A semidefinite hierarchy for containment of spectrahedra

Transcription

1 A semidefinite hierarchy for containment of spectrahedra Thorsten Theobald (joint work with Kai Kellner and Christian Trabandt, arxiv: )

2 Spectrahedra Given a linear pencil A(x) = A 0 + n i=1 x ia i, the set is called a spectrahedron. S A := {x R n : A(x) 0} Feasible sets of semidefinite programming 1 x 1 x 2 x 1 1 x 3 x 2 x Usual assumption: 0 int S A. ( W.l.o.g., pencil monic: A 0 = I k ).

3 Containment problems for polyhedra and spectrahedra Uniform general setup: Given two linear pencils A(x) S k [x] and B(x) S l [x], is S A S B?? Primary containment problem; larger classes available (up to homotheties, rotations,...) Decision version; optimization versions as well (e.g., smallest enclosing (centered) sphere, geometric radii,...)

4 Containment problems for polyhedra and spectrahedra Uniform general setup: Given two linear pencils A(x) S k [x] and B(x) S l [x], is S A S B?? Primary containment problem; larger classes available (up to homotheties, rotations,...) Decision version; optimization versions as well (e.g., smallest enclosing (centered) sphere, geometric radii,...)

5 Containment problems for polytopes For polytopes much work has been done (Freund, Orlin ( 86), Gritzmann, Klee ( 93),...): computational complexity strongly depends on type of the input: V-polytopes: given by the vertices H-polytopes: given as an intersection of halfspaces {x R n : b + Ax 0} Q P H V H P co-npc V P P P Q?

6 Hardness for polytopes and spectrahedra H V S H P co-npc co-np-hard V P P P S SDP co-np-hard co-np-hard The following problems are co-np-hard: 1 Deciding whether a spectrahedron is contained in a V-polytope. 2 Deciding whether an H-polytope or a spectrahedron is contained in a spectrahedron. This persists if the H-polytope is a standard cube or if the outer spectrahedron is a ball. (Cubes in spectrahedra already by Ben-Tal and Nemirovski.)

7 Constructive approaches Sufficient semidefinite criterion (Helton, Klep, McCullough 2013) Simplified proofs; exactness in several cases (Kellner, Trabandt, T.T. 2013) Recently: Broader picture: Hierarchy of PMIs (Polynomial Matrix Inequalities), PMI-relaxation and positivity of maps

8 Relaxations: The point of departure Let A(x) S k [x] and B(x) S l [x] be monic linear pencils (with A 0 = I k and B 0 = I l ). In the following, the indeterminate matrix C = ( ) k C ij ( Choi matrix ) i,j=1 is a symmetric kl kl-matrix where the C ij are l l-blocks. Consider: C = ( C ij ) k i,j=1 0, I l = k k C ii, p = 1,..., n : B p = a p ij C ij i=1 i,j=1 Theorem (Helton, Klep, McCullough 13) Let A(x) S k [x] and B(x) S l [x] be monic linear pencils. If this system is feasible then S A S B.

9 Sufficient criterion: Background Let A S k and B S l be linear subspaces. A linear map Φ : A B is called positive if every positive semidefinite matrix in A is mapped to a positive semidefinite matrix in B, i.e., Φ(A S + k ) B S+ l. d-positive if the map is positive. Φ d : R d d A R d d B, M A M Φ(A) completely positive if Φ d is positive for all d > 0.

10 Sufficient criterion: Once more the statement Let A(x) S k [x] and B(x) S l [x] be monic linear pencils (with A 0 = I k and B 0 = I l ). In the following, the indeterminate matrix C = ( ) k C ij ( Choi matrix ) i,j=1 is a symmetric kl kl-matrix where the C ij are l l-blocks. Generalized criterion: C = ( C ij ) k i,j=1 0, I l = k k C ii, p = 1,..., n : B p = a p ij C ij (1) i=1 i,j=1 Theorem (Helton, Klep, McCullough 13) Let A(x) S k [x] and B(x) S l [x] be monic linear pencils. If the system (1) is feasible then S A S B.

11 Elementary proof For x S A, the last two conditions imply n k n k B(x) = I l + x p B p = C ii + x p a p ij C ij p=1 i=1 p=1 i,j=1 k k = C ij (I k ) ij + C ij i,j=1 i,j=1 p=1 n x p a p ij = k (A(x)) ij C ij. i,j=1

12 Elementary proof For x S A, the last two conditions imply B(x) = k i,j=1 (A(x)) ij C ij. n k n k B(x) = I l + x p B p = C ii + x p a p ij C ij p=1 i=1 p=1 i,j=1 k k = C ij (I k ) ij + C ij i,j=1 i,j=1 p=1 n x p a p ij = k (A(x)) ij C ij. i,j=1

13 Elementary proof For x S A, the last two conditions imply B(x) = k i,j=1 (A(x)) ij C ij.

14 Elementary proof For x S A, the last two conditions imply B(x) = k i,j=1 (A(x)) ij C ij. Since A(x) and C are psd, the Kronecker product A(x) C R k 2 l k 2 l is psd. Consider the principal submatrix given by the (i, j)-th sub-block of every (i, j)-th block, ( (A(x)) ij C ij ) k i,j=1 S kl[x].

15 Elementary proof For x S A, the last two conditions imply B(x) = k i,j=1 (A(x)) ij C ij. Since A(x) and C are psd, the Kronecker product A(x) C R k 2 l k 2 l is psd. Consider the principal submatrix given by the (i, j)-th sub-block of every (i, j)-th block, ( (A(x)) ij C ij ) k i,j=1 S kl[x]. Setting I = [I l,..., I l ] T, for any v R l we have v T B(x)v = v T ( I T ( (A(x)) ij C ij ) k i,j=1 I ) v = ( v T... v T ) ( (A(x)) ij C ij ) k i,j=1 (v... v)t 0.

16 Extend to spectrahedra (Variation) Corollary The theorem still holds when system (1) is relaxed to the system C = ( ) k k k C ij i,j=1 0, I l C ii 0, p = 1,..., n : B p = a p ij C ij. i=1 i,j=1 (2)

17 Exact cases and quality Questions: When is the criterion exact? How good/useful is the criterion? Hierarchy? Given a linear pencil A(x) S k [x], call the linear pencil [ ] 1 0 Â(x) := S 0 A(x) k+1 [x] the extended linear pencil of S A = SÂ. The entries of Âp in Â(x) = Â0 + n p=1 x pâp are denoted by â p ij for i, j = 0,..., k.

18 Exact cases and quality Questions: When is the criterion exact? How good/useful is the criterion? Hierarchy? Given a linear pencil A(x) S k [x], call the linear pencil [ ] 1 0 Â(x) := S 0 A(x) k+1 [x] the extended linear pencil of S A = SÂ. The entries of Âp in Â(x) = Â0 + n p=1 x pâp are denoted by â p ij for i, j = 0,..., k.

19 Cases where exactness holds Let A(x) S k [x] and B(x) S l [x] be monic linear pencils. In the following cases the criteria (1) as well as (2) are necessary and sufficient for the inclusion S A S B : 1. if A(x) and B(x) are normal forms of ellipsoids (both centrally symmetric, axis-aligned semi-axes),

20 Cases where exactness holds Let A(x) S k [x] and B(x) S l [x] be monic linear pencils. In the following cases the criteria (1) as well as (2) are necessary and sufficient for the inclusion S A S B : 1. if A(x) and B(x) are normal forms of ellipsoids (both centrally symmetric, axis-aligned semi-axes), For semi-axis lengths a 1,..., a n : A(x) = I n+1 + n p=1 x p a p S p,n+1 with S p,n+1 S n+1 with 1 in the entries (p, n + 1), (n + 1, p).

21 Cases where exactness holds Let A(x) S k [x] and B(x) S l [x] be monic linear pencils. In the following cases the criteria (1) as well as (2) are necessary and sufficient for the inclusion S A S B : 1. if A(x) and B(x) are normal forms of ellipsoids (both centrally symmetric, axis-aligned semi-axes),

22 Cases where exactness holds Let A(x) S k [x] and B(x) S l [x] be monic linear pencils. In the following cases the criteria (1) as well as (2) are necessary and sufficient for the inclusion S A S B : 1. if A(x) and B(x) are normal forms of ellipsoids (both centrally symmetric, axis-aligned semi-axes), 2. if A(x) and B(x) are normal forms of a ball and an H-polyhedron, respectively,

23 Cases where exactness holds Let A(x) S k [x] and B(x) S l [x] be monic linear pencils. In the following cases the criteria (1) as well as (2) are necessary and sufficient for the inclusion S A S B : 1. if A(x) and B(x) are normal forms of ellipsoids (both centrally symmetric, axis-aligned semi-axes), 2. if A(x) and B(x) are normal forms of a ball and an H-polyhedron, respectively, Main exactness results: 3. if B(x) is the normal form of a polytope, 4. if Â(x) is the extended form of a spectrahedron and B(x) is the normal form of a polyhedron.

24 Cases where the criterion fails ( Let A(x) = I 3 + x B(x) = ) [ ] [ x Both define the unit disk, i.e., S A = S B. Proposition ( x ] + x 2 [ The containment question S B S A is certified by criterion (1), while the reverse containment question S A S B is not certified by the criterion. ]. ), Q: Does there exist a scaling factor r for one of the spectrahedra so that the containment criterion is satisfied after this scaling?

25 Cases where the criterion fails ( Let A(x) = I 3 + x B(x) = ) [ ] [ x Both define the unit disk, i.e., S A = S B. Proposition ( x ] + x 2 [ The containment question S B S A is certified by criterion (1), while the reverse containment question S A S B is not certified by the criterion. ]. ), Q: Does there exist a scaling factor r for one of the spectrahedra so that the containment criterion is satisfied after this scaling?

26 Scaled containment Theorem Let A(x) and B(x) be monic linear pencils such that S A is bounded. Then there exists a constant ν > 0 such that for the scaled spectrahedron νs A the inclusion νs A S B is certified by any of the criteria (1) and (2). For containment of cubes in spectrahedra there exists a quantitative version (Ben-Tal and Nemirovski).

27 Containment via PMI hierarchies S A S B x S A : B(x) 0 x S A : z T B(x)z 0 Lemma Let A(x) S k [x] and B(x) S l [x] with S A, and let r > 0. Then S A S B if and only if the infimum µ of the polynomial optimization problem µ = inf z T B(x)z s.t. A(x) 0 z B r (0) is zero, where B r (0) denotes the ball in R l with radius r > 0 centered at the origin.

28 Moment relaxation (Lasserre, Parrilo; Putinar, Henrion; Hol, Scherer..) Linearize polynomials in the PMI: b(x) = (1, x 1, x 2, x1 2, x 1 x 2, x2 2,...) ȳ = (1, y 10, y 01, y 20, y 11, y 02,...) p(x) = p T b(x) L ȳ (p) = p T ȳ

29 Moment relaxation (Lasserre, Parrilo; Putinar, Henrion; Hol, Scherer..) Linearize polynomials in the PMI: b(x) = (1, x 1, x 2, x 2 1, x 1 x 2, x 2 2,...) ȳ = (1, y 10, y 01, y 20, y 11, y 02,...) p(x) = p T b(x) L ȳ (p) = p T ȳ Model interplay of variables using moment matrix: 1 x 1 x 2 x y 10 y 01 y 20 M(ȳ) = L ȳ (b(x)b(x) T ) = x 1 y 10 y 20 y 11 y 30 x 2 y 01 y 11 y 02 y 21 x 1 2 y 20 y 30 y 21 y Model constraints using localization matrices: g(x) 0 M(gȳ) = Lȳ(g(x)b(x)b(x) T ) 0 A(x) 0 M(Aȳ) = Lȳ(b(x)b(x) T A(x)) 0

30 Moment relaxation (Lasserre, Parrilo; Putinar, Henrion; Hol, Scherer..) Linearize polynomials in the PMI: b(x) = (1, x 1, x 2, x1 2, x 1 x 2, x2 2,...) ȳ = (1, y 10, y 01, y 20, y 11, y 02,...) p(x) = p T b(x) L ȳ (p) = p T ȳ Truncate to matrices coming from monomials of degree at most 2t: Truncated moment matrix: M t (ȳ) Truncated localization matrices: M t (gȳ) M t (Aȳ) By restricting to moments up to a certain degree, we get a semidefinite program.

31 Moment relaxation (Lasserre, Parrilo; Putinar, Henrion; Hol, Scherer..) Linearize polynomials in the PMI: b(x) = (1, x 1, x 2, x1 2, x 1 x 2, x2 2,...) ȳ = (1, y 10, y 01, y 20, y 11, y 02,...) p(x) = p T b(x) L ȳ (p) = p T ȳ Relaxation for the containment PMI: µ = inf z T B(x)z s.t. A(x) 0 1 z T z 0 µ mom (t) = inf L ȳ (z T B(x)z) s.t. M t (ȳ) 0 M t 1 (A(x)ȳ) 0 M t 1 (1 z T zȳ) 0 Lemma The sequence µ mom (t) for t 2 is monotone increasing. If for some t the condition µ mom (t ) 0 is satisfied then S A S B.

32 Moment relaxation (Lasserre, Parrilo; Putinar, Henrion; Hol, Scherer..) Linearize polynomials in the PMI: b(x) = (1, x 1, x 2, x1 2, x 1 x 2, x2 2,...) ȳ = (1, y 10, y 01, y 20, y 11, y 02,...) p(x) = p T b(x) L ȳ (p) = p T ȳ Relaxation for the containment PMI: µ = inf z T B(x)z s.t. A(x) 0 1 z T z 0 µ mom (t) = inf L ȳ (z T B(x)z) s.t. M t (ȳ) 0 M t 1 (A(x)ȳ) 0 M t 1 (1 z T zȳ) 0 Theorem Let A(x) S k [x] be a linear pencil such that the spectrahedron S A is bounded. Then µ mom (t) µ as t.

33 Relations between relaxations Theorem Assume S A is bounded. For the properties (1) The SDFP (1) has a solution C 0, (2) The optimal value µ mom (2) 0 (and thus µ mom (t) 0 for all t 2), (3) The optimal value of the PMI is zero and thus S A S B, we have the implications (1) = (2) = (3). Exactness cases carry over

34 Relations between relaxations II Theorem (Longer version) Let S A is bounded and A 0,..., A n lin. independent. For the properties (1) ˆΦ AB is completely positive, (1 ) The SDFP (1) has a solution C 0, (2) The optimal value µ mom (2) 0 (and thus µ mom (t) 0 t 2), (3) The optimal value of the PMI is zero and thus S A S B, (3 ) ˆΦ AB is positive. we have the implications (1 ) (1) = (2) = (3) (3 ). ˆΦ : A B natural linear map, on extended linear pencil. (2) = (1) is open (initial relaxation step t = 2 exact ˆΦ AB is completely positive?) exactness of t-th relaxation step and (k + 2 t)-positivity?

35 Conclusion and question Containment problems and their complexity Sufficient criteria and PMI hierarchy for containment of spectrahedra Exact in certain cases (spectrahedron in polytope, for extended form: spectrahedron in polyhedron) Q: Further/more general quantitative versions (for certain classes)? Projections, rotations...