Hilbert s 17th Problem to Semidefinite Programming & Convex Algebraic Geometry Rekha R. Thomas University of Washington, Seattle
References Monique Laurent, Sums of squares, moment matrices and optimization over polynomials, IMA Volume, to appear Murray Marshall, Positive polynomials and sums of squares, AMS 2008 João Gouveia, Pablo Parrilo, Rekha Thomas, Theta bodies for polynomial ideals, preprint 2008.
Polynomial Optimization: p := inf{p(x) : x K} ( ) K = {x R n : g 1 0,...,g m 0,h 1 = 0,...,h s = 0} (basic semialgebraic set), p,g i,h j R[x] := R[x 1,...,x n ] K = R n unconstrained polynomial optimization m = 0 K =V R (I) real variety of I = h 1,...,h m R[x] non-convex optimization problem NP-hard even when deg(p) = 4 and K = R n. GOAL: Find a sequence of relaxations of ( ) by convex optimization problems that converges to ( ).
Examples (all NP complete) Partition problem: Given a 1,...,a n Z +, does there exist x {±1} n s.t. a i x i = 0? Yes p = 0 for p := ( n i=1 a ix i ) 2+ n i=1 (x2 i 1)2. Distance realization: Given distances d := (d i j ) R E, d is realizable in R k if there exists v 1,...,v n R k such that d i j = v i v j for all i j E. d realizable in R k p = 0 for p := i j E (d 2 i j k h=1 (x ih x jh ) 2) 2. 0/1 Integer programming: min{c x : Ax b, x 2 i = x i i [n]}
Semidefinite programming vector space: Symm(R n n ) trace product: A,B Symm(R n n ), A B := trace(ab) positive semidefinite: A 0 v t Av 0 for all v R n A = BB t for some B R n m all eigenvalues of A are non-negative semidefinite program (sdp): sup{c X : A j X = b j, j = 1,...,m, X 0} convex optimization, polynomial time algorithms Linear programming is SDP over diagonal matrices
Hilbert s 17th problem P n := {p R[x] : p(x) 0, x R n } non-negative polynomials Σ n := { h 2 j : h j R[x]} sum of squares (sos) of polynomials P n,d, Σ n,d polynomials of degree at most d inp n and Σ n Σ n P n and Σ n,d P n,d cones in R[x] Hilbert 1888: Σ n,d =P n,d n = 1 or d = 2 or (n,d) = (2,4). Hilbert s 17th problem: Is every p P n a sos of rational functions? Yes! Artin 1927 using Tarski s transfer principle and orderings on fields
Motzkin (1967): s(x,y) := 1 3x 2 y 2 + x 2 y 4 + x 4 y 2 s(x,y) P 2,6 \ Σ 2,6 a+b+c 3 3 abc a,b,c 0, a := 1, b := x 2 y 4, c := x 4 y 2 Blekherman (2006): for d fixed, many more nonnegative forms than sos forms as n Lasserre (2006): for n fixed, any nonnegative polynomial can be approximated arbitrarily closely by sos polynomials.
Testing for sos representations via SDP p = x 4 + 2x 3 y+3x 2 y 2 + 2xy 3 + 2y 4 sos A 0 s.t. p = (x 2,xy,y 2 ) a 11 a 12 a 12 a 22 a 13 a 23 a 13 a 23 a 33 x2 xy y 2 the sdp {A 0 : a 11 = 1,a 12 = 1,a 22 + 2a 13 = 3,a 23 = 1,a 33 = 2} is feasible ( ) 1 1 0 ex: A = B t 1 1 1 B for B = or B = 0 3/2 3/2 0 2 1 0 1/2 1/2 p = (x 2 + xy y 2 ) 2 +(y 2 + 2xy) 2 = (x 2 + xy) 2 + 3 2 (xy+y2 ) 2 + 1 2 (xy y2 ) 2
Sos relaxations for p := inf{p(x) : x K} p := sup{ρ : p ρ 0 on K} = sup{ρ : p ρ > 0 on K} K = R n : p sos := sup{ρ : p ρ is sos} CAUTION Motzkin polynomial: p sos = < p = 0 K = {x R n : g 1 0,...,g m 0} : p sos := sup{ρ : p ρ = s 0 + m i=1 s ig i, s i sos} to get sdps we look at successive truncations: p sos t := sup{ρ : p ρ = s 0 + m i=1 s ig i, s i sos, deg(s 0 ),deg(s i g i ) 2t} p sos t pt+1 sos psos p, lim t pt sos = p sos
Positivstellensatz (Krivine 1964, Stengle 1974) T := { s i (g i 1 1 g i 2 2 g i m m ) : (i 1,...,i m ) {0,1} m, s i sos} preorder non-negativity set of K = {x R n : g 1 0,...,g m 0}. Positivstellensatz: Given p R[x], (i) p > 0 on K p f = 1+g, f,g T (ii) p 0 on K p f = p 2k + g, f,g T (solves Hilbert s 17th prob) (iii) p = 0 on K p 2k T (iv) K = /0 1 T (compare: Hilbert s weak Nullstellensatz) Real Nullstellensatz: p vanishes on {x R n : h 1 = = h s = 0} p 2k + s = s j=1 u jh j for u j R[x],s sos. (Hilbert s strong NS)
Schmüdgen s & Putinar s refinements p := sup{ρ : p ρ > 0 on K} = sup{ρ : (p ρ) f = 1+g, f,g T }. Schmüdgen 1991: If K is compact then p > 0 on K p T. p Schm t = sup{ρ : (p ρ) = s i (g i 1 1 g i 2 2 g i m m ), s i sos, deg( ) t} (sdp, lim t p Schm t = p, involves 2 m terms!) Putinar 1993: If K is Archimedean then p > 0 on K p M. M := {s 0 + m i=1 s ig i : s 0,s i sos} quadratic module p Put t = sup{ρ : (p ρ) = s 0 + s i g i, s 0,s i sos, deg( ) t} (sdp, lim t p Put t = p, only m+1 terms!)
Moments of probability measures µ probability measure on R n : y α := R x α dµ moment of order α y := (y α : α N n ) moment sequence of µ K-moment problem: Characterize moment sequences of measures supported on K R n. y R Nn moment sequence: moment matrix: M(y) R Nn N n : M(y) (α,β) := y α+β truncated moment matrix: M t (y) principal submatrix of M(y) indexed by N n t
Moment relaxations for p := inf{p(x) : x K} y R Nn 2t (truncated) moment sequence of µ on K t d K, M t (y) 0, M t dk (g i y) 0, i = 1,...,m p = inf{ R K p(x)dµ : µ prob measure on K} = inf{p t y : y 0 = 1,y mom seq of µ on K} p mom := inf{p t y : y 0 = 1, M(y) 0, M(g i y) 0, i = 1,...,m} to get sdps we look at successive truncations: p mom t := inf{p t y : y 0 = 1, M t (y) 0, M t dk (g i y) 0, i, y R Nn 2t} p mom t pt+1 mom p mom p, lim t pt mom = p mom
Duality Recall:P = {p : p(x) 0, x R n }, Σ = { h 2 j : h j R[x]}. DefineM := {y R Nn : y mom seq of µ on R n } R[x] M := {y R Nn : M(y) 0} R[x] Haviland 1935P =M,P =M Berg,Christensen,Jensen 1979: Σ =M, Σ =M Corollary: (n = 1) M =M Hamburger s theorem K-versions of these dual cones sdp duality p sos t p mom t p sos p mom p
Theta Body of a Graph G = ([n],e) undirected graph STAB(G) := conv{χ S : S stable set in G} max stable set problem: max{ x i : x STAB(G)} (NP-hard) I G := x i : i [n], x i x j : i j E V (I G ) = {χ S : S stable set in G} Lovász 1980 TH(G) := {x R n : f(x) 0 f linear, f 1-sos mod I G } STAB(G) G perfect STAB(G) = TH(G) stable set problem solved in poly time if G perfect: max{ x i : x TH(G)} is an sdp
Lovász 1994: Which ideals I R[x] are perfect? ( f linear, f 0 onv R (I) f is 1-sos mod I). f R[x] k-sos mod I if f h 2 j mod I, deg(h j) k I is k-perfect if every linear f 0 mod I is k-sos mod I T H k (I) := {x R n : f(x) 0, f linear f k-sos mod I} TH 1 (I) TH 2 (I) conv(v R (I)) theta bodies for ideals Gouveia, Parrilo, T. 2008 T H k (I) is the closure of the projection of a spectrahedron I =I(S), S R n I k-perfect conv(v R (I)) = TH k (I) I =I(S), S finite I perfect for every facet inequality g(x) 0 of conv(s), S {g(x) = 0} {g(x) = c g } In this case, # facets, # vertices 2 n (both sharp)
y Convex Algebraic Geometry TH 1 (I) TH 2 (I) conv(v R (I)) Theta bodies approximate convex hulls of real varieties Compute theta bodies via sdp, convergence in many cases Gouveia, Parrilo, T. 2008 For I R[x], TH 1 (I) = {conv(v R (F)) : F convex quadric in I} Ex. S = {(0,0),(1,0),(0,1),(2,2)} 3 2.5 2 1.5 1 0.5 0 0.5 1 1 0.5 0 0.5 1 1.5 2 2.5 3 x