MIT Algebraic techniques and semidefinite optimization February 28, Lecture 6. Lecturer: Pablo A. Parrilo Scribe:???

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1 MIT 697 Algebraic techiques ad seidefiite optiizatio February 8, 6 Lecture 6 Lecturer: Pablo A Parrilo Scribe:??? Last week we leared about explicit coditios to deterie the uber of real roots of a uivariate polyoial Today we will expad o these thees, ad study two atheatical objects of fudaetal iportace: the resultat of two polyoials, ad the closely related discriiat The resultat will be used to decide whether two uivariate polyoials have coo roots, while the discriiat will give iforatio about the existece of ultiple roots Furtherore, we will see the itiate coectios betwee discriiats ad the boudary of the coe of oegative polyoials Besides the properties described above, a direct cosequece of their defiitios, there are ay other iterestig applicatios of resultats ad discriiat We describe a few of the below, ad we will ecouter the agai i later lectures, whe studyig eliiatio theory ad the costructio of cylidrical algebraic decopositios For uch ore iforatio about resultats ad discriiats, particularly their geeralizatios to the sparse ad ultipolyoial case, we refer the reader to the very readable itroductory article [Stu98] ad the book [CLO97] Resultats Cosider two polyoials p(x) ad q(x), of degree,, respectively We wat to obtai a easily checkable criterio to deterie whether they have a coo root, that is, there exists a x C for which p(x ) = q(x ) = There are several approaches, seeigly differet at first sight, for costructig such a criterio: Sylvester atrix: If p(x ) = q(x ) =, the we ca write the followig ( + ) ( + ) liear syste: p p p p + p(x )x x p + p(x )x x p p p(x )x x p p p p(x ) x q q q = = q(x )x q(x )x q x q(x )x q q q(x ) q q q This iplies that the atrix o the left had side, called the Sylvester atrix Syl x (p, q) associated to p ad q, is sigular ad thus its deteriat ust vaish It is ot too difficult to show that the coverse is also true; if det Syl x (p, q) =, the there exists a vector i the kerel of Syl x (p, q) of the for show i the atrix equatio above, ad thus a coo root x Root products ad copaio atrices: Let α j, β k be the roots of p(x) ad q(x), respectively By costructio, the expressio (α j β k ) j= k= vaishes if ad oly if there exists a root of p that is equal to a root of q Although the coputatio of this product sees to require explicit access to the roots, this ca be avoided Multiplyig by 6

2 a coveiet oralizatio factor, we have: p q (α j β k ) = p q(α j ) = p det q(c p ) j= k= j= = ( ) q p(β k ) = ( ) q det p(c q ) k= () Kroecker products: Usig a well kow coectio to Kroecker products, we ca also write () as p q det(c p I I C q ) Bézout atrix To be copleted ToDo If ca be show that all these costructios are equivalet They defie exactly the sae polyoial, called the resultat of p ad q, deoted as Res x (p, q): Res x (p, q) = det Syl x (p, q) = p det q(c p ) = ( ) q det p(c q ) = p q det(c p I I C q ) The resultat is a hoogeeous ultivariate polyoial, with iteger coefficiets, ad of degree + i the + + variables p j, q k It vaishes if ad oly if the polyoials p ad q have a coo root Notice that the defiitio is ot syetric i its two arguets, Res x (p, q) = ( ) Res(q, p) (of course, this does ot atter i checkig whether it is zero) Reark To copute the resultat of two polyoials p(x) ad q(x) i Maple, you ca use the coad resultat(p,q,x) I Matheatica, use istead Resultat[p,q,x] Discriiats As we have see, the resultat allow us to write a easily checkable coditio for the siultaeous vaishig of two uivariate polyoials Ca we use the resultat to produce a coditio for a polyoial to have a double root? Recall that if a polyoial p(x) as a double root at x (which ca be real or coplex), the its derivative p (x) also vaishes at x Thus, we ca check for the existece of a double root by coputig the resultat betweee a polyoial ad its derivative Defiitio The discriiat of a uivariate polyoial p(x) is defied as ( ) Dis x (p) := ( ) ( )/ Res x p(x), dp(x) dx Siilar to what we did i the resultat case, the discriiat ca also be obtaied by writig a atural coditio i ters of the roots α i of p(x): Dis x (p) = p (α j α k ) j<k If p(x) has degree, its discriiat is a hoogeeous polyoial of degree i its + coefficiets p,, p p 6

3 Exaple 3 Cosider the quadratic uivariate polyoial p(x) = ax + bx + c Its discriiat is: Dis x (p) = Res x (ax + bx + c, ax + b) = b ac a For the cubic polyoial p(x) = ax 3 + bx + cx + d we have 3 Applicatios 3 Polyoial equatios 3 Dis x (p) = 7a d + 8adcb + b c b 3 d ac Oe of the ost atural applicatios of resultats is i the solutio of polyoial equatios i two variables For this, cosider a polyoial syste p(x, y) =, q(x, y) =, () with oly a fiite uber of solutios (which is geerically the case) Cosider a fixed value of y, ad the two uivariate polyoials p(x, y ), q(x, y ) If y correspods to the y copoet of a root, the these two uivariate polyoials clearly have a coo root, hece their resultat vaishes Therefore, to solve (), we ca copute Res x (p, q), which is a uivariate polyoial i y Solvig this uivariate polyoial, we obtai a fiite uber of poits y i Backsubstitutig i p (or q), we obtai the correspodig values of x i Naturally, the sae costructio ca be used by coputig first the uivariate polyoial i x give by Res y (p, q) 3 3 Exaple Let p(x, y) = xy + 3y x x 3x y, q(x, y) = x y y 3 x + y + x y The resultat (i the x variable) is 8 6 Res x (p, q) = y(y + ) 3 (7y 5y 7 + 7y 5y 5 + 9y 6y 3 + 8y + ) Oe particular root of this polyoial is y 677, with the correspodig value of x Iplicitizatio of ratioal curves To be copleted ToDo 33 Rado atrices To be copleted ToDo The set of oegative polyoials Oe of the ai reasos why oegativity coditios about polyoials are difficult is because these sets ca have a quite coplicated structure, eve though they are always covex Recall fro last lecture that we have defied P R + as the set of oegative polyoials of degree It is easy to see that if p(x) is i the boudary of the set P, the it ust have a real root, of ultiplicity at least two Ideed, if there is o real root, the p(x) is i the strict iterior of P 6 3

4 b a -5 - Figure : The shaded regio correspods to the polyoial x + ax + b beig oegative The ubers idicate how ay real roots p(x) has b a - Figure : Regio of oegativity of the polyoial x + ax 3 + 6bx + ax +, ad uber of real roots (sall eough perturbatios will ot create a root), ad if it has a siple real root it clearly caot be oegative Thus, o the boudary of P, the discriiat of p(x) ust ecessarily vaish However, it turs out that Dis x (p) does ot vaish oly o the boudary, but it also vaishes at poits iside the set Why is this? Exaple 5 Cosider the uivariate polyoial p(x) = x + ax + b For what values of a, b does it hold that p(x) x R? Sice the leadig ter x has eve degree ad is strictly positive, p(x) is strictly positive if ad oly if it has o real roots The discriiat of p(x) is equal to 56 b (a b) Here is a slightly differet exaple, showig the sae pheoeo Exaple 6 As aother exaple, cosider ow p(x) = x + ax 3 + 6bx + ax + Its discriiat, i factored for, is equal to 56( + 3b + a)( + 3b a)( + a 3b) The correspodig oegativity regio ad uber of real roots are preseted i Figure As we ca see, this creates soe difficulties For istace, eve though we have a perfectly valid aalytic expressio for the boudary of the set, we caot have a good sese of how far we are fro the boudary by lookig at the absolute value of the discriiat Fro the atheatical viewpoit, there are a couple of (urelated?) reasos with these sets caot be directly hadled by stadard optiizatio, at least if we wat to keep the polyoial structure 6

5 5 3 t b a 6 Figure 3: A three diesioal covex set, described by oe quadratic ad oe liear iequality, whose projectio o the (a, b) plae is equal to the set i Figure Oe has to do with its algebraic structure, ad the other oe with covexity, ad i particular its facial structure Lea 7 (eg, [Ad3]) The set described i Figure is ot basic closed seialgebraic Reark 8 Notice that the covex sets described i Figures ad both have a ucoo feature They both have proper faces that are ot exposed, ie, they caot be isolated by a supportig hyperplae Ideed, i Figure the origi (, ) is a o exposed zero diesioal face, while i Figure the poit (, ) has the sae property A o exposed face is a kow obstructio for a covex set to be the feasible set of a seidefiite progra, see [RG95] Eve though these sets have these coplicatig features, it turs out that we ca ofte provide soe good represetatios These are orally give as a projectio fro higher diesioal spaces, where the object upstairs is uch ore sooth ad well behaved For istace, as a graphical illustratio, i Figure 3 we ca see a three diesioal covex set, whose projectio o the plae (a, b) is exactly the oe discussed i Exaple 5 ad Figure The presece of extraeous copoets of the discriiat iside the feasible set is a iportat roadblock for the availability of easily coputable barrier fuctios Ideed, every polyoial that vaishes o the boudary of the set P ust ecessarily have the discriiat as a factor This is a strikig differece with the case of the case of the oegative orthat or the PSD coe, where the stadard barriers are give (up to a logarith) by products of the liear costraits or a deteriat (which are polyoials) The way out of this proble is to produce o polyoial barrier fuctios, either by partial iiizatio fro a higher diesioal barrier (ie, projectio) or usig the uiversal barrier fuctio itroduced by Nesterov ad Neirovski I this directio, there have bee several research efforts that ai at directly characterizig barrier fuctios for the set of oegative polyoials (or related odificatios) Aog the, we etio the work of Kao ad Megretski [KM] ad Faybusovich [Fay], both of which produce barriers that rely o the coputatio of oe or ore itegral expressios Give the fact that these itegrals ust be coputed uerically, there is o clear cosesus yet o how useful this approach is i practical optiizatio probles A face of a covex set S is a covex subset F S, with the property that x, y S, (x + y) F x, y F A face F is exposed if it ca be writte as F = S H, where H is a supportig hyperplae of S 6 5

6 Figure : The discriiat of the polyoial x + ax 3 + 6bx + cx + The covex set iside the bowl correspods to the regio of oegativity There is a additioal oe diesioal copoet iside the set Refereces [Ad3] C Adradas Characterizatio ad descriptio of basic seialgebraic sets I Algorithic ad quatitative real algebraic geoetry (Piscataway, NJ, ), volue 6 of DIMACS Ser Discrete Math Theoret Coput Sci, pages Aer Math Soc, Providece, RI, 3 [CLO97] D A Cox, J B Little, ad D O Shea Ideals, varieties, ad algoriths: a itroductio to coputatioal algebraic geoetry ad coutative algebra Spriger, 997 [Fay] L Faybusovich Self cocordat barriers for coes geerated by Chebyshev systes SIAM J Opti, (3):77 78 (electroic), [KM] C Y Kao ad A Megretski A ew barrier fuctio for IQC optiizatio probles I Proceedigs of the Aerica Cotrol Coferece, [RG95] M Raaa ad A J Golda Soe geoetric results i seidefiite prograig J Global Opti, 7():33 5, 995 [Stu98] B Sturfels Itroductio to resultats I Applicatios of coputatioal algebraic geoetry (Sa Diego, CA, 997), volue 53 of Proc Sypos Appl Math, pages 5 39 Aer Math Soc, Providece, RI,