; Facial Structure of Cones Matrix with Fixed Principal Submatrix For Y S n, α {1,..., n}: Y[α] denotes principal submatrix formed from rows & cols with indices α. Sets with Fixed Principal Submatrices If α = k and Ȳ Sk, then: S n (α, Ȳ) := { Y S n : Y[α] = Ȳ}, S+ n (α, Ȳ) := { Y S+ n : Y[α] = Ȳ} i.e. the subset of matrices Y S n (Y S+) n with principal submatrix Y[α] fixed to Ȳ.

24 15 Preliminaries Basic Single Clique/Facial Reduction Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations D E k, α 1:n, α = k Define E n (α, D) := { D E n : D[α] = D }. Given D; find a corresponding B 0; find the corresponding face; find the corresponding subspace. if α = 1 : k; embed. dim of D is t r ] [ D D =,

25 Note that we add 1 k e to represent N (K ); then we use V to eliminate e to recover a centered face. 16 Preliminaries Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations BASIC THEOREM for Single Clique/Facial Reduction THEOREM 1: Single Clique/Facial Reduction Let: D := D[1:k] E k, k < n, with embedding dimension t r; B := K ( D) = ŪBSŪT B, ŪB M k t ], ŪT B ŪB = I t, S S++. t 1 Furthermore, let U B := [ŪB k e M k (t+1), [ ] UB 0 [ ] U U :=, and let V T e M n k+t+1 be 0 I U T e n k orthogonal. Then: face K ( E n (1:k, D) ) ( ) = US+ n k+t+1 U T S C = (UV)S+ n k+t (UV) T

26 For each clique α = k, we get a corresponding face/subspace (k r matrix) representation. We now see how to handle two cliques, α 1,α 2, that intersect. Preliminaries Sets for Intersecting Cliques/Faces Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations α 1 := 1:( k 1 + k 2 ); α 2 := ( k 1 + 1):( k 1 + k 2 + k 3 ) α 1 α 2 k 1 k 2 k3

27 18 Preliminaries Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations Two (Intersecting) Clique Reduction/Subsp. Repres. THEOREM 2: Clique/Facial Intersection Using Subspace Intersection { α1,α 2 1:n; k := α 1 α 2 For i = 1, 2: D i := D[α i ] E k i, embedding dimension t i ; B i := K ( D i ) = ŪiS i Ūi T, Ūi M k i t i, ŪT i Ū i = I ti, S i S t i ++ ; 1 U i := [Ūi e ] ki M k i (t i +1) ; and Ū M k (t+1) satisfies ([ ]) ([ ]) R(Ū) = R U1 0 R I k1 0, with 0 I k3 0 U ŪT Ū = I t+1 2 cont...

28 19 Preliminaries Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations Two (Intersecting) Clique Reduction, cont... THEOREM 2 Nonsing. Clique/Facial Inters. cont... cont... with ([ ]) ([ ]) R(Ū) = R U1 0 R I k1 0, with 0 I k3 0 U ŪT Ū = I t+1 ; 2 [Ū ] 0 let: U := M n (n k+t+1) and 0 I [ ] n k U V T e M n k+t+1 be orthogonal. Then U T e 2 i=1 face K ( E n (α i, D i ) ) ( ) = US+ n k+t+1 U T S C = (UV)S+ n k+t (UV) T

29 20 Preliminaries Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations Expense/Work of (Two) Clique/Facial Reductions Subspace Intersection for Two Intersecting Cliques/Faces Suppose: U 1 0 I 0 U 1 = U 1 0 and U 2 = 0 U 2 0 I 0 U 2 Then: U 1 U := U 1 or U := U 2 (U 2 ) U 1 U 1 (U 1 ) U 2 U 2 U 2 (Efficiently) satisfies: R (U) = R(U 1 ) R(U 2 )

30 Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations Two (Intersecting) Clique Reduction Figure Completion: missing distances can be recovered if desired.

31 Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations Two (Intersecting) Clique Explicit Delayed Completion COR. Intersection with Embedding Dim. r/completion Hypotheses of Theorem 2 holds. Let D i := D[α i ] E k i, for i = 1, 2, β α 1 α 2,γ := α 1 α 2, D := D[β], B := K ( D), Ū β := Ū(β,:), where Ū M k (t+1) satisfies ] intersection equation of Theorem 2. Let [ V M t+1 Ū T e ŪT e be orthogonal. Let Z := (JŪβ V) B((JŪβ V) ) T. If the embedding dimension for D is r, THEN t = r in Theorem 2, and Z S r + is the unique solution of the equation (JŪβ V)Z(JŪβ V) T = B, and the exact completion is D[γ] = K ( PP T) where P := UV Z 1 2 R γ r

32 Use R as lower bound in singular/nonrigid case. 23 Preliminaries Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations 2 (Inters.) Clique Red. Figure/Singular Case Two (Intersecting) Clique Reduction Figure/Singular Case C i C j

33 Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations Two (Inters.) Clique Explicit Compl.; Sing. Case COR. Clique-Sing.; Intersect. Embedding Dim. r 1 Hypotheses of previous COR holds. For i = 1, 2, let β δ i α i, A i := JŪδ i V, where Ū δi := Ū(δ i,:), and B i := K (D[δ i ]). Let Z S t be a particular solution of the linear systems A 1 ZA T 1 = B 1 A 2 ZA T 2 = B 2.. If the embedding dimension of D[δ i ] is r, for i = 1, 2, but the embedding dimension of D := D[β] is r 1, then the following holds. cont...

34 25 Preliminaries Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations 2 (Inters.) Clique Expl. Compl.; Degen. cont... COR. Clique-Degen. cont... The following holds: 1 dim N (A i ) = 1, for i = 1, 2. 2 For i = 1, 2, let n i N (A i ), n i 2 = 1, and Z := n 1 n2 T + n 2n1 T. Then, Z is a solution of the linear systems if and only if Z = Z + τ Z, for some τ R 3 There are at most two nonzero solutions, τ 1 and τ 2, for the generalized eigenvalue problem Zv = τ Z v, v 0. Set Z i := Z + 1 τ i Z, for i = 1, 2. Then the exact completion is one of D[γ] { K (Ū V Z i V T Ū T ) : i = 1, 2 }

35 26 Preliminaries Basic Single Clique Reduction Two Clique Reduction and EDM DELAYED Completion Completing SNL; DELAYED use of Anchor Locations Completing SNL (Delayed use of Anchor Locations) Rotate to Align the Anchor Positions [ ] P1 Given P = R n r such that D = K (PP T ) P 2 Solve the orthogonal Procrustes problem: min s.t. A P 2 Q Q T Q = I P T 2 A = UΣV T SVD decomposition; set Q = UV T ; (Golub/Van Loan, Algorithm ) Set X := P 1 Q

36 Algorithm: Four Cases Clique Union Clique Unions and Node Absorptions Results (low CPU time; high accuracy) Node Absorption Rigid Non-rigid C i Ci C j j

37 28 Preliminaries Clique Unions and Node Absorptions Results (low CPU time; high accuracy) ALGOR: clique union; facial reduct.; delay compl. Initialize: Find initial set of cliques. C i := { j : (D p ) ij < (R/2) 2}, Iterate Finalize for i = 1,...,n For C i C j r + 1, do Rigid Clique Union For C i N (j) r + 1, do Rigid Node Absorption For C i C j = r, do Non-Rigid Clique Union (lower bnds) For C i N (j) = r, do Non-Rigid Node Absorp. (lower bnds) When a clique containing all anchors, use computed facial representation and positions of anchors to solve for X

38 28 Preliminaries Clique Unions and Node Absorptions Results (low CPU time; high accuracy) ALGOR: clique union; facial reduct.; delay compl. Initialize: Find initial set of cliques. C i := { j : (D p ) ij < (R/2) 2}, Iterate Finalize for i = 1,...,n For C i C j r + 1, do Rigid Clique Union For C i N (j) r + 1, do Rigid Node Absorption For C i C j = r, do Non-Rigid Clique Union (lower bnds) For C i N (j) = r, do Non-Rigid Node Absorp. (lower bnds) When a clique containing all anchors, use computed facial representation and positions of anchors to solve for X

39 28 Preliminaries Clique Unions and Node Absorptions Results (low CPU time; high accuracy) ALGOR: clique union; facial reduct.; delay compl. Initialize: Find initial set of cliques. C i := { j : (D p ) ij < (R/2) 2}, Iterate Finalize for i = 1,...,n For C i C j r + 1, do Rigid Clique Union For C i N (j) r + 1, do Rigid Node Absorption For C i C j = r, do Non-Rigid Clique Union (lower bnds) For C i N (j) = r, do Non-Rigid Node Absorp. (lower bnds) When a clique containing all anchors, use computed facial representation and positions of anchors to solve for X

40 29 Preliminaries Clique Unions and Node Absorptions Results (low CPU time; high accuracy) Results - Data for Random Noisless Problems 2.16 GHz Intel Core 2 Duo, 2 GB of RAM Dimension r = 2 Square region: [0, 1] [0, 1] m = 9 anchors Using only Rigid Clique Union and Rigid Node Absorption Error measure: Root Mean Square Deviation RMSD = ( 1 n n i=1 p i p true i 2 ) 1/2

41 30 Preliminaries Clique Unions and Node Absorptions Results (low CPU time; high accuracy) Results - Large n (SDP size O(n 2 )) n # of Sensors Located n # sensors \ R CPU Seconds # sensors \ R RMSD (over located sensors) n # sensors \ R e 16 5e 16 6e 16 3e e 16 4e 16 3e 16 3e e 16 5e 16 4e 16 4e 16

42 31 Preliminaries Results - N Huge SDPs Solved Clique Unions and Node Absorptions Results (low CPU time; high accuracy) Large-Scale Problems # sensors # anchors radio range RMSD Time e 16 25s e 16 1m 23s e 16 3m 13s e 16 9m 8s Size of SDPs Solved: N = ( ) n (# vrbls) 2 E n (density of G ) = πr 2 ; M = E n ( E ) = πr 2 N (# constraints) Size of SDP Problems: M = [ 3, 078, , 315, , 709, , 969, 790 ] N = 10 9 [ ]

43 Locally Recover Exact EDMs 32 Nearest EDM Given clique α; corresp. EDM D ǫ = D + N ǫ, N ǫ noise we need to find the smallest face containing E n (α, D). min K (X) D ǫ s.t. rank (X) = r, Xe = 0, X 0 X 0. Eliminate the constraints: Ve = 0, V T V = I, K V (X) := K (VXV T ): Ur 1 argmin 2 K V (UU T ) D ǫ 2 F s.t. U M (n 1)r. The nearest EDM is D = K V (U r (U r ) T ).

44 Solve Overdetermined Nonlin. Least Squares Prob. 33 Newton (expensive) or Gauss-Newton (less accurate) ) F(U) := us2vec (K V (UU T ) D ǫ, min f(u) := 1 U 2 F(U) 2 Derivatives: gradient and Hessian ( ]) f(u)( U) = 2 K V [K V (UU T ) D ǫ U, U 2 f(u) = 2vec ( )) L U K V K V S Σ L U + K V (K V (UU T ) D ǫ Mat where L U ( ) = U T ; S Σ (U) = 1 2 (U + UT )

45 random noisy probs; r = 2, m = 9, nf = 1e 6 34 Using only Rigid Clique Union, preliminary results: remaining cliques n / R cpu seconds n / R max-log-error n / R Inf Inf

46 35 SDP relaxation of SNL is highly (implicitly) degenerate: The feasible set of this SDP is restricted to a low dim. face of the SDP cone, causing the Slater constraint qualification (strict feasibility) to fail We take advantage of this degeneracy by finding explicit representations of intersections of faces of the SDP cone corresponding to unions of intersecting cliques Without using an SDP-solver (eg. SeDuMi or SDPT3), we quickly compute the exact solution to the SDP relaxation

47 35 SDP relaxation of SNL is highly (implicitly) degenerate: The feasible set of this SDP is restricted to a low dim. face of the SDP cone, causing the Slater constraint qualification (strict feasibility) to fail We take advantage of this degeneracy by finding explicit representations of intersections of faces of the SDP cone corresponding to unions of intersecting cliques Without using an SDP-solver (eg. SeDuMi or SDPT3), we quickly compute the exact solution to the SDP relaxation

48 35 SDP relaxation of SNL is highly (implicitly) degenerate: The feasible set of this SDP is restricted to a low dim. face of the SDP cone, causing the Slater constraint qualification (strict feasibility) to fail We take advantage of this degeneracy by finding explicit representations of intersections of faces of the SDP cone corresponding to unions of intersecting cliques Without using an SDP-solver (eg. SeDuMi or SDPT3), we quickly compute the exact solution to the SDP relaxation

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54 Thanks for your attention! INFORMS, San Diego Oct , 2009 Explicit Sensor Network Localization using Semidefinite Programming and Facial Reduction Nathan Krislock and Henry Wolkowicz Dept. of Combinatorics and Optimization University of Waterloo