The Algebraic Structure of the p-order Cone

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1 The Algebraic Structure of the -Order Cone Baha Alzalg Abstract We study and analyze the algebraic structure of the -order cones We show that, unlike oularly erceived, the -order cone is self-dual for all greater than or equal to 1 We establish a sectral decomosition, consider the Jordan algebra associated with this cone, and rove that this algebra forms a Euclidean Jordan algebra with a certain inner roduct We generalize some imortant notions and roerties in the Euclidean Jordan algebra of the second-order cone to the Euclidean Jordan algebra of the -order cone Keywords th -order cones Second-order cones Euclidean Jordan algebras 1 Introduction The th -order cone otimization roblems are a class of convex otimization roblems, in which we minimize a linear function over a Cartesian roduct of th -order cones, where 1, The alications of the th -order cones lie in various real-world roblems More secifically, the alications of the th -order cone otimization with = 1 and = are highly diverse The references in this context are too numerous to be cited here When =, the th -order cone otimization reduces to the well-known and well-studied second-order cone otimization See Alizadeh and Goldfarb 1 for a comrehensive study of second-order cone otimization Alications of second-order cone otimization with = can be found in 1 5 The alications of the th -order cone otimization with 1,, are also diverse and can be found in 6 9 Based on the value of 1,, the contributions of the researchers who worked in the th - order cone otimization can be classified into three cases: For = 1 or, we get the roblem of minimizing 1-norms or minimizing -norms, which can be converted to a linear otimization roblem by using well-known transformations For =, we get a second-order cone otimization roblem which has been referentially considered in the literature The case 1,, has given much less attention than the above two cases because of treating the th -order cone as a non self-dual case, which in turn leads to lack of comutational efficiency if it is comared with the case of = Develoing otimization algorithms for the th -order cone otimization by generalizing that for the second-order cone otimization is challenging and difficult for different reasons The first reason is that it has been erceived so far that the th -order cone is self-dual when = only, and that the dual of the th -order cone is the q th -order cone, where q is the conjugate of The second reason is that some equalities and inequalities cannot be extended from the -norm to the -norm when is different from For instance, but not limited to: the fact that x T x equals x does not hold for the -norm when is different from The third reason of challenging and difficulty B Alzalg Deartment of Mathematics, Faculty of Science, The University of Jordan, Amman 1194, Jordan balzalg@jueduedu 1

2 of this generalization is that the Euclidean Jordan algebraic structure of the second-order cone is reviously established (see, for examle, the text of Faraut and Korányi 10, which is not the case for the th -order cone when is different from In this aer, we will see that the statement that the th -order cone is not self-dual when is different from has become no longer true and no longer of any benefit We rove that, for any 1,, the th -order cone is self-dual, under a certain inner roduct that we adot in this aer, and hence it is symmetric Examles of symmetric cones include: The nonnegative orthant cone in the sace of real numbers R, the second-order cone in the real vector sace R n, the cone of ositive semidefinite matrices in the sace of real symmetric matrices, the cone of ositive semidefinite matrices in the sace of comlex Hermitian matrices, and the cone of ositive semidefinite matrices in the sace of quaternion Hermitian matrices This aer views the th -order cones as a new aradigm of symmetric cones From the above oints of view, the rimary goal of this aer is to study, analyze and establish the algebraic structure of the th -order cone, for 1, We establish a sectral decomosition and the Jordan algebra associated with the th -order cone, and rove that this algebra forms a Euclidean Jordan algebra under the adoted inner roduct We then show that the cone of squares of the resulting Euclidean Jordan algebra is indeed the th -order cone itself We generalize some imortant algebraic notions and roerties in the Euclidean Jordan algebra of the second-order cone to the Euclidean Jordan algebra of the th -order cones So, most of our work can be viewed as a generalization of that in 1, Section 4 This aer is organized as follows Section contains some reliminaries, definitions and conventions that will be used throughout the aer Section 3 is devoted to rove that the th -order cone is symmetric In Section 4, we establish the articular Euclidean Jordan algebra associated with the th -order cone and some of its roerties Section 5 contains some concluding remarks Preliminaries and Definitions In this section, we introduce some definitions on the th -order cones, symmetric cones and Euclidean Jordan algebras that are needed for the statements of our results Our resentation in this section follows that of 11, Section Let x R n and 1 be a real number, the -norm of x is defined as ( n 1/ x := x i i=1 Note that when = 1 we get the taxicab norm, when = we get the Euclidean norm, and when aroaches the -norm aroaches the infinity norm which is defined as x := max{ x 1, x,, x n } For each vector x R n whose first entry is indexed with 0, we write x for the subvector consisting of entries 1, through n 1 (therefore x = (x 0, x T T R R n 1 We let E n denote the n th dimensional real vector sace R R n 1 whose vectors x are indexed from 0 Let K be a cone in R n The interior of K is denoted by int(k, and the boundary of K is denoted by Bd(K The th -order cone of dimension n is defined as P := {x E n : x 0 x }

3 We write x 0 to mean that x P, and x 0 to mean that x int(p (ie, x 0 > x We also write x y or y x to mean that x y 0, and x y or y x to mean that x y 0 A cone is said to be closed iff it is closed with resect to taking limits, solid iff it has a nonemty interior, ointed iff it does not contain two oosite nonzero vectors (so the origin is an extreme oint, and regular iff it is a closed, convex, ointed, solid cone Clearly, the th -order cone is regular Let V be a finite-dimensional Euclidean vector sace over R with inner roduct, The dual cone of a regular cone K V is denoted by K and is defined by K := {y V : x, y 0, x K} The cone K is said to be self-dual iff it coincides with its dual cone K, ie, K = K By GL(n, R we mean the general linear grou of degree n over R (ie, the set of n n invertible matrices with entries from R, together with the oeration of ordinary matrix multilication For a regular cone K V, we denote by Aut(K the automorhism grou of K, ie, Aut(K := {T GL(n, R : T (K = K} Definition 1 Let V be a finite-dimensional real Euclidean sace A regular K V is said to be homogeneous iff for each u, v int(k, there exists an invertible linear ma F : V V such that 1 F (K = K, ie, F is an automorhism of K, and F (u = v In other words, Aut(K acts transitively on int(k Definition A regular K is said to be symmetric iff it is both self-dual and homogeneous Let J be a finite-dimensional vector sace over R The vector sace J is called an algebra over R iff a bilinear ma : J J J exists Let x be an element in an algebra J, then for n we define x (n recursively by x (n := x x (n 1 Definition 3 Let J be a finite-dimensional R algebra with a bilinear roduct : J J J Then (J, is called a Jordan algebra iff for all x, y J 1 x y = y x (commutativity; x (x ( y = x ( (x y (Jordan s axiom However, if the algebra (J, is not commutative, then its binary oeration can be relaced by its commutative version where x y := x y+y x for all x, y E n The roduct x y between two elements x and y of a Jordan algebra (J, is called the Jordan multilication between x and y A Jordan algebra (J, has an identity element iff there exists a (necessarily unique element e J such that x e = e x = x for all x J A Jordan algebra (J, is not necessarily associative, that is, x (y z = (x y z may not hold in general However, it is ower associative, ie, x (n x (m = x (n+m for all integers n, m 1 Examle 1 It can be verified that the sace E n with the Jordan multilication x x y = T y x 0 ȳ + y 0 x (1 3

4 forms a Jordan algebra with the identity vector e := 1 0 Definition 4 A Jordan algebra J is called Euclidean iff there exists an inner roduct, on (J, such that for all x, y, z J 1 x, x > 0 x 0 (ositive definiteness; x, y = y, x (symmetry; 3 x, y z = x y, z (associativity That is, J admits a ositive definite, symmetric, quadratic form which is also associative Examle It is easy to verify that the sace E n, with the Jordan multilication defined in (1, is a Euclidean Jordan algebra under the standard inner roduct The sectral decomosition of x J is a decomosition of x into eigenvectors (c 1 (x and c (x together with its eigenvalues (λ 1 (x and λ (x so that x = λ 1 (xc 1 (x + λ 1 (xc (x The eigenvectors c 1 and c satisfy the roerties c 1 c = 0, c 1 = c 1, c = c, and e = c 1 + c A air {c 1, c } is a Jordan frame iff it satisfies the above roerties The elements x and y are simultaneously decomosed iff they share a Jordan frame, ie, x = λ 1 c 1 +λ c and y = ω 1 c 1 +ω c for a Jordan frame {c 1, c } In general, the number of eigenvectors (eigenvalues of an element x J might be greater than two, which is not the case in this aer Examle 3 The sectral decomosition of x E n associated with the second-order cone is obtained as follows: ( 1 x = (x 0 + x 1 x ( (x }{{} 0 x }{{} x x x λ 1 (x }{{} λ (x }{{} c 1 (x c (x We now define the cone of squares of a Euclidean Jordan algebra Definition 5 If J is a Euclidean Jordan algebra, then its cone of squares is the set K J := {x : x J } The following fundamental result gives a one-to-one corresondence between Euclidean Jordan algebras and symmetric cones Theorem 1 (Jordan algebraic characterization of symmetric cones, 10 A regular cone K is symmetric iff K = K J for some Euclidean Jordan algebra J Examle 4 The cone of squares of (E n,, with defined in (1, is the second-order cone Q n We now define three well-known mas: two linear mas from J into itself, namely L(x and Q x, and one quadratic ma from J J into J, namely Q x,y These imortant mas lay a crucial role in the develoment of the interior oint methods for conic otimization Definition 6 Let x and z be elements in a Jordan algebra J Then 4

5 1 The linear ma L(x : J J is defined by L(xy := x y (or equivalently, is defined by x y = (L(xy + L(yx/, for all y J The quadratic oerator Q(x, z : J J J is defined by Q(x, z := L(xL(z + L(zL(x L(x z 3 The quadratic reresentation of x, Q(x : J J, is defined by Q(x := L(x L(x = Q(x, x Note that L(xe = x, L(xx = x, and L(e = Q(e = I Examle 5 From (1, the exlicit formula of the L( oerator for the algebra of the second-order cone, the sace E n, can be immediately given by x0 x L(x = Arw(x = T x x 0 I Here Arw(x is the arrow-shaed matrix associated with the vector x E n oerator for the algebra of the second-order cone is given by Q(x, z = Arw(xArw(z + Arw(zArw(x Arw(x z x = T z (x 0 z T + z 0 x T x 0 z + z 0 x ( x z T + z x T (x 0 z 0 x T zi n 1 Hence, quadratic We can also easily verify that the quadratic reresentation for the algebra of the second-order cone is given by Q(x = Arw (x Arw(x x x = 0 x T x 0 x det(xi + x x T = xx T det(xr 3 Proving that the th -Order Cone is Symmetric In this section, we show that the th -order cone is a symmetric cone by roving two lemmas on the self-duality and homogeneity of the th -order cone Before this, we state and rove the following remark Remark 31 Under the standard inner roduct, the th -order cone P is not a self-dual cone for 1, (, Moreover, under the standard inner roduct, P = P q for 1,, where q is the conjugate of (ie, 1/ + 1/q = 1 Proof Let 1, We first rove that P q P Let x = (x 0 ; x P q, we show that x P by verifying that x T y 0 for any y P So let y = (y 0 ; ȳ P, then x T y = x 0 y 0 + x T ȳ x q ȳ + x T ȳ x T ȳ + x T ȳ 0, where the first inequality follows from the fact that x P q and y P, and the second one from Hölder s inequality Thus, P q P Now we show P P q Let y = (y 0 ; ȳ P, we need to show that y P q This is trivial if ȳ = 0 or = If ȳ 0 and 1 <, let u := (y 1 /q ; y /q ; ; y n 1 /q and consider x := ( u ; u P Then by using Hölder s inequality, where the equality is attained, we obtain 0 x T y = u y 0 u T ȳ = u y 0 u ȳ q = u (y 0 ȳ q This imlies that y 0 ȳ q, and therefore means that y P q Thus, P P q The roof is comlete 5

6 Note that the conjugate of is itself So, under the standard inner roduct, Remark 31 imlies that the th -order cone is self-dual when = only To establish the self-duality of P for all 1,, the inner roduct that we must adot with the th -order cone is not the standard inner roduct, but the inner roduct: ( x x, y := 0 y x x + ȳ ȳ x T ȳ, if x 0 and ȳ 0, ( x 0 y 0, if x = 0 or ȳ = 0, where x, y E n and 1, Note that x, x = x 0 + x (clearly, x, x = x 0 if x = 0 Note also that the inner roduct, is a generalization of the standard inner roduct (as x, y = x T y We have the following lemma Lemma 31 Under the inner roduct, defined in (, the th -order cone P is self-dual for any 1, Proof Let 1, We first show that P P Let x = (x 0 ; x P, we verify that x P by roving that x, y 0 for any y = (y 0 ; ȳ P This is trivial if x = 0 or ȳ = 0 If x 0 and ȳ 0, Then ( x x, y = x 0 y x + ȳ ȳ x T ȳ ( x x ȳ + 1 x + ȳ ȳ x T ȳ x ( ȳ x ȳ + x T ȳ x ȳ x ( ȳ x T ȳ + x T ȳ x ȳ 0, where the first inequality follows from the fact that x, y P, the second inequality follows from the fact that 1 ( a + b ab c d cd for any ositive integers a, b, c and d, and the third inequality follows from Cauchy-Schwartz inequality Thus, P P Now we rove P P Let y = (y 0 ; ȳ P, we need to show that y P This is trivial if ȳ = 0 If ȳ 0, consider x := ( ȳ ; ȳ P Then we obtain ( ( 0 x, y = x 0 y x x + ȳ ȳ x T ȳ = ȳ y 0 1 ȳ ȳ + ȳ ȳ ȳ T ȳ = ȳ (y 0 ȳ This gives that y 0 ȳ, and thus imlies that y P Therefore, P P The result is established We now rove the homogeneity of the th -order cone Throughout the roof, we will use an equivalent definition of the th -order cone Indeed we can redefine the th -order cone as P = {x E n : x T R (xx 0}, (3 6

7 where R (x is a matrix associated with x E n defined as 1 0 T R (x := 0 x x I n 1, if x 0; and R (x := R, if x = 0, (4 and R is the reflection matrix defined as 1 0 T R := 0 I n 1 (5 Note that the matrix R (x acts as a generalization of the reflection matrix R Lemma 3 The th -order cone P is homogeneous for any 1, Proof Let 1, and x E n Associated with x, we define G (x := {M R n n : M T R (xm = R (x} where R (x is the matrix defined in (4 We now rove that G (x is a grou Note that the set G (x is a subset of the general linear grou of degree n over R, denoted by GL(n, R, (ie, the set of n n invertible matrices with entries from R, together with the oeration of ordinary matrix multilication So to rove that G (x is a grou it is enough to show that it is a subgrou of the grou GL(n, R, ie, the set G (x is nonemty and closed under matrix multilication and inverses Clearly, G (x is nonemty because I n G (x Now, for any M 1, M G (x, we have M1 TR (xm 1 = R (x and M TR (xm = R (x, and hence and (M 1 M T R (x(m 1 M = M T (M T 1 R (xm 1 M = M T R (xm = R (x (M1 1 R (x(m1 1 = (M 1(R (x 1 M1 T 1 (M 1 (M1 TR (xm 1 1 M1 T 1 = = (M 1 M1 1 (x 1 (M1 T 1 M1 T 1 (R (x 1 1 = R (x This means that M 1 M and M 1 1 are also in G (x Thus, G (x is a grou Note that for every M G (x, we have (Mx T R (x(mx = x T R (xx As a result, each element of the grou G (x mas P onto itself, and so does the direct roduct H (x := 0, 1 G (x Now, to show that P is homogeneous, it remains to show that the grou H (x acts transitively on the interior of P To do so, it is enough to show that, for any x int(p, there exists an element in H (x that mas e to x This is trivial if x = 0 (in such a case x is nothing but a multile of e So, we assume that x 0 Note that in view of (3, the fact that x int(p imlies that x T R (xx > 0 So we may write x as x = λ ( x / x y with λ = x T R (xx and y E n Moreover, by 1, Theorem 30, there exists a reflector matrix P (hence P = I n 1 such that 0 P = x ȳ, r x 7

8 with r = x x ȳ Then we have y0 r = y0 x x ȳ = y T R (xy = 1 λ xt R (xx = 1 As a result, there exists s 0 such that y 0 = cosh s and r = sinh s Now, we define the matrices 1 0 T ˆP := 0 P x x cosh s 0 T x sinh s x and H (x, s := 0 I n 0 x sinh s 0 T cosh s x Since ˆP, H (x, s G (x, we have that ˆP H (x, s G (x, and therefore λ ˆP H (x, s H (x The result follows by observing that The roof is comlete λ ˆP H (x, s e = λ x x y = x Deending on Lemmas 31 and 3, we establish the following imortant result which is one of our main results in this aer This theorem will lay a crucial role in the develoment of the interior oint methods for the th -order cone otimization as a new aradigm of symmetric otimization Theorem 31 The th -order cone P is a symmetric cone for any 1, As early mentioned, we will establish a articular Euclidean Jordan algebra associated with the th -order cone Based on this Euclidean Jordan algebra and the one-to-one corresondence between Euclidean Jordan algebras and symmetric cones, we can rovide another way to rove Theorem 31 as we will see in the next section 4 The Euclidean Jordan Algebra of the th -Order Cone Let x, y E n and 1, We define the ma : E n E n E n as x y = x, y x 0 y 1 + y 0 x 1 x 0 y + y 0 x x 0 y n 1 + y 0 x n 1 = x, y x 0 ȳ + y 0 x, (6 where, is the inner roduct defined in ( In this section, we draw a sequence of observations leading us to conclude that the algebra (E n, is not only a Jordan algebra, but also a Euclidean Jordan algebra We also resent some notions and concets associated with the th -order cone, for any 1, All these notions and 8

9 concets resented in this section can be viewed as a generalization of those of notions and concets associated with the second-order cone that we resented in the examles of Section To start with, note that the vector 1 e := E n 0 is the unique vector such that x e = x for all x E n The vector e is called the identity of the algebra (E n, for any 1, Let x, y, z E n be simultaneously decomosed, it is not hard to see that x (αy + βz = αx y + βx z and (αy + βz x = αy x + βz x, for any α, β R For any x, y E n, we have that x y = y x (commutativity, and x (x y = x (x y (Jordan s axiom This roves that (E n, is a Jordan algebra for any 1, Let x (E n, The th -order sectral decomosition of x into eigenvectors (c 1 (x and c (x together with eigenvalues (λ 1 (x and λ (x is obtained as follows: ( 1 1 x = (x 0 + x }{{} x λ 1 (x x }{{} c 1 (x + (x 0 x }{{} λ (x ( 1 1 x x }{{} Note that c 1 (x and c 1 (x lay the role of basis for the set consisting of all vectors which are simultaneously decomosed with the vector x Note also that c 1 (x + c (x = e c 1 (x = c 1(x c 1 (x = c 1 (x, c (x = c (x c (x = c (x, c 1 (x c (x = 0, c 1 (x, c (x = 0, c 1 (x, c (x Bd(P, λ 1 (c 1 (x = λ 1 (c (x = 1, λ (c 1 (x = λ (c (x = 0, c 1 (x = Rc (x, and c (x = Rc 1 (x, where R is the reflection matrix defined in (5 We now define the trace and determinant of x as c (x trace(x := λ 1 (x + λ (x = x 0, and det(x := λ 1 (xλ (x = x 0 x (7 Note that trace(e =, det(e = 1, and x, y = 1 trace(x y For any continuous real-valued function f, the image of x under f is defined as f(x := f(λ 1 (xc 1 (x + f(λ (xc (x 9

10 Therefore, we can now redefine x as With a little calculation, we obtain x := (λ 1 (x c 1 (x + (λ (x c (x = x 1 = 1 λ 1 (x c 1(x + 1 λ (x c (x = 1 det(x x 0 + x x 0 x x0 x = 1 R x, (8 det(x which is called the inverse of x (rovided that x is invertible, ie, det(x 0 Note that x x 1 = e Note also that, for any nonnegative integer n, m 1, we have x n x m = (λ n 1 (xc 1(x + λ n (xc (x (λ m 1 (xc 1(x + λ m (xc (x = λ n+m 1 (xc 1 (x + λn+m (xc (x + (λn 1 (xλm (x + λm 1 (xλn (x c 1(x c (x = λ n+m 1 (xc 1 (x + λ n+m (xc (x = x n+m Therefore, the algebra (E n, is ower associative for any 1, The following theorem is one of our main results in this aer Theorem 41 The algebra (E n, is a Euclidean Jordan algebra for any 1, Proof We have seen that (E n, is a Jordan algebra Now we need to show that (E n, admits a ositive definite, symmetric, quadratic form which is also associative Let 1, and consider the inner roduct, defined in ( It is straightforward to show that x, x > 0 for all x 0, and x, y = y, x for all x, y E n Now, we need to show that x, y z = x y, z for x, y, z E n This is trivial if x = 0, ȳ = 0 or z = 0 If x 0, ȳ 0 and z 0, we have x, y x y, z = z0, x 0 ȳ + y 0 x z 1 = x, y 1 z0, + x, y z0, x 0 ȳ z y 0 x z ( ( = z 0 x, y + 1 ȳ ȳ + z z x 0 ȳ T z + 1 x x + z z y 0 x T z ( ( = x 0 y 0 z ȳ ȳ + z z x 0 ȳ T z + 1 x x + z z y 0 x T z ( + 1 x x + ȳ ȳ z 0 x T ȳ ( ( = x 0 y, z + 1 x x + z z y 0 x T z + 1 x x + ȳ ȳ z 0 x T ȳ 1 x0 =, y, z 1 x0 +, y, z x y 0 z x z 0 ȳ x0 y, z =, x y 0 z + z 0 ȳ = x, y z Therefore, the inner roduct, is ositive definite, symmetric and associative, and hence the Jordan algebra (E n, is Euclidean The roof is comlete 10

11 As long as (E n, is Euclidean and as far as Theorem 31 is concerned, 11, Theorem tells us that another way to rove Theorem 31 is roving the following result which characterizes the th -order cones Theorem 4 The cone of squares of (E n, is the th -order cone P for 1, Proof Let 1, We need to show that P = K where K is the cone of squares of E n, under the Jordan multilication, which is defined as K := {ξ : ξ E n } = {ξ ξ : ξ E n } = { ξ 0 + ξ ξ 0 ξ : ξ E n } Let x K, then there exists y E n such that x = (y 0 + ȳ, y 0 ȳ T T It follows that x 0 = y 0 + ȳ y 0 ȳ = x, where the inequality is obtained by observing that (y 0 ȳ 0 This means that x P and hence K P Now, we rove that P K Let x P We need to rove that x = y for some y in the algebra (E n, Equivalently, we need to show that the system of n equations x 0 = y 0 + ȳ, x 1 = y 0 y 1, x = y 0 y, x n 1 = y 0 y n 1, (9 has at least one real solution Assuming at first that y 0 0, then we have y i = x i y 0, for i = 1,,, n 1, and hence ȳ = 1 y 0 x Substituting this in the first equation of (9 we get the following quartic equation 4y 4 0 4x 0 y 0 + x = 0 This equation has u to four solutions, namely x 0 ± x 0 x y 0 = ± Since x 0 x, all these four solutions are real Note that vectors x Bd(P (where x 0 = x have only two square roots, one of which is in Bd(P Vectors x int(p have four square roots, excet for multiles of the identity (wherever x = 0 In such a case if y 0 = 0, then y i can be arbitrarily chosen, as long as ȳ = 1 which gives x = e The identity has infinitely many square roots (assuming n > for which two of them are ±e, and all others are of the form (0; w with w = 1 Thus, every x P has a unique square root in P This establishes the result The arrow-shaed matrix associated with each vector x in the Euclidean Jordan algebra (E n, and the quadratic reresentation of x are essential concets in the theory of Jordan algebras and lay a significant role for designing interior oint algorithms for solving symmetric otimization roblems 11 11

12 The arrow-shaed matrix Arw (x associated with each vector x (E n, is defined as x x 0 x x T, if x 0; Arw (x := x x 0 I n 1 x 0 I n, if x = 0 (10 Using Schur comlement, it is easy to see that the vector x 0 (x 0 if and only if the matrix Arw (x is ositive semidefinite (ositive definite Therefore, th -order cone otimization is a secial case of semidefinite otimization and it includes second-order cone otimization as a secial case However, like the case of second-order cone otimization, th -order cone otimization deserves its own study as a new aradigm of symmetric otimization and its articular Euclidean Jordan algebra must be indeendently develoed Note that Arw (e = I n, Arw (xe = x, Arw (xx = x x = x, and, more generally, that x y = 1 (Arw (xy + Arw (yx (11 Therefore, we define the linear ma L : (E n, (E n, associated with the th -order cone P as L (x := Arw (x for x in the algebra (E n, Let x and y be two vectors in the algebra (E n, It is not hard to see that the quadratic ma Q (, : (E n, (E n, (E n, associated with the th -order cone P is given by x 0 y 0 Q (x, y = y 0 x x, y ȳ x 0 ȳ T x + y ȳ 0 x T ( x x 0 ȳ + y 0 x x 0 y 0 I n x ( xȳ T + ȳ x T x T, if x 0 & ȳ 0; ȳi n 1 x 0 y 0 x 0 ȳ x y 0 x T x x 0 y 0 I n 1 ȳ x 0 ȳ T ȳ x 0 y 0 I n 1 x + ȳ ȳ, if x 0 & ȳ = 0;, if x = 0 & ȳ 0; x 0 y 0 I n, if x = 0 & ȳ = 0 (1 Hence, the quadratic reresentation of x is given by Q (x = x 0 + x x 0 x x x 0 x T x det(xi n 1 + x x x T x = xx T J (x det(xr, if x 0; x 0 I n, if x = 0, (13 where R is the matrix defined in (5, and J (x is the matrix: 1 0 T J (x := 0 x x I n 1, if x 0; and J (x := I n, if x = 0 (14 1

13 Note that Q (e = I n, Q (xe = x and Q (xx 1 = x Furthermore, one can also rove that Q (x 1 = (Q (x 1, hence Q (xq (x 1 = I n The following theorem will lay an imortant role in the develoment of interior oint algorithms for th -order cone otimization (see 11, Section 3 This theorem is the counterart of 1, Theorem 9 Theorem 43 Let z E n be such that z 0 Then Q (z (P = P for any 1, Proof We first rove that Q (z (P P Let x P and y = Q (zx We need to show that y P From (13, we have y = Q (zx = zz T J (zx det(zrx = z T J (zzx det(zrx = (α γrx, where α = z T J (zz and γ = det(z Accordingly, we get It follows that y 0 = (α γ x 0, and ȳ = (α + γ x (15 det(y = y0 ȳ = (α γ x 0 (α + γ x = (4α 4αγ + γ x 0 (4α + 4αγ + γ x = (4α + γ (x 0 x 4γα(x 0 + x (4α + γ det(x 0, where the first inequality follows from the fact that γ = det(z = z 0 z < 0 (as z 0, α = z T J (zz = z 0 + z 0, and x 0 + x 0, and the last inequality follows from the fact that det(x = x 0 x 0 (as x 0 Since det(y = λ 1 (yλ (y 0, either y P or y P That is, either λ 1 (y and λ (y are both nonnegative, or λ 1 (y and λ (y are both nonositive Therefore, to show that y P, it suffices to show that λ 1 (y + λ (y 0 By the definition of the trace and using (15, we have λ 1 (y + λ (y = trace(y = 1 y 0 = 1 (α γ x 0 0 Thus Q (z (P P, and the first inclusion is established The other inclusion is easy In fact, since z 0, we have det(z = λ 1 (zλ (z < 0, therefore we also have det(z 1 1 = λ 1 (zλ (z < 0, which in turn imlies that z 1 0 Thus, from the first inclusion, we get Q z 1(P P, which imlies that P = Q (z(q (z 1( ( P = Q (z Q (z 1 (P Q (z (P The roof is comlete Likewise, we can also rove the following corollary Corollary 41 Let z E n be such that z 0 Then Q (z(int (P = Int (P for any 1, 13

14 5 Conclusions Contrary to the beliefs in the otimization community (see for examle 6 8,13 17, we have roved that the th -order cone is self-dual and hence symmetric for any 1, To the best of our knowledge, this aer is the first of its kind treats the th -order cone as a symmetric cone Like the second-order cone, the th -order cone warrants its own study and its Euclidean Jordan algebra must be indeendently develoed In this aer, we have studied and analyzed the algebraic structure of the th -order cone We have established a sectral decomosition associated with this cone, set u the corresonding Jordan algebra, and formed the corresonding Euclidean Jordan algebra Some imortant notions and roerties in the Euclidean Jordan algebra of the second-order cone are generalized to the Euclidean Jordan algebra of the th -order cone It is our firm belief that the results of this aer are very crucial in designing and analyzing interior-oint algorithms for the th -order cone otimization, and the reason for that is because the roerties of the Euclidean Jordan algebra associated with the th -order cones (as symmetric cones can be used as a fundamental toolbox for establishing the comlexity analysis and the olynomiality roofs and for the th -order cone otimization Our current research is using the machinery of Euclidean Jordan algebra associated with the th -order cone established in this aer to develo rimal-dual ath following interior oint algorithms for solving otimization roblems over th -order cones References 1 Alizadeh, F, Goldfarb, D: Second-order cone rogramming Math Program Ser B 95, 3 51 (003 Lobo, MS, Vandenberghe, L, Boyd, S, Lebret, H: Alications of second-order cone rogramming Linear Alg A 84, ( Alzalg, B: Stochastic second-order cone rogramming: Alication models Alied Math Modeling 36, (01 4 Maggioni, F, Potra, F, Bertocchi, M: Stochastic second-order cone rogramming in mobile ad hoc networks J Otim Theory Al 143, (009 5 Bertazzi L, Maggioni F: Solution aroaches for the stochastic caacitated traveling salesmen location roblem with recourse J Otim Theory Al (015 6 Krokhmal, P, Soberanis, P: Risk otimization with order conic constraints: A linear rogramming aroach Eur J Oer Res 01, (010 7 Xue, G, Ye, Y: An efficient algorithm for minimizing a sum of norms SIAM J Otim 10:, (000 8 Kloft, M, Brefeld, U, Sonnenburg, S, Zien, A: l -norm multile kernel learning J Mach Learn Res 1, (011 9 Krokhmal, P: Higher moment risk measures Quant Financ 7, ( Faraut, J, Korányi, A: Analysis on Symmetric Cones Oxford University Press, Oxford, UK (

15 11 Schmieta, SH, Alizadeh, F: Extension of rimal-dual interior oint methods to symmetric cones Math Program Ser A 96, (003 1 Watkins, DS: Fundamentals of Matrix Comutations nd edn Wiley, New York (00 13 Nesterov, Y: Towards non-symmetric conic otimization Otim Methods Softw 7, (01 14 Glineur, F, Terlaky, T: Conic formulation for l -norm otimization J Otim Theory Al 1, ( Vinel, A, Krokhmal, P: On valid inequalities for mixed integer order cone rogramming J Otim Theory Al 160, ( Vinel, A, Krokhmal, P: Polyhedral aroximations in -order cone rogramming Otim Methods Softw 9, ( Gotoh, J, Uryasev, S: Two airs of families of olyhedral norms versus l -norms: roximity and alications in otimization Math Program Ser A 96, 1 41 (015 15