SOS hierarchies we could have had in the 1920s

Transcription

1 SOS hieachies we could have had in the 1920s Ami Ali Ahmadi Pinceton, ORFE Affiliated membe of PACM, COS, MAE, CSML Geogina Hall Pinceton, ORFE-->INSEAD ISMP 2018, Bodeaux 1

2 How to pove positivity? Is p x > 0 on {g 1 x 0,, g m x 0}? Why pove positivity? (Tight) lowe bounds fo polynomial minimization poblems Infeasibility cetificates fo systems of polynomial inequalities {g 1 x 0, g 2 x 0,, g m x 0} g 1 x > 0 on {g 2 x 0,, g m x 0} empty Dynamics and contol (Lyapunov functions) Stats/ML (Geogina s talk) 2

3 Positivstellensätze Atin Stengle p x > 0, x R n p x > 0, x S = x g i x 0} th centuy Schmudgen If p x > 0, x S, then p x = σ 0 x + i σ i x g i x + x g i x g j x +, whee σ 0, σ i, sos ij σ ij If p x 0, x R n, then sos q s.t. p q sos. (Requies some compactness assumptions) Putina If p x > 0, x S, then p x = σ 0 x + i σ i x g i (x), whee σ 0, σ i ae sos Seach fo these sos polynomials (when degee is fixed) --->SDP. 3

4 Motivation/Outline Can we get away with less? (in ode to poduce conveging hieachies of lowe bounds fo polynomial optimization poblems) Q1: Do we eally need Stengle, Schmudgen, Putina,? Can we only use cetificates of global positivity? (e.g., Atin s) Pat I: A meta-theoem Q2: Do we eally need SDPs (o convex optimization)? Pat II: An optimization-fee Positivstellensatz Caveat: Have you theoetical hat on! 4

5 A meta-theoem fo poducing hieachies Theoem: Let {K n,2d } be a sequence of sets of homogeneous polynomials in n vaiables and of degee 2d. If (1) K n,2d 0 P n,2d and s n,2d pd in K n,2d (2) p > 0 N s. t. p K n,2d (3) K n,2d K +1 n,2d (4) p K n,2d p + εs n,2d K n,2d, ε [0,1], then, K n,2d P n,2d POP min p(x) x Rn s. t. g i x 0, i = 1,, m 2d = maximum degee of p, g i R: adius of the feasible set = opt. val. max γ s. t. f γ z 1 s n+m+3,4d z K n+m+3,4d γ whee f γ is a fom in n + m + 3 vaiables of degee 4d which can be witten down explicitly fom p, g i, R. 5

6 What is f γ? + Poof idea (1/2) min p(x) x Rn s. t. g i x 0, i = 1,, m 2d = maximum degee of p, g i R: adius of the feasible set p x > γ on x g i x 0} f is positive definite (pd) 6

7 Poof idea (2/2) p x > γ on x g i x 0} f is positive definite (pd) (1) K n,2d 0 P n,2d and s n,2d pd in K n,2d (2) p > 0 N s. t. p K n,2d (3) K n,2d K +1 n,2d (4) p K n,2d p + εs n,2d K n,2d, ε [0,1] K n,2d P n,2d POP min p(x) x Rn s. t. g i x 0, i = 1,, m = opt. val. max γ s. t. f γ z 1 s n+m+3,4d z K n+m+3,4d γ 7

8 Families of cones that satisfy (1)-(4) 0 (1) K n,2d P n,2d and s n,2d pd in K n,2d (2) p > 0 N s. t. p K n,2d (3) K n,2d K +1 n,2d (4) p K n,2d p + εs n,2d K n,2d, ε [0,1] K n,2d P n,2d Examples: Atin cones : A n,2d = p p q is sos fo some sos q of degee 2} Reznick cones : R n,2d = p p x x n is sos} (both lead to SDP-based hieachies fo polynomial optimization) 8

9 Enough to poduce a conveging hieachy fo POPs Atin Stengle p x > 0, x R n p x > 0, x S = x g i x 0} th centuy Schmudgen If p x > 0, x S, then p x = σ 0 x + i σ i x g i x + x g i x g j x +, whee σ 0, σ i, sos ij σ ij If p x 0, x R n, then sos q s.t. p q sos. (Requies some compactness assumptions) Putina If p x > 0, x S, then p x = σ 0 x + i σ i x g i (x), whee σ 0, σ i ae sos 9

10 Pat II 10

11 An optimization-fee Positivstellensatz (1/2) p x > 0, x x R n g i x 0, i = 1,, m} 2d =maximum degee of p, g i N such that f v 2 w 2 1 i v i 2 w i 2 2 d i v i 4 + w i 4 d i v i 2 + i w i 2 2 has nonnegative coefficients, whee f is a fom in n + m + 3 vaiables and of degee 4d, which can be explicitly witten fom p, g i and R. 11

12 An optimization-fee Positivstellensatz (2/2) Poof sketch: p x > 0 on x g i x 0} f v 2 w i v 4 4 d i w i i v i i w 2 i v 2 2 i w 2 d i i N s. t. f v 2 w 2 1 i i i i i v 2 2 i w 2 d i i v 4 4 d i + w i i v 2 2 i + i w 2 i p x > 0 on x g i x 0} f is pd Result by Polya (1928): has has coefficients coefficients f even and pd N such that f z i z i 2 has nonnegative coefficients. Make f(z) even by consideing f v 2 w 2. We lose positive definiteness of f with this tansfomation. Add the positive definite tem 1 2 i v i 4 + w i 4 d to f(v 2 w 2 ) to make it positive definite. Apply Polya s esult. The tem 1 i v i 2 w i 2 2 d ensues that the convese holds as well. As a coollay, gives LP/SOCP-based conveging hieachies 12

13 LP/SOCP-based altenatives to SOS 13 [AAA, Majumda]

14 LP and SOCP-based conveging hieachies fo POPs min p(x) x Rn s. t. g i x 0, i = 1,, m = opt. val. sup γ s.t. f γ v 2 w 2 1 i(v i 2 w i 2 ) 2 d i v i 4 + w i 4 d q(v, w) is s/dsos q(v, w) is s/dsos of degee 2, whee f γ is a fom in n + m + 3 vaiables and of degee 4d γ 2d = maximum degee of p, g i Unde compactness assumptions, i.e., x g i x 0} B(0, R) Fo lage enough 14

15 The main takeaway Positivstellensatze by Atin (1927) and Polya (1928) ae enough to poduce a conveging hieachy of lowe bounds fo polynomial optimization poblems with a bounded feasible set. The latte only involves polynomial multiplication. Want to know moe? 15

16 You ae codially invited Pinceton Day of Optimization Septembe 28, Thank you.