Dimension reduction for semidefinite programming

Transcription

1 1 / 22 Dimension reduction for semidefinite programming Pablo A. Parrilo Laboratory for Information and Decision Systems Electrical Engineering and Computer Science Massachusetts Institute of Technology Joint work with Frank Permenter (MIT) arxiv: CDC Las Vegas

2 2 / 22 Semidefinite programs (SDPs) minimize Tr CX subject to X A S n + Formulated over vector space S n of n n symmetric matrices. variable X S n A S n an affine subspace, C S n cost matrix S n + cone of psd matrices Efficiently solvable in theory; in practice, solving some instances impossible unless special structure is exploited.

3 Dimension reduction Reformulate problem over subspace S S n intersecting set of optimal solutions minimize Tr CX subject to X A S n + minimize subject to Tr CX X A S n + S (Reformulation) S A S n + opt. solns where S n + S equals product K i K m of simple cones. Reduction methods: symmetry reduction and facial reduction 3 / 22

4 4 / 22 Symmetry reduction (MAXCUT relaxation example ) minimize Tr CX subject to X A S n + A := {X S n : X ii = 1} C := adjacency matrix

5 Symmetry reduction (MAXCUT relaxation example ) minimize Tr CX subject to X A S n + A := {X S n : X ii = 1} C := adjacency matrix range P A S n + opt. solns (e.g., Schrijver 79; Gatermann-P. 03) Idea: find special projection map P P(X) optimal when X optimal. P explicitly constructed from automorphism group of graph. Range block-diagonal a direct-sum of matrix algebras. 4 / 22

6 5 / 22 Facial reduction H S n + A minimize Tr CX subject to X A S n + First, find face of S n + containing feasible set. There exists a hyperplane H containing A. S n + H a face isomorphic to S d + for d < n. Face S n + H contains feasible set A S n +.

7 5 / 22 Facial reduction H S n + A minimize Tr CX subject to X A S n + First, find face of S n + containing feasible set. There exists a hyperplane H containing A. S n + H a face isomorphic to S d + for d < n. Face S n + H contains feasible set A S n +. Next, reformulate SDP over face: minimize subject to Tr CX X A S n + H

8 Facial reduction H S n + A minimize Tr CX subject to X A S n + First, find face of S n + containing feasible set. There exists a hyperplane H containing A. S n + H a face isomorphic to S d + for d < n. Face S n + H contains feasible set A S n +. Next, reformulate SDP over face: minimize subject to Tr CX X A S n + H Borwein-Wolkowicz 81; Pataki 00; Permenter-P / 22

9 6 / 22 Application specific approaches Facial reduction: MAXCUT (Anjos, Wolkowicz) QAP (Zhao,Wolkowicz) Sums-of-squares optimization (Permenter-P., Waki-Muramatsu) Matrix completion (Krislock,Wolkowicz)... Symmetry reduction: MAXCUT (earlier example), QAP (de Klerk, Sotirov); Markov chains (Boyd et al.); codes (Schrijver; Laurent)...

10 7 / 22 Our approach This talk: new reduction method subsuming symmetry reduction Notion of optimal reductions. A general purpose algorithm with optimality guarantees Jordan algebra interpretation; hence, easy extension to symmetric cone optimization (e.g., LP, SOCP). Combinatorial refinements for computational efficiency

11 8 / 22 How does symmetry reduction work? Given SDP min X A S n + Tr CX, method finds special orthogonal projection P : S n S n range of P A S n + opt. solns If X feas./optimal, P(X) feas./optimal.

12 8 / 22 How does symmetry reduction work? Given SDP min X A S n + Tr CX, method finds special orthogonal projection P : S n S n range of P A S n + opt. solns If X feas./optimal, P(X) feas./optimal. P satisfies following conditions: P(A) A, P(S n +) S n +, P(C) = C

13 How does symmetry reduction work? Given SDP min X A S n + Tr CX, method finds special orthogonal projection P : S n S n range of P A S n + opt. solns If X feas./optimal, P(X) feas./optimal. P satisfies following conditions: P(A) A, P(S n +) S n +, P(C) = C Hence, if X feasible then P(X) feasible with equal cost: 8 / 22

14 9 / 22 Example: a MAXCUT SDP relaxation minimize Tr CX subject to X A S n + A := {X S n : X ii = 1} C := adjacency matrix

15 9 / 22 Example: a MAXCUT SDP relaxation minimize Tr CX subject to X A S n + A := {X S n : X ii = 1} C := adjacency matrix

16 9 / 22 Example: a MAXCUT SDP relaxation minimize Tr CX subject to X A S n + A := {X S n : X ii = 1} C := adjacency matrix Let G denote group of permutation matrices (automorphisms) G := {U a permutation matrix : U T CU = C}

17 9 / 22 Example: a MAXCUT SDP relaxation minimize Tr CX subject to X A S n + A := {X S n : X ii = 1} C := adjacency matrix Let G denote group of permutation matrices (automorphisms) G := {U a permutation matrix : U T CU = C} Taking P(X) := 1 G U G UT XU, desired conditions hold: P(S n +) S n + P(A) A, P(C) = C

18 9 / 22 Example: a MAXCUT SDP relaxation minimize Tr CX subject to X A S n + A := {X S n : X ii = 1} C := adjacency matrix Let G denote group of permutation matrices (automorphisms) G := {U a permutation matrix : U T CU = C} Taking P(X) := 1 G U G UT XU, desired conditions hold: P(S n +) S n + P(A) A, P(C) = C Hence, range of P contains solutions: when X feasible, P(X) feasible with equal cost.

19 10 / 22 Our approach: optimize over projections Given SDP min X A S n + C, X, find map P that solves minimize subject to rank P P(C) = C, P(I) = I P(A) A P(S n +) S n + P : S n S n an orthogonal projection.

20 10 / 22 Our approach: optimize over projections Given SDP min X A S n + C, X, find map P that solves minimize subject to rank P P(C) = C, P(I) = I P(A) A P(S n +) S n + P : S n S n an orthogonal projection. Main properties: Can be solved in polynomial time (!) Range of P structured: a Jordan subalgebra of S n. S n + range P equals a product of symmetric cones.

21 11 / 22 Invariance characterization of optimal subspace Theorem (Permenter-P.) Orthogonal projection P : S n S n solves minimize subject to rank P P(C) = C, P(I) = I P(A) A P(S n +) S n + iff the range of P solves minimize dim S subject to S {I, X L, C} S P L (S) S {X 2 : X S}, where A = X L + L, and X L is the min-norm point of A.

22 12 / 22 Subspace optimization and solution algorithm minimize subject to dim S S C, X L, I S P L (S) S {X 2 : X S} S span{c, X L, I} repeat S S + P L (S) S S + span{x 2 : X S} until converged.

23 12 / 22 Subspace optimization and solution algorithm minimize subject to dim S S C, X L, I S P L (S) S {X 2 : X S} S span{c, X L, I} repeat S S + P L (S) S S + span{x 2 : X S} until converged. Properties of algorithm: Optimal subspace contains each iterate (induction)

24 12 / 22 Subspace optimization and solution algorithm minimize subject to dim S S C, X L, I S P L (S) S {X 2 : X S} S span{c, X L, I} repeat S S + P L (S) S S + span{x 2 : X S} until converged. Properties of algorithm: Optimal subspace contains each iterate (induction) Computes ascending chain of subspaces terminates.

25 12 / 22 Subspace optimization and solution algorithm minimize subject to dim S S C, X L, I S P L (S) S {X 2 : X S} S span{c, X L, I} repeat S S + P L (S) S S + span{x 2 : X S} until converged. Properties of algorithm: Optimal subspace contains each iterate (induction) Computes ascending chain of subspaces terminates. At termination, subspace feasible; hence, optimal.

26 12 / 22 Subspace optimization and solution algorithm minimize subject to dim S S C, X L, I S P L (S) S {X 2 : X S} S span{c, X L, I} repeat S S + P L (S) S S + span{x 2 : X S} until converged. Properties of algorithm: Optimal subspace contains each iterate (induction) Computes ascending chain of subspaces terminates. At termination, subspace feasible; hence, optimal. Properties of minimization problem: Feasible set closed under intersection (lattice)

27 Subspace optimization and solution algorithm minimize subject to dim S S C, X L, I S P L (S) S {X 2 : X S} S span{c, X L, I} repeat S S + P L (S) S S + span{x 2 : X S} until converged. Properties of algorithm: Optimal subspace contains each iterate (induction) Computes ascending chain of subspaces terminates. At termination, subspace feasible; hence, optimal. Properties of minimization problem: Feasible set closed under intersection (lattice) A unique solution. 12 / 22

28 13 / 22 Combinatorial descriptions Great! Now, we can easily compute the optimal subspace S.

29 13 / 22 Combinatorial descriptions Great! Now, we can easily compute the optimal subspace S. But, often want/need additional properties (e.g., dense subspaces may not be very efficient). Can tradeoff dimension with sparsity of a basis?

30 13 / 22 Combinatorial descriptions Great! Now, we can easily compute the optimal subspace S. But, often want/need additional properties (e.g., dense subspaces may not be very efficient). Can tradeoff dimension with sparsity of a basis? Yes! Three kinds of sparse bases for S: E.g., Partition subspaces: defined by a partition of [n] [n]. Coordinate subspaces: defined by a sparsity pattern Combinatorial subspaces: orthogonal basis of 0/1 matrices a a b a a b b b c vs. a b 0 b c d vs. a 0 b 0 a c b c b

31 14 / 22 Finding optimal structured subspaces The main algorithm can be adapted to compute the optimal subspace for each of these three cases.

32 14 / 22 Finding optimal structured subspaces The main algorithm can be adapted to compute the optimal subspace for each of these three cases. Key property (again): lattice structure (closedness under intersection)

33 14 / 22 Finding optimal structured subspaces The main algorithm can be adapted to compute the optimal subspace for each of these three cases. Key property (again): lattice structure (closedness under intersection) E.g., for partition subspaces, instead of optimizing over lattice of subspaces, use the lattice of partitions: minimize subject to dim S S C, X L, I S P L (S) S {X 2 : X S} S is a partition subspace P Part{C, X L, I} repeat P refine(p, P L ) P refine(p, X X 2 ) until converged.

34 14 / 22 Finding optimal structured subspaces The main algorithm can be adapted to compute the optimal subspace for each of these three cases. Key property (again): lattice structure (closedness under intersection) E.g., for partition subspaces, instead of optimizing over lattice of subspaces, use the lattice of partitions: minimize subject to dim S S C, X L, I S P L (S) S {X 2 : X S} S is a partition subspace P Part{C, X L, I} repeat P refine(p, P L ) P refine(p, X X 2 ) until converged. Great! But there s more...

35 15 / 22 Decomposition via Jordan algebras Given SDP min X A S n + C, X, we ve found a subspace invariant under X X 2 containing optimal solutions: S A S n + opt. solns S {X 2 : X S}

36 15 / 22 Decomposition via Jordan algebras Given SDP min X A S n + C, X, we ve found a subspace invariant under X X 2 containing optimal solutions: S A S n + opt. solns S {X 2 : X S} Subspaces invariant under X X 2 have decomposition S S S = Q QT, S m S i are simple Jordan algebras

37 Decomposition via Jordan algebras Given SDP min X A S n + C, X, we ve found a subspace invariant under X X 2 containing optimal solutions: S A S n + opt. solns S {X 2 : X S} Subspaces invariant under X X 2 have decomposition S S S = Q QT, S m S i are simple Jordan algebras Number of distinct eigenvalues of generic element equals rank of S i a complexity measure. 15 / 22

38 16 / 22 Minimizing dimension optimizes decomposition minimize subject to dim S S X L, C, I S P L (S) S {X 2 : X S}, All feasible subspaces have decomp. S = d S i=1 S i. In what sense does solution S optimize the ranks of each S i?

39 16 / 22 Minimizing dimension optimizes decomposition minimize subject to dim S S X L, C, I S P L (S) S {X 2 : X S}, All feasible subspaces have decomp. S = d S i=1 S i. In what sense does solution S optimize the ranks of each S i? Thm. (Permenter-P.): S minimizes i rank S i and max i rank S i

40 Minimizing dimension optimizes decomposition minimize subject to dim S S X L, C, I S P L (S) S {X 2 : X S}, All feasible subspaces have decomp. S = d S i=1 S i. In what sense does solution S optimize the ranks of each S i? Thm. (Permenter-P.): S minimizes i rank S i and max i rank S i Majorization inequalities hold, i.e., for each m 1 m m rank Si rank S i i=1 (ranks sorted in decreasing order) i=1 16 / 22

41 17 / 22 Majorization example Subspaces (parametrized by u i and v i ) and their rank vectors u 1 u u 2 u u u 5 u u 6 u 7 v 1 v v 2 v v 4 v 5 v v 5 v 7 v v 6 v 8 v 9 r u = (2, 1, 2) r v = (2, 3)

42 17 / 22 Majorization example Subspaces (parametrized by u i and v i ) and their rank vectors u 1 u u 2 u u u 5 u u 6 u 7 v 1 v v 2 v v 4 v 5 v v 5 v 7 v v 6 v 8 v 9 r u = (2, 1, 2) r v = (2, 3) Vector r u = (2, 2, 1) majorized by r v = (3, 2, 0): 2 3, ,

43 18 / 22 Jordan algebras Jordan algebras are commutative algebras satisfying Jordan identity (X Y ) X 2 = X (Y X 2 ) The vector space S n a Jordan algebra if equipped with product X Y := 1 (XY + YX) 2 The subalgebras of S n precisely the sets closed under squaring map X X 2 since XY + YX = (X + Y ) 2 X 2 Y 2. Structure theorem of Jordan-von Neumann-Wigner describes subalgebras of S n...

44 Decomposition of S S n + If S S n a Jordan subalgebra, it equals direct-sum m i=1 S i, where each S i is isomorphic to one of the following: Algebra of Hermitian matrices with real, complex or quaternion entries A spin-factor algebra Implies cone-of-squares S S n + isomorphic to product of PSD cones with real/complex/quaternion entries Lorentz cones Yields reformulation of original SDP over this product minimize Tr CX subject to X A S n + minimize Tr CX subject to X A T (K 1 K m ) }{{} S S n + 19 / 22

45 20 / 22 Computational results Comparison with reduction method of de Klerk 10 survey (generating *-algebras from data): instance S S data hamming_7_5_ hamming_8_3_ hamming_9_5_ hamming_9_ hamming_10_ Table list dimension of our subspace S S n and subspace S data S n found by generating *-algebra. Decomposing S yields a linear program.

46 Results: SOSOPT (Seiler 13) Demo scripts Script Name n (before) n (after) sosoptdemo2 13, 3 3, 2 3, 1 7 sosoptdemo , 1 10 gsosoptdemo1 9, 5 6, 3 2, 2 IOGainDemo_3 15, 8 10, 5 2, 3 Chesi(1 2)_IterationWithVlin 9, 5 6, 3 2, 2 Chesi3_GlobalStability 14, 5 8, 6, 3, 2 Chesi(3 4)_IterationWithVlin 9, 5 6, 3 2, 2 Chesi(5 6)_Bootstrap 19, 9 13, 6 2, 3 Chesi(5 6)_IterationWithVlin 19, 9 13, 6 2, 3 Coutinho3_IterationWithVlin 9, 5 6, 3 2, 2 HachichoTibken_Bootstrap 19, 9 12, 7, 6, 3 HachichoTibken_IterationWithVlin 19, 9 12, 7, 6, 3 Hahn_IterationWithVlin 9, 5 6, 3, 3, 2 KuChen_IterationWithVlin 19, 9 13, 6 2, 3 Parrilo1_GlobalStabilityWithVec 3, 2 2, 1 3 Parrilo2_GlobalStabilityWithMat 3, 2 2, 1 3 VDP_IterationWithVball 5, 4 3 2, 2, 1 VDP_IterationWithVlin 9, 5 6, 3 2, 2 VDP_LinearizedLyap 9, 5 6, 3 2, 2 VannelliVidyasagar2_Bootstrap 19, 9 13, 6 2, 3 VannelliVidyasagar2_IterationWithVlin 19, 9 13, 6 2, 3 VincentGrantham_IterationWithVlin 9, 5 6, 3 2, 2 WTBenchmark_IterationWithVlin 19, 9 13, 6 2, 3 21 / 22

47 22 / 22 Conclusions New reduction method for SDP. Generalizes symmetry reduction and *-algebra-methods Fully algorithmic, don t need to compute automorphisms! Yields optimal block-diagonalization (majorization) Can exploit combinatorial description of subspace Through Jordan algebra theory, extends to LP/SOCP/...

48 22 / 22 Conclusions New reduction method for SDP. Generalizes symmetry reduction and *-algebra-methods Fully algorithmic, don t need to compute automorphisms! Yields optimal block-diagonalization (majorization) Can exploit combinatorial description of subspace Through Jordan algebra theory, extends to LP/SOCP/... Preprint at arxiv:

49 22 / 22 Conclusions New reduction method for SDP. Generalizes symmetry reduction and *-algebra-methods Fully algorithmic, don t need to compute automorphisms! Yields optimal block-diagonalization (majorization) Can exploit combinatorial description of subspace Through Jordan algebra theory, extends to LP/SOCP/... Preprint at arxiv: Thanks for your attention!