Exact algorithms for linear matrix inequalities

Transcription

1 0 / 12 Exact algorithms for linear matrix inequalities Mohab Safey El Din (UPMC/CNRS/IUF/INRIA PolSys Team) Joint work with Didier Henrion, CNRS LAAS (Toulouse) Simone Naldi, Technische Universität Dortmund SMAI-MODE, 2016

2 1 / 12 Spectrahedra and LMI A 0, A 1,..., A n are m m real symmetric matrices Spectrahedron: S = {x R n : A(x) = A 0 + x 1 A x na n 0} It is basic semi-algebraic since, if det(a(x) + ti m) = f m(x) + f m 1 (x)t + + f 1 (x)t m 1 + t m then S = {x R n : f i (x) 0, i = 1,..., m}. A(x) 0 is called an LMI.

3 1 / 12 Spectrahedra and LMI A 0, A 1,..., A n are m m real symmetric matrices Spectrahedron: S = {x R n : A(x) = A 0 + x 1 A x na n 0} It is basic semi-algebraic since, if det(a(x) + ti m) = f m(x) + f m 1 (x)t + + f 1 (x)t m 1 + t m then S = {x R n : f i (x) 0, i = 1,..., m}. A(x) 0 is called an LMI. SDP : linear optimization over S (i.e. over LMI) 1 x 1 x 2 S = (x 1, x 2, x 3 ) R 3 : x 1 1 x 3 0 x 2 x 3 1 Figure: The Cayley spectrahedron

4 2 / 12 Why exact algorithms? 1. It is Hard to compute low-rank solutions to SDP Figure: Low-rank points : they minimize a cone of linear forms Figure: SEDUMI returns a floating point approximation of (0, 0) when maximizing x 2 2. The interior of S can be empty Interior point algorithms could fail 1 0 x 1 (1 x 2 4) x 1 x 2 x (1 x 2 4) x 3 x 4

5 2 / 12 Why exact algorithms? 1. It is Hard to compute low-rank solutions to SDP 2. The interior of S can be empty Interior point algorithms could fail x 1 (1 x 2 4) x 1 x 2 x (1 x 2 4) x 3 x Main motivations for the design of exact algorithms: 1. Can we manage algebraic constraints such as rank defects? 2. Can we handle degenerate non-full-dimensional examples? 3. Consequence: The output is a point whose coordinates may be real algebraic numbers {( q1 (t) (q, q 0, q 1,..., q n) Q[t] q 0 (t),..., qn(t) ) } : q(t) = 0 q 0 (t)

6 3 / 12 State of the art Decision / Sampling problem for real algebraic or semi-algebraic sets Cylindrical Algebraic Decomposition Tarski (1948), Seidenberg, Cohen,... Collins (1975) in O((2m) 22n+8 m 2n+6 ),... Critical Points Method local extrema of algebraic maps f on S Grigoriev, Vorobjov (1988) first singly exp: m O(n2 ) Renegar (1992), Heintz Roy Solernó (1989,1993), Basu Pollack Roy (1996,... ) linear exponent m O(n) Polar varieties local extrema of linear projections π on S Bank, Giusti, Heintz, Mbakop, Pardo (1997,... ) Safey El Din, Schost (2003,2004) regular in O(m 3n ), singular in O(m 4n ) The goal was: Better results for spectrahedra? How to take advantage of the structure?

7 4 / 12 Complexity of SDP Special case of SDP Khachiyan, Porkolab (1996) decide LMI-feasibility in time O(nm 4 ) + m O(min{n,m2 }) on (lm O(min{n,m2 }) )-bit numbers (l = input bit-size)

8 4 / 12 Complexity of SDP Special case of SDP Khachiyan, Porkolab (1996) decide LMI-feasibility in time O(nm 4 ) + m O(min{n,m2 }) on (lm O(min{n,m2 }) )-bit numbers (l = input bit-size) Positive aspects: 1. No assumptions, Deterministic 2. Binary complexity

9 4 / 12 Complexity of SDP Special case of SDP Khachiyan, Porkolab (1996) decide LMI-feasibility in time O(nm 4 ) + m O(min{n,m2 }) on (lm O(min{n,m2 }) )-bit numbers (l = input bit-size) Positive aspects: 1. No assumptions, Deterministic 2. Binary complexity Main drawbacks: 1. It relies on Quantifier Elimination 2. Too large constant in the exponent

10 5 / 12 Low rank positive semidefinite matrices Define: For any A(x) (not nec. symmetric): D r = {x C n : rank A(x) r} For A(x) symmetric, and S : r(a) = min{rank A(x) x S } So one has nested sequences D 0 D m 1 D 0 R n D m 1 R n

11 5 / 12 Low rank positive semidefinite matrices Define: For any A(x) (not nec. symmetric): D r = {x C n : rank A(x) r} For A(x) symmetric, and S : r(a) = min{rank A(x) x S } So one has nested sequences D 0 D m 1 D 0 R n D m 1 R n Smallest Rank Property Henrion-N.-Safey El Din 2015 A(x) symmetric, and S. Let C be a conn. comp. of D r(a) R n s.t. C S. Then C S. In particular C D r(a) \ D r(a) 1.

12 5 / 12 Low rank positive semidefinite matrices Define: For any A(x) (not nec. symmetric): D r = {x C n : rank A(x) r} For A(x) symmetric, and S : r(a) = min{rank A(x) x S } So one has nested sequences D 0 D m 1 D 0 R n D m 1 R n Smallest Rank Property Henrion-N.-Safey El Din 2015 A(x) symmetric, and S. Let C be a conn. comp. of D r(a) R n s.t. C S. Then C S. In particular C D r(a) \ D r(a) 1. Either the set S is empty

13 5 / 12 Low rank positive semidefinite matrices Define: For any A(x) (not nec. symmetric): D r = {x C n : rank A(x) r} For A(x) symmetric, and S : r(a) = min{rank A(x) x S } So one has nested sequences D 0 D m 1 D 0 R n D m 1 R n Smallest Rank Property Henrion-N.-Safey El Din 2015 A(x) symmetric, and S. Let C be a conn. comp. of D r(a) R n s.t. C S. Then C S. In particular C D r(a) \ D r(a) 1. Either the set S is empty Or r(a) is well-defined C D r(a) : C S

14 5 / 12 Low rank positive semidefinite matrices Smallest Rank Property Henrion-N.-Safey El Din 2015 A(x) symmetric, and S. Let C be a conn. comp. of D r(a) R n s.t. C S. Then C S. In particular C D r(a) \ D r(a) 1. Either the set S is empty Or r(a) is well-defined C D r(a) : C S

15 6 / 12 Problem statement Emptiness of spectrahedra Given A(x) symmetric, with entries in Q, compute a finite set meeting S = {x R n : A(x) 0}, or establish that S is empty. In other words: Decide the feasibility of an LMI A(x) 0. Particular instance of: Decide the emptiness of semi-algebraic sets.

16 6 / 12 Problem statement Emptiness of spectrahedra Given A(x) symmetric, with entries in Q, compute a finite set meeting S = {x R n : A(x) 0}, or establish that S is empty. In other words: Decide the feasibility of an LMI A(x) 0. Particular instance of: Decide the emptiness of semi-algebraic sets. Sample points on S + Smallest Rank Property = Sample points on D r(a) R n

17 Problem statement Emptiness of spectrahedra Given A(x) symmetric, with entries in Q, compute a finite set meeting S = {x R n : A(x) 0}, or establish that S is empty. In other words: Decide the feasibility of an LMI A(x) 0. Particular instance of: Decide the emptiness of semi-algebraic sets. Sample points on S + Smallest Rank Property = Sample points on D r(a) R n Real root finding on determinantal varieties Given any A(x) with entries in Q, compute a finite set meeting each connected component of D r R n = {x R n : rank A(x) r}. Particular instance of: Sampling real algebraic sets. 6 / 12

18 7 / 12 Strategy 1. The Smallest Rank Property ( C D r(a) : C S ) allows to reduce: Sampling/Optimization over One semi algebraic set Sampling/Optimization over Many algebraic sets This is somehow typical in PO. Ex. Polar Varieties for PO: Safey El Din, Greuet

19 7 / 12 Strategy 1. The Smallest Rank Property ( C D r(a) : C S ) allows to reduce: Sampling/Optimization over One semi algebraic set Sampling/Optimization over Many algebraic sets This is somehow typical in PO. Ex. Polar Varieties for PO: Safey El Din, Greuet But in our case the structure is preserved!

20 7 / 12 Strategy 1. The Smallest Rank Property ( C D r(a) : C S ) allows to reduce: Sampling/Optimization over One semi algebraic set Sampling/Optimization over Many algebraic sets This is somehow typical in PO. Ex. Polar Varieties for PO: Safey El Din, Greuet But in our case the structure is preserved! 2. For r = 1,..., m 1 compute sample points in D r R n 3. Output the minimum rank on S with a sample point.

21 7 / 12 Strategy 1. The Smallest Rank Property ( C D r(a) : C S ) allows to reduce: Sampling/Optimization over One semi algebraic set Sampling/Optimization over Many algebraic sets This is somehow typical in PO. Ex. Polar Varieties for PO: Safey El Din, Greuet But in our case the structure is preserved! 2. For r = 1,..., m 1 compute sample points in D r R n 3. Output the minimum rank on S with a sample point. Sampling determinantal varieties Either the empty list iff D r R n = Or (q, q 1,..., q n) Q[t] s.t. C D r R n t : x(t) C

22 7 / 12 Strategy 1. The Smallest Rank Property ( C D r(a) : C S ) allows to reduce: Sampling/Optimization over One semi algebraic set Sampling/Optimization over Many algebraic sets This is somehow typical in PO. Ex. Polar Varieties for PO: Safey El Din, Greuet But in our case the structure is preserved! 2. For r = 1,..., m 1 compute sample points in D r R n 3. Output the minimum rank on S with a sample point. Sampling determinantal varieties Either the empty list iff D r R n = Or (q, q 1,..., q n) Q[t] s.t. C D r R n t : x(t) C Emptiness of spectrahedra Either the empty list iff S = Or (q, q 1,..., q n) Q[t] s.t. t : x(t) S

23 8 / 12 Incidence varieties and critical points 1st step Lifting of the determinantal variety: y 1,1... y 1,m r A(x) Y (y) = A(x).. = 0. π1 (x2, y) C π1 y m,1... y m,m r x1 U Y (y) = I m r If A is generic, the lifted algebraic set V r is smooth and equidimensional 2nd step Compute critical points of the map π(x, y) = a 1 x a nx n on V r : When a 1.. a n are generic, there are finitely many critical points. 3rd step Intersect with any fiber of π and call point 2

24 8 / 12 Incidence varieties and critical points 1st step Lifting of the determinantal variety: y 1,1... y 1,m r A(x) Y (y) = A(x).. = 0. π1 (x2, y) C π 1 1 y m,1... y m,m r x1 U Y (y) = I m r If A is generic, the lifted algebraic set V r is smooth and equidimensional 2nd step Compute critical points of the map π(x, y) = a 1 x a nx n on V r : When a 1.. a n are generic, there are finitely many critical points. 3rd step Intersect with any fiber of π and call point 2

25 9 / 12 Computing critical points on incidence varieties A(X)Y = 0, UY = I m r F 1 (X, Y ) = = F m(m r) (Y ) = = F m(m r)+(m r) 2 = 0 Lagrange System L.jac(F, [X, Y]) = [a 0]

26 Computing critical points on incidence varieties Multi-Linear System F(X, A(X)Y Y) = = 0, 0, G(X, UY L) = = I m r 0, H(Y, L) = 0 All equations have multi-degree (1, 1, 0) or (1, 0, 1) or (0, 1, 1) F 1 (X, Y ) = Impact = F m(m r) on multi-linear/sparsity (Y ) = = F m(m r)+(m r) 2 = 0 structure on the number of solutions Multi-linear Bézout bounds Symbolic-homotopy algorithms Lagrange System cubicl.jac(f, complexity [X, in Y]) these = [abounds 0] Symbolic Newton iteration (Hensel lifting) Jeronimo/Matera/Solerno/Waissbein 9 / 12

27 Complexity bounds Complexity for Sampling determinantal varieties ( ) 6 O (n + m 2 r 2 ) 7 n + m(m r) n Complexity for Emptiness of spectrahedra O n ( ) ( ) 6 m (n + p r + r(m r)) 7 p r + n r n r r(a) O (k) = O(k log c k) c N with p r = (m r)(m + r + 1)/2. Remarkable aspects: Explicit constants in the exponent When m is fixed, polynomial in n Strictly depends on r(a) 10 / 12

28 11 / 12 SPECTRA: a library for real algebraic geometry and optimization What is SPECTRA? A MAPLE library, freely distributed Depends on Faugère s FGB for computations with Gröbner bases Addressed to researchers in Optimization, Convex alg. geom., Symb. comp. (m, r, n) RAGLIB SPECTRA deg (3, 2, 8) (3, 2, 9) (4, 2, 5) (4, 2, 6) (4, 2, 7) (4, 2, 8) (4, 2, 9) (4, 3, 10) (4, 3, 11) (5, 2, 9) (6, 5, 4) RAGLIB = Real algebraic geometry library - SPECTRA = new algorithms - deg = degree of Rational Parametrization - Time in seconds - = more than 2 days Download a beta version: homepages.laas.fr/snaldi/software.html

29 12 / 12 Scheiderer s spectrahedron f = u u 1 u u 4 2 3u 2 1u 2 u 3 4u 1 u 2 2u 3 + 2u 2 1u u 1 u u 2 u u 4 3 One can write f = v A(x)v with v = [u 2 1, u 1 u 2, u 2 2, u 1 u 3, u 2 u 3, u 2 3] 1 0 x 1 0 3/2 x 2 x 3 0 2x 1 1/2 x 2 2 x 4 x 5 x A(x) = 1 1/2 1 x 4 0 x 6 0 x 2 x 4 2x x 5 3/2 x 2 2 x 4 0 x 5 2x 6 1/2 x 3 x 5 x 6 1/2 1/2 1 What information can be extracted? No matrices of rank 1 s.t. A(x) 0 f g 2 Two matrices of rank 2 s.t. A(x) 0 f = g1 2 + g2 2 = g3 2 + g4 2 No matrices of rank 3 s.t. A(x) 0 f h1 2 + h2 2 + h3 2

30 12 / 12 Perspectives 1. Remove genericity assumptions on the input linear matrix A 2. Use of numerical homotopy for studying incidence varieties 3. Theoretical toolbox for analyzing singularities of determinantal varieties Surprising applications in optimal control techniques for the contrast imaging problem in medical imagery joint work with B. Bonnard, J.-C. Faugère, A. Jacquemard, T. Verron.