Convexity and SOS-Convexity

Transcription

1 Conveit and SOS-Conveit Amir Ali Ahmadi Pablo A. Parrilo Laborator for Information and Decision Sstems Massachusetts Institute of Technolog SIAM Conference on Alied Algebraic Geometr NCSU, October 0

2 Deciding Conveit [Magnani, Lall, Bod] Given a multivariate olnomial, how to decide if it is conve? How to search and otimize over a famil of conve olnomials? Alications: Global otimization Conve data fitting, conve enveloes, Launov analsis, defining norms

3 Alication in Launov analsis Suose we are given m discrete dnamical sstems: Suose we can find a conve common Launov function: Then, an other dnamical sstem is also globall asmtoticall stable. Proof:

4 Outline A word on comleit of deciding conveit SOS-Conveit: an algebraic SDP based relaation for conveit Equivalent characterizations Conve but not sos-conve olnomials Comlete characterization of the ga between conveit and sos-conveit An oen roblem! 4

5 Comleit of deciding conveit Inut to the roblem: an ordered list of the coefficients all rational Degree d odd: trivial d=, i.e., = T Q+q T +c : check if Q is PSD d=4, first interesting case Question of N. Z. Shor: What is the comleit of deciding conveit of a multivariate olnomial of degree four? aeared on a list of seven oen roblems in comleit of numerical otimization in 99, [Pardalos, Vavasis] Our recent result: roblem is strongl NP-hard Same result for strong, strict, quasi, seudo-conveit *Ahmadi, Olshevsk, Parrilo, Tsitsiklis, 0+

6 Nonnegativit and Sum of Squares Deciding if a olnomial is nonnegative sd: NP-hard when d 4 Deciding if a olnomial is a sum of squares sos: an SDP sos sd obvious, but what about the converse? 6

7 From Logicomi sd=sos? Polnomials n,d 4 6 es es es es es no 3 es no no 4 es no no Motzkin 967: Hilbert s 888 Paer Forms n,d 4 6 es es es es es es 3 es es no 4 es no no Robinson 973: 7

8 SOS-Conveit Defn. [Helton, Nie]: A olnomial sos-conve if its Hessian factors as T H M M is for a ossibl nonsquare olnomial matri We call such matrices an sos-matri. Equivalent Defn. Deciding sos-conveit: an SDP! 8

9 9 Equivalent characterizations of conveit T, H T, 0 Basic definition: First order condition: Second order condition: Algebraic sos-based relaations: siml relace with sos [0,], enough

10 0 Each condition can be SOS-ified equivalent to SOS-Conveit: H an SOS-matri [0,], SOS g SOS g T, SOS H g T, Basic definition: First order condition: Second order condition: SOS g, A B C A M M H T *Ahmadi, Parrilo, 0+

11 Conve olnomials that are not sos-conve? [0,], g, g T H g T, B C A A conve but not sos-conve ol would be a ol such that: are sd but not sos.

12 The first countereamle Claim: is conve: H is PSD is not sos-conve: H M T M A homogeneous olnomial in 3 variables, of degree *Ahmadi, Parrilo, 09+

13 Conve but not SOS-Conve, lower degree? = *Ahmadi, Parrilo, 0+ To elain where this comes from, need to talk first about biquadratic forms 3

14 Biquadratic forms and biquadratic Hessian forms Biquadratic form: where is a matri whose entries are quadratic forms Eamle: the celebrated Choi biquadratic form with Biquadratic Hessian form: Secial biquadratic form where Choi matri is not a valid Hessian: is a valid Hessian 4

15 From biquadratic forms to biquadratic Hessian forms We give a constructive rocedure to go from an biquadratic form to a biquadratic Hessian form b doubling the number of variables, such that: Proves NP-hardness of checking conveit of quartic forms! nonnegativit of biquadratic forms known to be NP-hard; *Gurvits, 03+, *Ling et al., 09+, reduction from CLIQUE via Motzkin-Straus Gives an elicit method to construct quartic conve but not sos-conve olnomials! 5

16 Reduction on an instance A 66 Hessian with quadratic form entries

17 Conve but not SOS-Conve Quartic = *Ahmadi, Parrilo, 0+ Has the Hessian we just resented 7

18 Conveit = SOS-Conveit? Polnomials Forms n,d 4 6 n,d 4 6 es es es es es es es es no es es es 3 es no no 3 es es no 4 es no no 4 es no no *Ahmadi, Parrilo, 0+ Thm: Ever conve ternar quartic is SOS-conve. *Ahmadi, Blekherman, Parrilo, 0+ 8

19 Conve=SOS-conve for bivariate quartics Thm the biform theorem : Let be a form that is quadratic in v for fied u and a form of however large degree in u for fied v. Then, f is sd iff it is sos. 9

20 Conve=SOS-conve for ternar quartic forms =,, 3 form of degree 4 Let H be its Hessian biquadratic form Biquadratic from in 6 variables can be sd but not sos e.g., Choi But H has a secial smmetr: H= H Even with this smmetr, can have a sd but not sos biquadratic form: 'A = *Ahmadi, Blekherman, Parrilo, 0+ But H has et a little more structure 0

21 Minimal countereamle: n=3, d=6 Not sos-conve: Conve:

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23 sd=sos? conve=sos-conve? n,d 4 6 es es es es es no 3 es no no 4 es no no A mere coincidence? Long stor short Polnomials Forms n,d 4 6 es es es es es es 3 es es no 4 es no no 3

24 Nonnegativit and conveit: some similarities Both never hold for odd degree olnomials Nonnegativit=conveit for quadratic forms Both NP-hard for degree 4 or larger Nonnegativit = SOS if and onl if conveit=sos-conveit Perhas there are more and deeer connections? 4

25 Oen roblem Find a conve form that is not SOS. Blekherman 009 has shown that the eist! Has imlications for algebraic methods in olnomial otimization Conve forms are nonnegative sos-conve sos So, such a olnomial must necessaril be conve but not sos-conve Our countereamles ass this necessar condition but the olnomials themselves are sos 5

26 Thank ou for our attention! Questions? Want to know more? htt://aaa.lids.mit.edu 6