Symbolic computation in hyperbolic programming

Transcription

1 Symbolic computation in hyperbolic programming Simone Naldi and Daniel Plaumann MEGA 2017 Nice, 14 June 2017

2 1 Polyhedra P = {x R n : i, l i (x) 0} Finite # linear inequalities : l i (x) 0, i = 1,..., m One can associate a diagonal matrix : A(x) = diag(l i (x)) f = l 1 (x)l 2 (x) l m (x) = det A(x) Feasibility Problem: Is P =? ( x R n s.t. l i (x) 0, i) Linear Programming (LP): Compute inf l(x) s.t. x P Algebraic degree = 1 (solutions are rational ) Combinatorics of boundary active constraints l i

3 2 Spectrahedra Start with a symmetric linear matrix : A(x) = A 0 + x 1 A x n A n Infinite # linear ineq. y T A(x)y 0 Finite # Non-linear inequalities : {Principal Minors of A(x)} 0 S = {x R n : A(x) 0} f = det A(x) Feasibility Problem (LMI): x R n s.t. A(x) 0 Semidefinite Progr. (SDP): Compute inf l(x) s.t. x S Algebraic degree = δ(size,var,rank) ( irrational sol.) Combinatorics of boundary co-rank of A(x)

4 3 Hyperbolicity cones C(f, e) hyperbolicity cone Input data : a polynomial and a vector f R[x 1,..., x n ] d, e R n Finite # Non-linear inequalities : Coeff. of t f(te x) f hyperbolic w.r. to e Feasibility Problem? By definition e Int(C(f, e))!! Hyperbolic Progr. (HP): Compute inf l(x) s.t. x C(f, e) Algebraic degree =?(deg,n,mult) ( irrational solutions) Combinatorics of boundary multiplicity of x

5 4 Recap Cone Polynomial Optimization Boundary Hyperbolic HP Multiplicity (Slide 6) (Slide 7) f = det A(x) SDP Co-rank of A(x) f = l i (x) LP Active constraints Two different approaches: Interior approach (e.g. classical interior-point methods) Boundary approach (more suitable for algebraic methods) General goal: certification of information on the solution

6 5 Certify what? For sums of squares (SOS), one would like to know (1) how many squares you need and (2) where the SOS-certificate lives. For instance, the polynomial f = a 4 + ab 3 + b 4 3a 2 bc 4ab 2 c + 2a 2 c 2 + ac 3 + bc 3 + c 4 Q[a, b, c] is a SOS in R[a, b, c], but not in Q[a, b, c]. Actually it is SOS in some F[a, b, c] with F Q algebraic. What is F : Q? We associate to f a linear matrix A(x), and solve A(x) 0. Then r = rank A(x ) = The length of f = f f 2 r Algebraic degree of entries of A(x ) = F : Q Numerical IP methods will not give these information

7 6 Hyperbolic polynomials... Definition of hyperbolic polynomial f R[x] d is hyperbolic w.r.t. e = (e 1,..., e n ) R n if f(e) 0 (we suppose w.l.o.g. f(e) = 1) a R n t ch a (t) := f(t e a) has only real roots If such e exists, f is called a hyperbolic polynomial. Fundamental examples: General case (here d = 4) : (1) f = x 1 x d ch a (t) = i(te i a i ) (2) f = det X, X sym. ch a (t) = det(ti d A) R 3 P 2 Brändén (2010): hyperb. polynomials without determinantal representations

8 7... Cones and Multiplicity Hyperbolicity cone The hyperbolicity cone of f R[x] d (w.r.t. e) is C(f, e) = {a R n : ch a (t) = 0 t 0} Multiplicity: For a R n, we define mult(a) := multiplicity of 0 as root of ch a (t) = f(te a) Multiplicity set: For m d, Γ m = {a R n : mult(a) m} Remark: The set Γ m is real algebraic. Indeed, if ch a (t) = t d + g 1 (a)t d g d 1 (a)t + g d (a) then Γ m = {a : g i (a) = 0, i d m + 1}

9 8 Motivation in real algebraic geometry Generalized Lax conjecture Every hyperbolicity cone is a spectrahedron, that is A 1,..., A n such that C(f, e) = {x R n : A 1 x A n x n 0} By Brändén s result, Lax conjecture cannot be proved by proving that every hyperbolic polynomial admits a determinantal representation. Kurdyka, Paunescu (2015) Characterisation of Nuij-type perturbations of hyperbolic polynomials Leykin, Plaumann (2012) Determinantal representations via homotopy Kummer (2015) Partial results towards Lax conjecture, for hyperbolic polynomials without real singularities.

10 9 This work: Algebraic approach to HP We prove that every hyperbolic programming problem is equivalent to linear optimization over some multiplicity locus. Moreover, computing max mult(x) over C(f, e) is equivalent to the computation of witness points on connected components of the multiplicity loci (classical problem in R.A.G.) All the reduced problems have simply exponential complexity with respect to the number n of variables. Reference arxiv:

11 10 A test on a hyperbolic quartic Optimizing random linear forms yields solutions of multiplicity 1 for 64% of the times multiplicity 2 for 36% of the times One can get rational solutions (multiplicity 2) : x 1 = 0 x 2 = 1/2 x 3 = 0 or irrational smooth boundary solutions (mult 1) : x 1 [ , ] x 2 [ , ] x 3 [ , ]

12 11 Renegar s derivative cones Based on the remark that f hyperbolic w.r. to e D e f = i e i f x i still hyperbolic This gives a nested sequence of convex hyperbolicity cones: C(f, e) C(D e f, e) C(D (d 1) e f, e) (the last one being a half-space), giving a sequence of lower bounds for the linear function to optimize: inf l(a) C(f,e) inf l(a) C(D e f,e) inf C(D (d 1) e f,e) l(a)

13 12 The quartic Quartic hyperbolic polynomial

14 12 The quartic + First derivative

15 13 Renegar s derivative cones (continued) Why is Renegar s method useful from an effective viewpoint? At each step of the relaxation, the degree of the polynomial decreses by 1 The algebraic degree of the solution might decrease, the multiplicity decreases (the solution becomes smoother at each step) One of the C(D e (j) f, e) could be a section of the PSD cone (solution set of a LMI), in which case a lower bound can be computed by solving a single SDP: E.g. Sanyal (2013) constructs explicit spectrahedral representations for derivative cones of polyhedra

16 14 Planar 3 ellipse Here I minimize a random linear function over the 3-ellipse l = 55x 94y + 87z 56, f has degree 8 = 2 3, and: C(f, e) = {P R 2 : d(p, P 1 ) + d(p, P 2 ) + d(p, P 3 ) c} Deriv. Min Value Mult. Degree time(s) Still OK for 4 ellipse, becoming hard for 5 ellipse

17 15 Summary Hyperbolic polynomials (resp. HP) is a rich class of real polynomials, defining highly-structured optimization problems. It is important to develop an effective approach to hyperbolic polynomials, independent on their determinantal representability to hyperbolicity cones, independent on Lax conjecture Algebraic alternatives to interior-point methods for polynomial optimization Complexity of HP: polynomial in the number of variables with fixed degree? True for generic SDP

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21 Historical motivation The Cauchy problem: Given f R[x] d and Ω R n open : Given p C (Ω) compute u C (Ω) such that f( 1,..., n )u = p Theorem (Lax, Mizohata). Decompose f = i d f i with f i R[x] i. If the Cauchy problem is well-posed (existence/uniqueness of solutions) then f d is hyperbolic. Example: The Wave operator in ( t 2 i i 2 )u = p corresponds to the polynomial f = x 2 n+1 n i=1 x 2 i, which is hyperbolic in direction e = (0,..., 0, 1). Its hyp. cone is the second-order (or Lorentz) cone C(f, e) = {x R n+1 : x n+1 x x2 n}

22 Open questions Complexity of HP: polynomial in the number of variables with fixed degree? True for generic SDP Can we combine exact and numerical methods for SDP or HP, still being able to certify output information? For instance, can we perturb a spectrahedron, preserving the same rank/multiplicity structure of the boundary? How about hyperbolicity cones?