Convergence of the Lasserre Hierarchy of SDP Relaxations for Convex Polynomial Programs without Compactness

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1 Convergence of he Lasserre Hierarchy of SDP Relaxaions for Convex Polynomial Programs wihou Compacness V. Jeyakumar, T. S. Phạm and G. Li Revised Version: Sepember 18, 2013 Absrac The Lasserre hierarchy of semidefinie programming SDP relaxaions is a powerful scheme for solving polynomial opimizaion problems wih compac semi-algebraic ses. In his paper, we show ha, for convex polynomial opimizaion, he Lasserre hierarchy wih a slighly exended quadraic module always converges asympoically even in he case of non-compac semi-algebraic feasible ses. We do his by exploiing a coerciviy propery of convex polynomials ha are bounded below. We furher esablish ha he posiive definieness of he Hessian of he associaed Lagrangian a a saddle-poin raher han he objecive funcion a each minimizer guaranees finie convergence of he hierarchy. We obain finie convergence by firs esablishing a new sum-of-squares polynomial represenaion of convex polynomials over convex semi-algebraic ses under a saddle-poin condiion. Keywords: Convex polynomial opimizaion, sums-of-squares of polynomials, posiivsellensaz, represenaions, semidefinie programming AMS subjec class: 90C60, 90C56, 90C26 The auhors are graeful o he referee and he ediors for heir consrucive commens and helpful suggesions which have conribued o he final preparaion of he paper. Research was parially suppored by a gran from he Ausralian Research Council Deparmen of Applied Mahemaics, Universiy of New Souh Wales, Sydney 2052, Ausralia. v.jeyakumar@unsw.edu.au Cener of Research and Developmen, Duy Tan Universiy, K7/25, Quang Trung, Danang, Vienam, and Deparmen of Mahemaics, Universiy of Dala, 1, Phu Dong Thien Vuong, Dala, Vienam. sonp@dlu.edu.vn. This work was carried ou while he auhor was a visior o he School of Mahemaics and Saisics, Universiy of New Souh Wales, Sydney, Ausralia. This work is also parially funded by Vienam Naional Foundaion for Science and Technology Developmen NAFOSTED, Gran Deparmen of Applied Mahemaics, Universiy of New Souh Wales, Sydney 2052, Ausralia. g.li@unsw.edu.au 1

2 1 Inroducion Consider he polynomial opimizaion problem: min x R n{fx g ix 0, i = 1, 2,..., m}, 1.1 where f, g 1,..., g m are polynomials on R n and K := {x R n g i x 0, i = 1, 2,..., m} is a non-empy basic closed semi-algebraic se. A powerful approach o solving problem 1.1 globally is he Lasserre s hierarchy of semidefinie programming SDP relaxaions [16, 17]. Under he sandard archimedean assumpion see Definiion 2.2, he sequence of opimal values of he Lasserre hierarchy of SDP relaxaions converges o he opimal value of he original problem 1.1 and we can also ge a sequence of poins in R n converging o a minimizer of he problem 1.1 see [15, 17, 22, 28]. The archimedean assumpion guaranees ha he feasible se K is compac. The proof of convergence relies on he powerful sum-of-squares polynomial represenaion of posiive polynomials over compac semi-algebraic ses from real algebraic geomery [24, 27]. Moreover, i has recenly been shown ha finie convergence occurs for Lasserre s hierarchy generically see [21]. On he oher hand, i is known ha, under he archimedean assumpion, Lasserre s hierarchy has finie convergence whenever f, g 1,..., g m are convex polynomials on R n and he Hessian of he objecive funcion f is posiive definie a each minimizer [14, 16, 17], requiring sric convexiy of f see Lemma 2.1 in Secion 2. However, o he bes knowledge of he auhors, no much is known abou convergence of Lasserre s hierarchy for problem 1.1 in he case where K is no compac and f, g 1,..., g m are convex polynomials on R n. In his paper, wihou assuming compacness of K, we show ha, if f, g 1,..., g m are convex polynomials on R n hen he Lasserre hierarchy of SDP relaxaions wih an exended quadraic module, generaed in erms of boh he convex polynomial objecive funcion f and he convex polynomials g i, i = 1, 2,..., m, converges asympoically. We prove his by exploiing a coerciviy propery of convex polynomials ha are bounded below. In addiion, if he Hessian of he associaed Lagrangian funcion of 1.1 is posiive definie a a saddle-poin of he Lagrangian, hen we show ha our hierarchy has finie convergence. We derive he finie convergence resul by firs proving ha a convex polynomial wih posiive definie Hessian a a single poin is sricly convex and coercive, and hen esablishing ha he posiive definieness of he Hessian of he Lagrangian a a saddle-poin guaranees a sum-of-squares represenaion of a convex polynomial over a convex no necessarily compac semi-algebraic se. The significance of our sumof-squares polynomial represenaion is ha i allows us o consruc a hierarchy of SDP approximaions in erms of quadraic modules raher han pre-orderings [3, 4, 5] even in he case of convex polynomial programs wih non-compac feasible 2

3 ses. Also, our represenaion exends he corresponding known represenaions of convex polynomials over compac feasible ses, given in [14, 16]. A necessary condiion for finie convergence of he Lasserre hierarchy has recenly been given in [21, Proposiion 3.4], where i was shown ha if he Lasserre hierarchy has finie convergence hen he well-known Kurash-Kuhn-Tucker KKT firs-order opimaliy condiion holds a every minimizer of he given non-convex polynomial opimizaion problem whenever he hierachy achieves is opimal value. Indeed [21, Proposiion 3.4] holds for polynomial opimizaion problems, where he feasible ses are no necessarily compac ses. As a consequence of [21, Proposiion 3.4], we obain ha he exisence of a saddle-poin of he Lagrangian funcion of problem 1.1 a each minimizer is necessary for finie convergence of our hierarchy of SDP relaxaions whenever our hierarchy achieves is opimal value. 2 Asympoic Convergence wihou Compacness We begin by fixing noaion, definiions and preliminaries. Throughou his paper R n denoes he Euclidean space wih dimension n. The inner produc in R n is defined by x, y := x T y for all x, y R n. The non-negaive orhan of R n is denoed by R n + and is defined by R n + := {x 1,..., x n R n x i 0}. Denoe by R[x] he ring of polynomials in x := x 1, x 2,..., x n wih real coefficiens. For a polynomial f wih real coefficiens, we use deg f o denoe he degree of f. A symmeric n n marix A is said o be posiive definie, denoed by A 0, whenever x T Ax > 0 for all x R n, x 0. The gradien and he Hessian of a real polynomial f R[x] a a poin x are denoed by fx and 2 fx respecively. Moreover, for a funcion L: R n R m R, we use 2 xxlx, λ o denoe he Hessian of L a x, λ R n R m wih respec o he variable x. We say ha a real polynomial f R[x] is sum-of-squares SOS if here exis real polynomials f j, j = 1,..., r, such ha f = r j=1 f 2 j. The se of all sum-of-squares real polynomials is denoed by Σ 2. Definiion 2.1. Quadraic module A quadraic module generaed by polynomials g 1,..., g m R[x] is defined as M g 1,..., g m := {σ 0 σ 1 g 1 σ m g m σ i Σ 2, i = 0, 1,..., m}. A quadraic module is a subse of polynomials ha are non-negaive on he se {x R n g i x 0, i = 1,..., m} and i possess a very nice cerificae for his propery. The following archimedean condiion see [17, 19] has played a key role in he sudy of polynomial opimizaion. Definiion 2.2. Archimedean The quadraic module M g 1,..., g m is called Archimedean if here exiss p M g 1,..., g m such ha {x : px 0} is compac. 3

4 When he quadraic module M g 1,..., g m is Archimedean, we have he following imporan characerizaion of posiiviy of a polynomial over a semialgebraic se. Lemma 2.1. Puinar posiivsellensaz [24] Le f, g i, i = 1,..., m, be real polynomials wih K := {x : g i x 0, i = 1,..., m}. Suppose ha M g 1,..., g m is Archimedean. If fx > 0 for all x K, hen f M g 1,..., g m. Now, consider he convex polynomial programming problem, discussed in he Inroducion: f := min x R n{fx g ix 0, i = 1, 2,..., m}, 2.2 where f, g 1,..., g m are convex polynomials on R n and K := {x R n g 1 x 0,..., g m x 0}. Le c R be such ha c > fx 0 for some x 0 K. For each ineger k, we define he runcaed quadraic module M k generaed by he polynomials c f and g 1,..., g m as M k := {σ 0 σ i g i + σc f σ, σ 0, σ 1,..., σ m Σ 2 R[x], deg σ 0 2k, deg σ i g i 2k and deg σc f 2k}. Consider he following SDP relaxaion problem of 2.2 P k fk := sup{µ f µ M k }. 2.3 µ R I is well known ha compuing he supremum fk can be equivalenly reformulaed as a semidefinie programming problem see [15], [17], [19]. Moreover, i can be easily verified ha fk fk+1 f. Noe ha he relaxaion problem 2.3 is he Lasserre hierarchy of SDP relaxaions of he convex program min x R n{fx : g ix 0, i = 1,..., m, fx c 0}. Definiion 2.3. Coerciviy A polynomial f : R n R is coercive whenever he lower level se {x R n : fx α} is a possibly empy compac se, for all α R. The following useful coerciviy propery of a convex polynomial, ha is bounded below, allows us o esablish ha he Lasserre hierarchy of SDP relaxaions of Problem 2.2 has an asympoic convergence in he sense ha f k f as k. 4

5 Lemma 2.2 Coerciviy and Convex Polynomials. Le h R[x] be a convex polynomial which is bounded below on R n. Then here exis an orhogonal n n marix A and a coercive polynomial g : R l R, 1 l n, such ha hax = hax 1,..., x l,..., x n = gx 1,..., x l, x = x 1,..., x l,..., x n T R n. In paricular, h aains is infimum on R n. Proof. The proof is given in he Appendix. Recall ha for a funcion f : R n R and a se C R n, argmin x C fx is defined by argmin x C fx = {a C : fa = inf x C fx}. The following known exisence resul of a convex polynomial program will also be useful for he proof of he asympoic convergence. Lemma 2.3. [1] Le f 0, f 1,..., f m R[x] be convex polynomials. Le C := {x R n : f i x 0, i = 1,..., m}. Suppose ha inf f 0x >. Then, argmin x C f 0 x. x C Theorem 2.1 Asympoic Convergence. Le f and g 1,..., g m R[x] be convex polynomials. Le x be a minimizer of he convex polynomial opimizaion problem 2.2. Then, lim k f k = f. Proof. [Posiiviy of Approximae Lagrangian by Convex Programming Dualiy]. Le ɛ > 0. We firs prove ha here exiss λ R m + such ha fx fx + λ i g i x + ɛ > 0, x R n. 2.4 To see his, noe ha f fx 0 on K, where K := {x R n g 1 x 0,..., g m x 0}. Then, f + ɛ fx > 0 on K. So, here exiss δ > 0 such ha f + ɛ fx > 0 on K δ, where K δ := {x R n g 1 x δ,..., g m x δ}. [Oherwise, we can find a sequence {δ k } R +, δ k 0 and {x k } R n such ha g i x k δ k, i = 1, 2,..., m and fx k + ɛ fx 0. Then, 0 inf x,z 1,...,z m { zi 2 fx + ɛ fx 0, g i x z i 0, i = 1,..., m} δk 2 = mδk 2 0, as k. So, from Lemma 2.3 ha here exis y R n and z = z1,..., zm R m such ha fy + ɛ fx 0, g i y zi 0, i = 1,..., m, and m z i 2 = 0. Thus, fy + ɛ fx 0 and g i y 0, i = 1, 2,..., m. This conradics he propery ha f + ɛ fx > 0 on K]. 5

6 Now, by Lemma 2.3, f aains is minimizer a w K δ wih fw +ɛ fx > 0. As g i x 0 < δ, i = 1, 2,..., m, he Slaer condiion holds for he consrains, g 1 x δ,..., g m x δ. So, by he convex programming dualiy [8, 9, 10], here exis λ i 0, i = 1, 2,..., m such ha, for all x R n, fx + m λ ig i x δ fw. This gives us ha, for all x R n, fx + m λ ig i x fw + m λ iδ fw > fx ɛ. Thus, our claim 2.4 holds. [Asympoic Represenaion by Puinar Posiivsellensaz]. Define h : R n R by hx := fx fx + λ i g i x + ɛ, x R n. Clearly, h is a convex polynomial which is posiive over R n. Lemma 2.2 shows ha here exis an orhogonal n n marix A and a coercive polynomial g : R l R such ha hax 1,..., x l,..., x n = gx 1,..., x l, x = x 1,..., x l,..., x n R n. 2.5 Le T = {x R n : hx c fx + ɛ}. Clearly, T is nonempy as x 0 T. As g is coercive on R l, i follows from 2.5 ha S := {x R l gx 1,..., x l c fx + ɛ} is a nonempy and compac se. Since h is posiive, i follows from 2.5 ha g > 0 over R l, and in paricular g > 0 over S. Le px := gx c + fx ɛ, for x R l. Then, he quadraic module M p is Archimedean as p M p and {x : px 0} = S is compac. Now, from he Puinar Posiivsellensaz Lemma 2.1, we ge ha here exis sum-of-squares polynomials σ 0, σ 1 over R l such ha g = σ 0 + σ 1 c fx g + ɛ. From 2.5, we see ha, for each x = x 1,..., x l, x l+1,..., x n R n, hax = gx 1,..., x l. So, for each x = x 1,..., x l, x l+1,..., x n R n, hax = σ 0 x 1,..., x l + σ 1 x 1,..., x l c fx hax + ɛ. Then, for each z R n, hz = σ 0 A 1 z 1,..., A 1 z l + σ1 A 1 z 1,..., A 1 z l c fx hz + ɛ. Using he definiion of h, we obain ha, for each z R n, fz fx + λ i g i z + ɛ m = σ 0 A 1 z 1,..., A 1 z l + σ1 A 1 z 1,..., A 1 z l c fz λ i g i z. 6

7 Thus, for each z R n, fz fx + ɛ 2.6 = σ 0 A 1 z 1,..., A 1 z l + σ1 A 1 z 1,..., A 1 z l c fz σ 1 A 1 z 1,..., A 1 z l λi + λ i g i z, where z σ i A 1 z 1,..., A 1 z l, i = 0, 1, are sum-of-squares polynomials and λ i 0, for i = 1, 2,..., m. [Convergence from Asympoic Represenaion]. Equaion 2.6 shows ha, for each ɛ > 0, f f + ɛ M g 1,..., g m, c f. So, for each ɛ > 0, here exiss k N such ha f ɛ f k. This ogeher wih he fac ha f k f k+1 f gives us ha lim k f k = f. 3 Represenaions and Finie Convergence In his secion, we presen new represenaion resuls for non-negaiviy of convex polynomials over convex semi-algebraic ses. For relaed resuls, see [6, 7, 20, 23, 28, 29] and oher references herein. The following Lemma on sric convexiy and coerciviy of convex polynomials plays a key role in proving he desired represenaion of convex polynomials and also he finie convergence of our hierarchy. Lemma 3.1 Hessian Condiion for Coerciviy and Sric Convexiy. Le f R[x] be a convex polynomial. If 2 fx 0 0 a some poin x 0 R n, hen f is coercive and sricly convex on R n. Proof. A simple proof is given in he Appendix. Le f and g 1,..., g m R[x] be convex polynomials wih K := {x R n g i x 0, i = 1,..., m}. Suppose ha argmin x K fx and ha here exiss x argmin x K fx. Then, convex programming dualiy [8, 9, 10, 11] shows ha if here exiss x 0 R n such ha g i x 0 < 0 for i = 1,..., m, hen here exiss λ R m + such ha x, λ is a saddle-poin of he Lagrangian funcion Lx, λ := fx + m λ ig i x in he sense ha, x R n λ R m + Lx, λ Lx, λ Lx, λ. 3.7 Theorem 3.1 Represenaion of Convex Polynomials. Le f and g 1,..., g m R[x] be convex polynomials wih K := {x R n g i x 0, i = 1,..., m} =. Le L: R n R m + R be he Lagrangian funcion defined by Lx, λ := fx + m λ ig i x. If he Lagrangian funcion L has a saddle-poin x, λ K R m + wih 2 xxlx, λ 0, hen, for any c R wih c > fx, we have f fx M g 1,..., g m, c f. 7

8 Proof. Since x, λ is a saddle-poin of he Lagrangian funcion L and x K, i follows ha, for each x R n, Lx, λ Lx, λ = fx and x is a minimizer of f over K. Le hx := Lx, λ fx = fx fx + λ i g i x, x R n. Clearly h is a convex polynomial and hx 0, for all x R n. Moreover, i is easy o check ha hx = 0 = inf x R n hx; in paricular, hx = 0. By a direc calculaion, we see ha he Hessian 2 h of h a x is posiive definie. Now, Lemma 3.1 shows us ha he polynomial h is sricly convex and coercive, which implies ha x is he unique minimizer of h on R n and ha is a nonempy compac se. Since S := {x R n hx c fx } hx + c fx = c fx λ i g i x Mc f, g 1,..., g m and S = {x R n hx + c fx 0} is compac, i follows ha he quadraic module Mc f, g 1,..., g m is Archimedean. We apply [26, Corollary 3.6] see also [25, Example 3.18] o conclude ha here exis sum-of-squares polynomials σ 0, σ 1 Σ 2 such ha, for each x R n, So, for each x R n, hx = σ 0 x + σ 1 x hx + c fx. fx fx = σ 0 λ i + λ i σ 1 g i x + σ 1 c fx. Hence, he conclusion follows. Example 3.1 Imporance of posiive definie Hessian of L a a saddle poin for represenaion. Le p R[x] be a convex form i.e., homogeneous polynomial on R n wih degree a leas 4 which is no a sum-of-squares polynomial. See [2] for he exisence of such polynomials. Le f, g be convex polynomials on R n R defined by fx, y := px and gx, y := y 2 1. Then, f is no sricly convex. Le f := min x K fx, y, where K := {x, y R n R gx, y 0} = R n [ 1, 1]. Then f = 0 because f0, 1 = 0, f0, 1 = 0 and f is convex. Consider he corresponding Lagrangian L: R n+1 R + R defined by Lx, y, λ := fx, y + λgx, y. Clearly, x, y, λ := 0, 1, 0 is a saddle poin of L as Lx, y, λ = Lx, y, λ = 0 fx, y = Lx, y, λ for all x R n and λ R +. Moreover, as 8

9 2 px = 2 p0 = 0, he Hessian of he Lagrangian funcion L is no posiive definie a he poin x, y, λ. We now show ha he represenaion of Theorem 3.1 fails. To see his, noe ha he quadraic module M1 x 2 R[x] is Archimedean. So, here exiss c > fx, y = 0 such ha c p M1 x 2 for example, see [19, Corollary 5.2.4]. On he conrary, suppose ha he represenaion of Theorem 3.1 holds. Then, fx, y = σ 0 x, y σ 1 x, ygx, y + σx, yc fx, y, for all x, y R n R, for some sum-of-squares polynomials σ, σ 0, σ 1 in he ring R[x, y]. Leing y = 1 and noing ha gx, 1 = 0, we see ha, for all x R n px = fx, 1 = σ 0 x, 1 + σx, 1c fx, 1 = σ 0 x, 1 + σx, 1c px. So, p Mc p R[x]. Then we have p M1 x 2. By Proposiion 4 in De Klerk, Lauren, and Parrilo [13], a form belongs o he quadraic module M1 x 2 if and only if i is a sum-of-squares polynomial. This conradics our assumpion ha he polynomial p is no a sum-of-squares. Thus, he represenaion fails in his case. As an easy applicaion of Theorem 3.1, we obain he following represenaion under he Archimedean assumpion. For relaed resuls, see [16, Theorem 3.4] and [14, Corollary 3.3]. Corollary 3.1 Represenaion wih Archimedean Condiion. Le f, g 1,..., g m be convex polynomials, and le K := {x R n g i x 0, i = 1,..., m}. Suppose ha he following assumpions hold: i There exiss x 0 R n such ha g i x 0 < 0 for i = 1,..., m; ii 2 xxlx, λ 0 a a saddle-poin x, λ R n R m + funcion L; of he Lagrangian iii The quadraic module M g 1,..., g m is Archimedean. Then, f fx M g 1,..., g m. Proof. The assumpion iii implies ha he se K is compac, and so argmin x K fx. The assumpion i guaranees ha here exiss λ R m + such ha x, λ is a saddle-poin of he Lagrangian funcion L. Le c N be an arbirary naural number saisfying c > fx. I follows from Theorem 3.1 ha f fx M g 1,..., g m, c f. On he oher hand, by aking c large enough, if necessary, from he assumpion iii we may assume ha c f M g 1,..., g m. Therefore f fx M g 1,..., g m, c f = M g 1,..., g m, which complees he proof. 9

10 Remark 3.1 Comparisons wih known recen resuls. In he special case where he Hessian 2 f of he objecive funcion f is posiive definie a a minimizer x argmin x K fx, hen he Slaer condiion ensures ha here exiss λ R m + such ha x, λ is a saddle-poin of he Lagrangian funcion L, and so, he Hessian 2 xxl of L is posiive definie a x, λ. Hence, i is easy o see ha he above corollary exends he represenaion resuls for convex polynomial opimizaion esablished in [16, Theorem 3.4] and [14, Corollary 3.3]. The following simple one dimensional example illusraes ha our represenaion resul can be applied o he case where he Hessian 2 f of he objecive funcion f is no posiive definie a a minimizer. Example 3.2. Verifying represenaion: Non-posiive definieness case of he Hessian 2 f Le fx = x and gx = x 2 1. Then, K := {x R gx 0} = [ 1, 1]. Clearly, argmin x Kfx = { 1} and f is no posiive definie a he unique minimizer x := 1. On he oher hand, direc verificaion shows ha x, λ := 1, 1 is a saddle poin of he Lagrangian funcion Lx, λ := 2 fx + λgx = x + λx 2 1, and 2 xxlx, λ 0. Moreover, Slaer condiion is saisfied and he quadraic module M g is Archimedean. So, i follows from he previous corollary ha f fx = f + 1 M g. Indeed, f fx = x + 1 = 1 x x2 M g. As we see in he following heorem, under he Slaer condiion and he posiive definieness of he Hessian of f a a minimizer, we obain a sharper represenaion han he one in Theorem 3.1. Theorem 3.2 Sharp Represenaion wih posiive definie 2 fx. Le f and g 1,..., g m R[x] be convex polynomials wih K := {x R n g i x 0, i = 1,..., m}. Le argmin x K fx and x argmin x K fx. If here exiss x 0 R n such ha g i x 0 < 0, for i = 1,..., m and if 2 fx 0 hen, for any c > fx, here exis sum-of-squares polynomials σ 0, σ 1 Σ 2 and Lagrange mulipliers λ i 0, i = 1, 2,..., m such ha f fx = σ 0 λ i g i + σ 1 c f. Proof. The Slaer condiion and convex programming dualiy guaranee ha here exiss λ R m + such ha x, λ is a saddle-poin of he Lagrangian funcion Lx, λ := fx + m λ ig i x. So, for each x R n, Lx, λ Lx, λ = fx. Then, hx := Lx, λ fx = fx fx + λ i g i x 0, x R n. Now, as 2 fx 0, Lemma 3.1 shows ha f is a sricly convex and coercive polynomial. Then, he convex se S := {x R n fx c} is nonempy and 10

11 compac. On he oher hand, as c f Mc f and {x R n c fx 0} = S is compac, i follows ha he quadraic module Mc f is Archimedean. Now, as h 0 on S, [26, Corollary 3.6] see also [25, Example 3.18] guaranees ha here exis sum-of-squares polynomials σ 0, σ 1 Σ 2 such ha, for each x R n, hx = σ 0 x + σ 1 xc fx. So, for each x R n, fx fx = σ 0 m λ i g i x + σ 1 c fx. Hence, he conclusion follows. Remark 3.2 Consrain qualificaions. In Corollary 3.1 and Theorem 3.2, we have used he Slaer condiion for guaraneeing he exisence of a saddle-poin. For oher general consrain qualificaions ensuring he exisence of a saddle poin of he Lagrangian funcion, see [10, 11, 12]. Definiion 3.1. Finie convergence We say ha he hierarchy of SDP relaxaions P k of problem 2.2 has finie convergence whenever fk = f for some ineger k. We now show ha our hierarchy of SDP relaxaions of problem 2.2 has finie convergence under suiable condiions. Theorem 3.3 Finie convergence of our hierachy. For he problem 2.2, le f and g 1,..., g m R[x] be convex polynomials. Le L: R n R m + R be he Lagrangian funcion defined by Lx, λ = fx + m λ ig i x. Assume ha he Lagrangian funcion L has a saddle-poin x, λ K R m + wih 2 xxlx, λ 0. Then here exiss an ineger k such ha fk = f and he problem P k achieves is opimal value. Proof. We know ha fk f for all k 1. On he oher hand, i follows from Theorem 3.1 ha here exis sum-of-squares polynomials σ, σ 0, σ 1,..., σ m Σ 2 such ha f f = σ 0 σ 1 g 1 σ m g m + σc f. Hence f k = f for some k N. As x, λ K R m + is a saddle-poin of L, x is minimizer of Problem 2.2 and f = fx which is also a soluion of P k. Remark 3.3. I is worh noing ha in he case where 2 fx is posiive definie a a minimizer x of Problem 2.2, using Theorem 3.2, one can esablish finie convergence of a sharper form of approximaion problem 2.3, where σ i, i = 1, 2,..., m, in he runcaed quadraic module M k are replaced by he Lagrange mulipliers, λ i, i = 1, 2,..., m, associaed wih he minimizer x. The following example shows ha he finie convergence in he preceding heorem may fail if he saddle-poin condiion does no hold a a minimizer. 11

12 Example 3.3 Imporance of Saddle-poin Condiion for Finie Convergence. Consider he minimizaion problem min{fx, y gx, y 0 }, 3.8 where fx, y = x 2 +y 2 +x+y, gx, y = x 2 +y 2 and K := {x, y R 2 gx, y 0}. Clearly, he unique minimizer of 3.8 is x, y = 0, 0, f := fx, y = 0 and 2 fx, y = diag2, 2 0. I is easy o check ha he saddle-poin condiion is no saisfied a x, y = 0, 0. Now, le c be a real number such ha c > f = 0. For each k, he kh-order relaxaion problem of 3.8 is sup{µ f µ M k }, µ R where M k := {σ 0 σ 1 g+σc f σ, σ 0, σ 1 Σ 2, deg σ 0 2k, deg σ 1 g 2k, deg σc f 2k}. We now show ha he finie convergence fails. We esablish his by he mehod of conradicion. Suppose ha problem 3.8 has finie convergence. Then, here exiss k 0 N, σ, σ 0, σ 1 Σ 2 wih deg σ 0 2k 0, deg σ 1 g 2k 0 and deg σc f 2k 0 such ha f = f f = σ 0 σ 1 g + σc f. This gives us ha, for each x, y R 2, 1+σx, y+σ1 x, y x 2 +y 2 +1+σx, yx+y = σ 0 x, y+cσx, y Leing x, y = 1, 1 in 3.9, where k N, yields k k Then, 1 + σ 1 k, 1 k + σ 1 1 k, 1 k 2 k σ 1 1 k, 1 k 2 k σ 1 k, 1 k + σ 1 1 k, 1 k k1 + σ 1 1 k, 1 k, k which is impossible as he lef hand side converges o 1 + σ0, 0 + σ 1 0, 0. As a consequence of [21, Proposiion 3.4], we obain ha he exisence of a saddlepoin of he Lagrangian funcion of problem 2.2 a each minimizer is necessary for finie convergence of our hierachy of SDP relaxaions whenever he SDP relaxaion P k of problem 2.2 achieves is opimal value fk, for all k sufficienly large. Proposiion 3.1 Necessiy of Saddle-poin for Finie Convergence. For problem 2.2, le f and g 1,..., g m R[x] be convex polynomials. Suppose ha he SDP relaxaion P k of problem 2.2 achieves is opimal value fk, for all k sufficienly large. If he saddle-poin condiion 3.7 fails a a minimizer of problem 2.2, hen he hierarchy of SDP relaxaions P k of problem 2.2 does no have finie convergence. 12

13 Proof. By conrary, assume ha he hierarchy of SDP relaxaions P k of problem 2.2 has finie convergence. Le x K wih f := fx = min x K fx. Then, x is a minimizer of he following convex program: min x R n{fx : g ix 0, i = 1,..., m, fx c 0} 3.10 and P k is he Lasserre hierarchy of SDP relaxaions of 3.10, where c is a real number such ha c > fx 0 for some x 0 K = {x : g i x 0, i = 1,..., m}. Now, by our assumpion, fk 0 = f for some ineger k 0. Since f k f and he sequence {f k } is monoonically increasing, we ge ha fk = f for all k k 0. By increasing k 0 if necessary, we can assume wihou loss of generaliy ha problem P k0 achieves is opimal value fk 0. So, i follows from [21, Proposiion 3.4] ha he firs-order necessary opimaliy condiions hold for problem 3.10 a x. Thus, here exis λ R m and γ R such ha fx + λ i g i x + γ fx = 0, λ i g i x = 0, γ fx c = 0. As c > fx 0 fx, γ = 0. So, fx + m λ i g i x = 0. Hence, by convexiy of f + m λ i g i, we ge ha, for each x R n, fx + λ i g i x fx + λ i g i x. I is now easy o check ha x, λ is a saddle-poin of he Lagrangian funcion of problem Appendix:Proofs of Lemma 2.2 and Lemma 3.1 Proof of Lemma 2.2. Le E h := {d R n hx + d = hx, R and x R n }. Then, i is easy o verify direcly ha E h is a subspace of R n. Le l := n dim E h, and le e 1,..., e n R n be an orhonormal basis such ha span{e l+1,..., e n } = E h and span{e 1,..., e l } = Eh, where E h is he orhogonal complemen of E h. Le A := [e 1,..., e n ]. Then, A is an orhogonal marix. Define g : R l R by l gx 1,..., x l := h x ie i. Then, g is a convex polynomial which is bounded below on R l. Moreover, for all x R n, we have n l n l hax = h x i e i = h x i e i + x i e i = h x i e i = gx 1,..., x l, i=l+1 where he hird equaliy follows by he fac ha n i=l+1 x ie i E h. 13

14 To verify ha g is indeed coercive, we assume, on he conrary, ha S := {x : gx α} is unbounded for some α R. Le {a k } S such ha a k + as k. Le a R l. Then, by passing o subsequence if necessary, we may assume a ha k a a k v 0. Le 0. For sufficienly large k, we have 0 < < 1, a a k a and so g a + a k a a k a = g 1 1 a k a a k a max{ga, α}. a + ga + a k a a k a k a gak Leing k, we ge ha ga + v max{ga, α} for all 0. By assumpion, g is bounded below. So, ga+v is eiher a consan or a polynomial wih even degree 2. I hen follows ha g akes a consan value on {a + v : 0} for all a R l. Then, for all 0 and for any a R l, ga v = ga v + v = ga. Thus, ga = ga + v for all a R l and R Le ṽ := v T, 0,..., 0 T R n and d := Aṽ = l v ie i Eh. Since v 0, d 0. Moreover, for all x R n and R, hx + d = haa 1 x + ṽ = gz + v = gz = hx, where z = A 1 x 1,..., A 1 x l R l. So, by definiion, d E h. Consequenly, we obain ha d E h Eh \{0}, which is impossible. Hence, g is coercive. Since he polynomial g is coercive, here exiss z := z1,..., zl Rl such ha gz = inf z R l gz. Le x := A z 0 = z 1 e zl e l Eh Rn. Then, hx = inf x R n hx = gz. Proof of Lemma 3.1. Coerciviy Le c be any real number such ha c fx 0. To prove coerciviy of f on R n, we show ha he se S := {x R n fx c} is compac. On he conrary, suppose ha here exiss a sequence {a k } k 0 S such ha a k as k. Wihou los of generaliy, we may assume ha here exiss v 0 such ha a k x 0 v := lim k a k x 0. Le 0. For sufficienly large k, we have 0 < f x 0 + a k x 0 = f 1 a k x 0 a k x 0 1 c. a k x 0 14 a k x 0 < 1, and so x 0 + fx 0 + a k x 0 a k a k x 0 fa k

15 Leing k, we ge fx 0 + v c, for all 0. On he oher hand, as he Hessian 2 fx 0 is posiive definie, 2 fx 0 v, v > 0 and so, for each R, fx 0 + v = fx 0 + fx 0, v fx 0 v, v 2 + higher order erms in. Hence, he one dimensional convex polynomial fx 0 + v is of even degree 2. This is a conradicion since fx 0 + v c for all 0. Sric Convexiy We esablish sric convexiy of f by he mehod of conradicion and suppose ha f is no sricly convex. Then, here exis x, y R n, x y, and 0 0, 1 such ha f1 0 x + 0 y = 1 0 fx + 0 fy. Define h : [0, 1] R by h = f1 x + y 1 fx fy. Then, h is a convex polynomial, h 0, for each [0, 1] and h 0 = 0 = max [0,1] h. As h is a convex funcion on [0, 1], i aains is maximum on he exreme poins of [0, 1], and so, f1 x + y = 1 fx + fy, [0, 1]. Now, define a polynomial ϕ on R by ϕλ := f x+λy x, λ R. Clearly, ϕ is affine on [0, 1], and moreover, i is coercive on R because f is coercive on R n, shown above. We show ha ϕ is indeed affine over R. Le he degree of he one-dimensional polynomial ϕ be d. Then, for each λ R, ϕλ = ϕ0 + ϕ 0λ + ϕ 0 2 λ ϕd 0 λ d. d! As ϕ is affine over [0, 1], ϕ i 0 = 0 for i = 2,..., d, and so, ϕλ = ϕ0 + ϕ 0λ. Hence, ϕ is affine over R. This conradics he fac ha ϕ is coercive on R. Remark 4.1. The conclusion of Lemma 3.1 may also be derived from error bound resuls of convex polynomials see e.g [18, 30] and oher references herein. However, for he sake of simpliciy and self-conainmen, we have given an elemenary direc proof for Lemma 3.1. References [1] E. G. Belousov, and D. Klae, A Frank-Wolfe ype heorem for convex polynomial programs, Comp. Opim. & Appl., , [2] G. Blekherman, Nonnegaive polynomials and sums of squares, Amer. Mah. Soc ,

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