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Self-scaled barrier functions on symmetric cones and their classification

Self-scaled barrier functions on self-scaled cones were introduced through a set of axioms in 1994 by Y.E. Nesterov and M.J. Todd as a tool for the construction of long-step interior point algorithms. This paper provides firm foundation for these objects by exhibiting their symmetry properties, their intimate ties with the symmetry groups of their domains of definition, and subsequently their decomposition into irreducible parts and algebraic classification theory. In a first part we recall the characterisation of the family of self-scaled cones as the set of symmetric cones and develop a primal-dual symmetric viewpoint on self-scaled barriers, results that were first discovered by the second author. We then show in a short, simple proof that any pointed, convex cone decomposes into a direct sum of irreducible components in a unique way, a result which can also be of independent interest. We then show that any self-scaled barrier function decomposes in an essentially unique way into a direct sum of self-scaled barriers defined on the irreducible components of the underlying symmetric cone. Finally, we present a complete algebraic classification of self-scaled barrier functions using the correspondence between symmetric cones and Euclidean Jordan algebras.


Introduction
In recent years a theory of interior-point methods for linear, semidefinite, and second-order cone programming has been developed within the unified framework of self-scaled conic programming. The origins of this theory can be traced to the works [14,15,4]. The importance of the problems which can be cast in this framework, and the fact that it is possible to develop efficient long-step interior-point methods for these problems have contributed to the popularity of the subject in the optimization community and beyond.
In order to facilitate our exposition, we consider the following pair of convex programs in conic duality Here E is a finite dimensional Euclidean space equipped with an inner product ·, · , L is a linear subspace of E, and L ⊥ its orthogonal complement. The cone K is a regular (closed, convex, pointed, solid) cone, x 0 ∈ int(K), and s 0 ∈ int(K * ), where K * is the dual cone Interior-point algorithms can be used to solve (1.1) over any regular cone, provided one has a self-concordant barrier function F (x) defined over the interior int(K) of K. The reader may consult the authoritative monographs [13,17] for a detailed treatment of self-concordant functions and interior-point methods. In a generic self-concordant barrier function, one has control over the behaviour of the Hessians F ′′ (y) only when y lies in the local ball y : F ′′ (x)(y − x), y − x < 1 , leading to short-step interior-point methods. Although these methods have a polynomial runningtime guarantee, they tend to be less efficient linear programming solvers in practice than long-step interior-point methods. The theoretical basis for this latter type of algorithm is the fact that the self-concordant barrier function ln x i has additional properties which make it possible to control it in all of int(K).
In [14], Nesterov and Todd isolate two properties of the barrier − n i=1 ln x i which they identify as being responsible for making the long-step approach succeed in linear programming. They generalise these properties (see (1.4) and (1.5)) and call self-concordant barrier functions satisfying these conditions self-scaled. Since these properties also impose certain conditions on the domain of definition of such functions, Nesterov and Todd call the closures of such domains self-scaled cones. For convenience, we recall these concepts here: Definition 1.1. Let K ⊆ E be a regular cone. A self-concordant barrier function F : int(K) → R is called self-scaled if F ′′ (x) is non-singular for every x ∈ K, F is logarithmically homogeneous, that is, there exists a constant ν > 0 such that ∀x ∈ int(K), t > 0, (1.3) and if F satisfies the following two properties F ′′ (w)x ∈ int(K * ), ∀x, w ∈ int(K), (1.4) F * F ′′ (w)x = F (x) − 2F (w) − ν, ∀x, w ∈ int(K). (1.5) If K allows such a barrier function, then K is called a self-scaled cone.
The dual barrier F * : int(K) → R that appears in the last Axiom (1.5) is defined as F * (s) := sup − x, s − F (x) : x ∈ int(K) . Theorem 3.1 in [14] states that (1.4) can be strengthened to Theorem 1.2. If x ∈ int(K) and y ∈ int(K * ), then there exists a unique point w ∈ int(K) such that Moreover, if w ∈ int(K) then F ′′ (w)(K) = K * .
The point w is called the scaling point of x and y. The last statement is a consequence of the first part of the theorem and of Equation (3.2) from Nesterov and Todd's paper [14]. We reproduce this formula here for convenience: See also Lemma 2.3 below, where this formula reappears and where we give a proof of this important identity.
We would like to mention that Rothaus [18], using rather elementary tools, proves a number of results which are useful in Section 3 of this paper. Two key results are [18], Theorem 3.12 and Corollary 3.15, p. 205. These results imply Theorem 1.2 for the universal barrier function, a special self-concordant barrier function defined by Nesterov and Nemirovskii [13] which is further discussed below. Theorem 1.2 is an independently discovered extension of Rothaus's result to all self-scaled functions. It can be shown that, when suitably modified, all results of Section III in [18] can be extended to general self-scaled barriers. In particular this is true for Theorem 1.2.
Nesterov and Todd [14,15] demonstrate that self-scaled barrier functions can indeed be used to develop various long-step interior-point methods for linear optimization over self-scaled cones, in particular for semidefinite programming and for convex quadratic programming with convex quadratic constraints.
Inspired by the paper of Vinberg [20], O. Güler [4] develops the relationship between the universal barrier function of Nesterov and Nemirovskii [13] and the characteristic function of the cone K, introduced in 1957 by Koecher, see [10]. In particular, Güler [4] shows that if K is a regular cone, then the universal barrier function U (x) for K satisfies the equation for some constants c 1 > 0 and c 2 .
Through Güler's paper [4] the concepts of homogeneous cones, homogeneous self-dual cones (or symmetric cones), Euclidean Jordan algebras and Siegel domains as well as the classification theory of symmetric cones and Euclidean Jordan algebras, known to mathematicians since 1960 and 1934 respectively, were first introduced into the in the interior-point literature. The interested reader can find a complete treatment of these classification results in the book of J. Faraut and A. Korányi [1]. See also [10] for a different treatment of some of the same topics. Because of their importance for this paper, we recall some of the concepts mentioned above.
The cone K is called homogeneous if Aut(K) acts transitively on K, that is, given arbitrary points x, y ∈ K, there exists a map A ∈ Aut(K) such that Ax = y. The cone K is called self-dual if E can be endowed with an inner product such that K * = K where K * is defined with respect to this inner product, see (1.2). The cone K is called homogeneous self-dual if K is both homogeneous and self-dual.
Homogeneous self-dual cones are also called symmetric cones in [1], a terminology which we shall adopt in this paper.
The motivation behind [14,15] in contrast to [4] is rather different: While the first two papers deal with long-step interior-point methods and regular cones on which such methods can be designed, the latter one deals with the universal barrier function and the symmetry properties of regular cones, both in the group theoretic and the duality theoretic sense. Shortly after the announcement of the paper [14] Güler [3] discovers that the families of self-scaled cones and of symmetric cones are identical, thus establishing a connection between these two previously distinct ideas.
As mentioned earlier, symmetric cones are fully classified in the theory of Euclidean Jordan algebras, see [10,1] and the references therein. According to this theory, each symmetric cone has a unique decomposition into a direct sum of elementary building blocks, so-called irreducible symmetric cones, of which there exist only five types. Three examples of symmetric cones are of particular interest to the optimization community: The non-negative orthant K = R n + , the cone K = Σ + n of n × n symmetric, positive semidefinite matrices over the real numbers, and the Lorentz cone K = τ x ∈ R n+1 : τ ≥ x 2 . The general self-scaled conic optimization problems associated with these cones are respectively linear programming, semidefinite programming and second order cone programming. The latter can be seen as a reformulation of convex quadratic programming with convex quadratic constraints. Considering more general symmetric cones, one can treat linear optimization problems with mixed linear, semidefinite and convex quadratic constraints in a single unified framework.
Motivated by [4] and by the fact that only a small number of examples of self-scaled barrier functions are explicitly known, Hauser develops a partial algebraic classification theory for selfscaled barrier functions in a chapter of his thesis [7] and later announces a report [8] based on these result. Hauser shows that any self-scaled barrier over a symmetric cone K has an essentially unique decomposition into a direct sum of self-scaled barriers defined on the irreducible summands of K. Using this decomposition, he classifies in particular the family of isotropic self-scaled barrier functions which are characterised by rotational invariance. The insight gained from a lemma leads Hauser to conjecture that all self-scaled barrier functions defined on irreducible symmetric cones must be isotropic. Hauser points out that this conjecture, if true, would settle the classification problem of general self-scaled barrier functions. This conjecture also implies that all self-scaled barriers over irreducible symmetric cones are of the form c 1 ln ϕ K + c 2 , where c 1 > 0 and c 2 are constants. It follows from this theory that the full set of self-scaled barrier functions is readily known, and that all of these functions are just minor transformations of the universal barrier function.
In a second report [9], Hauser proves this conjecture in the special case where K is the positive semidefinite cone, i.e., he shows that all self-scaled barrier functions for use in semidefinite programming are isotropic and essentially identical to the standard logarithmic barrier function. Starting from first principles, Hauser shows that the orientation preserving part of the automorphism group of the positive semidefinite symmetric cone is generated by the Hessians of an arbitrary self-scaled barrier function defined on this cone, see [9], Corollary 4.3. Hauser's solution of the isotropy conjecture in this special case relies on Proposition 3.3, which also forms the key mechanism in the proof of Corollary 4.3 in [9].
Hauser's Corollary 4.3 is essentially a rediscovery of a result by Koecher, Theorem 4.9 (b), pp. 88-89 [10], for the special case of the cone of positive semidefinite symmetric matrices. Shortly after the announcement of Hauser's report [9] in March of 2000, Y. Lim [11] settles the general case of the isotropy conjecture by generalising Hauser's proof while refereeing the paper. Subsequently, both O. Güler [5] and S. Schmieta [19] independently of each other and independently of Y. Lim prove the isotropy conjecture in the general irreducible case. It is interesting to note that all three approaches to the general case of the conjecture, as well as Hauser's approach to the special case relevant to semidefinite programming, rely on the same deeper principle provided by Koecher's Theorem 4.9 (b) cited above. See also Remark 5.2 of this article.
The present article is a revision of Hauser's original report [8] which is based on parts of his thesis [7], while incorporating the solution to the general problem using Güler's approach. Schmieta's report [19] constitutes the first document where a proof of the general classification Theorem 5.5 became publicly available.
The rest of the paper is organised as follows. In Section 2 we reconsider self-scaled cones and self-scaled barriers from a symmetric point of view. Section 3 is devoted to certain properties of self-scaled barriers which link self-scaled barriers to the symmetry group of their domain of definition. These results are needed in later sections. In Section 4, we show that any pointed, convex cone has a unique decomposition as a direct sum of irreducible components. This result, of which we manage to locate only technically more involved generalisations, may be of independent interest. We therefore include a simple proof. We then use this decomposition result to show that any self-scaled barrier defined on a symmetric cone K decomposes in an essentially unique way into a direct sum of self-scaled barriers defined on the irreducible components of K, which also shows that the irreducible components are symmetric cones themselves. This decomposition reduces the problem of classifying self-scaled barriers to the case where the domain of definition is irreducible, a problem we solve in Section 5. Theorem 5.5 constitutes the main and final result of this paper. We thus present all the essential elements of the theory of self-scaled barrier functions in a single document.
The following basic properties of ν-self-concordant logarithmically homogeneous barrier functions and their duals will be used frequently in later sections. These properties are easy consequences of logarithmic homogeneity, see [13] or [14]: Proposition 1.4. Let F be a ν-self-concordant logarithmically homogeneous barrier function on the regular cone K ⊂ E, and let x ∈ int(K), s ∈ int(K * ). Then

A Symmetric View on Self-Scaledness
In this section we undertake a study of self-scaled cones and barrier functions while emphasising their symmetry properties in a duality-theoretic sense.
Let F be a self-scaled barrier function on a regular cone K in a finite dimensional Euclidean space E equipped with an inner product ·, · . With a given arbitrary point e ∈ int(K) we associate an inner product The following result is due to Güler [3]: Proof. We have where the third equality follows from Theorem 1.2. Note that Theorem 1.2 also shows that, given any two points u, v ∈ int(K), we can find a (unique) For the second assertion, note that if s ∈ K * e = K, then where the second and last equalities follow from (2.1) and (1.5), respectively. Consequently, F is self-scaled under ·, · e . Remark 2.2. Theorem 2.1 shows that from here on we may assume without loss of generality that K is a symmetric cone and that there exists a (unique) point e ∈ int(K) such that F ′′ (e) = I.
Together with Equation (1.5) this implies that and invoking (1.5) once more this implies the identity Note that (2.3) is a criterion that involves only the primal barrier F . Indeed, this identity allows one to characterise self-scaled barrier functions without invoking F * , see Lemma 2.5 below. Changing a barrier function by an additive constant is of no real consequence, as interior-point methods rely on gradient and Hessian information. Therefore, we can assume without loss of generality that F (e) = 0. For the same reason, Equation (2.2) makes it possible to think of F and F * as the same function. Hence, we no longer need to distinguish between primal and dual quantities -between F and F * , the primal and dual scaling points and so forth.
We next prove a property of the Hessian F ′′ (w) which will become an essential tool for our classification of self-scaled barriers. For all y ∈ int(K) let us define Expanding both sides of this equation and comparing the t 2 terms one gets In the proof above, we need only the weaker condition  [16], Güler [5] proves that this is indeed the case. Subsequently, S. Schmieta [19] independently discovers the same result, following essentially the same steps. Schmieta uses this fact as an essential tool to classify self-scaled barriers. As it turns out, the Jordan algebra connected to F is already discovered by McCrimmon in his thesis [12], even without the assumption of convexity for F . His proof in turn is a generalisation of Koecher's ideas [10] on ω-domains. Reading both works is instructive in delineating the role of convexity.
The following result provides an alternative definition of self-scaled barrier functions.
Lemma 2.5. Let K be a regular, self-dual cone. A logarithmically homogeneous self-concordant barrier function F on int(K) is self-scaled if and only if
Using logarithmic homogeneity alone one can prove that where the last equality follows from Equations (2.6) and (2.7). This concludes the proof.
Note that together with Equation (2.2), Lemma 2.5 implies that we can replace Axiom (1.5) of the original definition of a self-scaled barrier function by the requirement F * (F ′′ (w)x) = F (x) + C(w) for some constant C(w) which depends on w. This fact is already known, see [17].

Group-Theoretic Aspects of Self-Scaledness
In this section we explore the relationship between the Hessians of self-scaled barrier functions and the symmetry group of their domain of definition. Though we present these results primarily for the purposes of later sections they are also of independent interest.
The universal barrier function U (x) defined in (1.8) plays an important role in the context of this section. The choice of the inner product ·, · used in the definition of the characteristic function ϕ K (x) via (1.7) is irrelevant, since ϕ K changes only by an additive constant under a change of ·, · . Güler [4] shows that the universal barrier function U (x) is self-scaled, see Equation (13) and Theorem 4.4 in [4]. For all x ∈ int(K) let and let f ∈ int(K) be the point characterised by the equation Remark 3.1. It follows from Theorem 1.2 that f is unique. The existence of such a point is also well known, see for example page 17 of [1].
The point f is the "unit" associated with the self-scaled barrier U (x), see [1] Proposition I.3.5, p. 14, and it is also the unit of the Jordan algebra associated with U .
The following lemma is Theorem 3.17, pp. 205-206 in [18]. We include a short proof of this result because these ideas play an important role in later sections. See also [1], Proposition I.4.3, p. 18 for a different approach to proving this result. Proof. If A ∈ Aut(K), then for every vector h ∈ E. That is, A * Q(Af ) −1 A = I, or Q(Af ) = AA * , see for example [4], Equation (11). This shows that A is orthogonal if and only if I = Q(Af ). The uniqueness of f implies that this condition is equivalent to Af = f .
Next we note that the elements of Aut(K) have a unique polar decomposition, see [18], Theorem 3.18, p. 206. For the sake of completeness we give a short proof. is orthogonal, and we have I = HH * = Q −1 2 Q 2 1 Q −1 2 , or Q 2 2 = Q 2 1 . Since Q 1 and Q 2 are symmetric, we have Q 1 = Q 2 and H 1 = H 2 .
The following result will play a key role in Section 5 where we classify self-scaled barriers.
Lemma 3.4. The sets of inverse Hessians of F and U coincide, that is, and for all x ∈ int(K) it is true that
The point e −1 is the inverse of e in the Jordan algebra associated with U (x). Note that Proposition 1.4 iii) shows that e −1 = −F ′ (e).
Proof. If v ∈ int(K), then Lemma 3.3 implies that we can write P (v) = Q(u)H for some u ∈ int(K) and H ∈ O Aut(K) . By the uniqueness of the polar decompositions P (v) = P (v) I and P (v) = Q(u)H it must be true that P (v) = Q(u). Thus, Conversely, let u ∈ int(K). By Theorem 1.2, there exists a point v ∈ int(K) such that P (v)f = Q(u)f . But this implies that Q(u) −1 P (v)f = f . By virtue of Lemma 3.2 H := Q(u −1 )P (v) is therefore orthogonal. This means that P (v) has the polar decompositions P (v) = P (v) I and P (v) = Q(u)H. The uniqueness part of Lemma 3.2 then implies that Q(u) = P (v) and H = I. This proves the first statement of the lemma.
Now let x ∈ int(K) and define u by x = Q(u)f , see Theorem 1.2. We have where the second equality follows from the fundamental Formula (2.4). In a similar vein, taking the first part of this lemma into account we obtain These two equations imply that Q(u) = Q(x) 1/2 and In particular, setting x = e yields I = Q(e) 1/2 P (f )Q(e) 1/2 , that is, P (f )Q(e) = I, and P (f ) = Q(e) −1 = Q(e −1 ). The lemma follows, since this implies that Although it does not have a direct bearing on later results, the following proposition already shows that the self-scaled barrier function F is intimately connected to the universal barrier function.

Decomposition of Cones and Barrier Functions
In this section, we prove two related results. A cone is called pointed if it does not contain any whole lines. First, we show that any pointed, convex cone decomposes into a direct sum of indecomposable or irreducible components in a unique fashion. This result, which is of independent interest, is essentially a special case of Corollary 1 in [2]. Gruber's paper is the earliest mentioning of this result we could find, though it may have been derived several times independently. Gruber's original result addresses a more general affine setting which renders his proof more technically involved. Therefore, we include a simple and accessible proof. Second, we use this decomposition to write any self-scaled barrier function defined on the interior of a symmetric cone K as a direct sum of self-scaled barriers defined on the irreducible components of K.
Recall that the Minkowski sum of a set {A i } k i=1 of subsets of E is defined as If all of the A i are linear subspaces {0} = E i ⊆ E which satisfy E = E 1 + · · · + E k and E i ∩ ( j =i E j ) = {0}, then we say that the sum E = E 1 + · · · + E m is direct and write to denote this relationship between K and {K i } m i=1 . K is called indecomposable or irreducible if it cannot be decomposed into a non-trivial direct sum.

Let us defineÊ
This implies that K i = π Ei K is a convex cone. Similarly, we have We first prove a useful technical result: Lemma 4.2. Let K be a pointed, convex cone which decomposes into the direct sum (4.1). If x ∈ K i is a sum x = x 1 + · · · + x k of elements x j ∈ K, then each x j ∈ K i . Proof. We have 0 = πÊ i x = πÊ i x 1 + . . . + πÊ i x k . Each termx j := πÊ i x j ∈K i ⊆ K, therefore we havex j ∈ K and −x j = l =jx l ∈ K. Since K contains no lines it must be true thatx j = 0, that is, x j = π Ei x j ∈ K i , (j = 1, . . . , k). Theorem 4.3. Let K ⊆ E be a decomposable, pointed, convex cone. The irreducible decompositions of K are identical modulo indexing, that is, the set of cones {K i } m i=1 is unique. Moreover, the subspaces E i corresponding to the non-zero cones K i are also unique. In particular, if K is solid, then all the cones K i are non-zero and the subspaces E i are unique.
Proof. Suppose that K admits two irreducible decompositions Note that each non-zero summand in either decomposition of K must lie in span(K) and that the subspace corresponding to each zero summand must be one-dimensional, for otherwise the summand would be decomposable. This implies that the number of zero summands in both decompositions is codim span(K) . We may thus concentrate our efforts on span(K), that is, we can assume that K is solid and all the summands of both decompositions of K are non-zero. By (4.1), each x ∈ C j ⊆ K has a unique representation x = x 1 + · · · + x m where x i = π Ei x ∈ K i ⊆ K. Also, Lemma 4.2 implies that x i ∈ C j , and hence x i ∈ K i ∩ C j . Consequently, every x ∈ C j lies in the set (K 1 ∩ C j ) + · · · + (K m ∩ C j ). Conversely, we have K i ∩ C j ⊆ C j , implying that (K 1 ∩ C j ) + · · · + (K m ∩ C j ) ⊆ C j . Therefore, we have , and the intersection of any two distinct summands in the last sum is the trivial subspace {0}. The above decompositions of F j and C j are thus direct sums. Since C j is indecomposable, exactly one of the summands in the decomposition of C j is non-trivial. Thus, C j = K i ∩ C j , and hence C j ⊆ K i for some i. Arguing symmetrically, we also have K i ⊆ C l for some l, implying that C j ⊆ C l . Therefore, j = l for else C j ⊆ F j ∩ F l = {0}, contradicting our assumption above. This shows that C j = K i . The theorem is proved by repeating the above arguments for the coneK i = ⊕ k =i K k = ⊕ l =j C l . Next, we show that self-scaled barrier functions have irreducible decompositions as well. Let F be defined on int(K) where K is a symmetric cone with irreducible decomposition (4.1). For i = 1, . . . , m, let F i be a function defined on ri(K i ), the relative interior of , then we say that F is the direct sum of the F i and write Theorem 4.4. Let K be a symmetric cone with irreducible decomposition (4.1). The irreducible components K i are symmetric cones. Let F (x) be a self-scaled barrier for K. There exist selfscaled barrier functions F i for the cones K i such that The functions F i are unique up to additive constants.
Proof. Recall that the universal barrier function U (x) is given in (1.8). Changing the inner product used in the definition of the characteristic function ϕ K (x) changes U (x) only by an additive constant, hence we may assume that x, y = m i=1 x i , y i Ei for the purposes of this definition. Here, ·, · Ei is an inner product defined on E i chosen so that U ′′ i (f i ) = id Ei for some elements f i ∈ ri(K i ) where U i denotes the universal barrier function defined on ri(K i ). Then we have , in full consistency with our previous notation. Moreover, K is self-dual under ·, · , since K * = Q(f )K = K. Hence, we may choose the vector e ∈ int(K) specified in Remark 2.2 as the unique element in int(K) such that F ′′ (e) = I under ·, · , see Remark 3.1.
Thus, U can be written as the direct sum U (x) = ⊕ m i=1 U i (x i ) and Q(x) has block structure corresponding to the subspaces Consequently, Equation (3.1) implies that P (x) also has the same block structure, where , π Ei being the projection at the beginning of this section.
So far we know that P (x) has block structure corresponding to the direct sum E = ⊕ m i=1 E i , but it is not a priori clear that P i (x i ) −1 is the Hessian of a function defined on ri(K i ). Let the spaceŝ E i be defined as earlier in this section, and let us consider the vector fields v i : x → π Ei F ′ (x), defined on int(K) and taking values in E i for all i, i = 1, . . . , m. We claim that v i depends only on x i = π Ei x. In fact, for any two vectors x, y ∈ int(K) such that which shows our claim. Hence, the quotient vector fieldŝ are well defined and can be identified with vector fieldsv i defined on the cones ri(K i ). The direct sum of these vector fields amounts to the gradient field F ′ being conservative, thev i must be conservative too, implying that these are the gradient vector fields of some functions F i defined on ri(K i ) which are uniquely determined up to additive constants. We may choose these constants so that Using Equation (4.3), it is straightforward to check that the F i are self-concordant, see [13]. Applying Proposition 1.4 i) and iv) to F , using Equation (4.4) and considering variations of x ∈ int(K) only in the part x i = π Ei x, we obtain that This shows that the functions F i are ν i -logarithmically homogeneous. It is a well-known fact that any logarithmically homogeneous self-concordant function is also a barrier function, see for example [13] or [17]. It remains to show that the functions F i are self-scaled. As previously noted, Condition (2.5) is satisfied, since P i (x i ) ∈ Aut(K i ) for all i. Finally, Condition (2.6) holds for F i because we can apply this condition to F , choosing w = w i ⊕ ⊕ j =i e i and x = x i ⊕ ⊕ j =i e i and using the block structures of F and F ′′ .
Note that the irreducible components K i of K must be symmetric cones, since the F i are selfscaled barriers defined on ri(K i ). The symmetry of the K i can also be directly derived from the block structure of Q(x) = ⊕ m i=1 Q i (x i ) and the fact that the set of cone automorphisms Q(x) : x ∈ int(K) acts transitively on K.
The decomposition Theorem 4.4 shows that for the purposes of classifying self-scaled barriers we may concentrate our efforts on irreducible cones.

Classification of Self-Scaled Barriers
In this section, we give a complete classification of self-scaled barrier functions on the symmetric cone K.
The definition of a self-scaled barrier function F requires that F changes only by an additive constant under the action of symmetric cone automorphisms {P (u) : u ∈ int(K)}, see Equation (2.3). However, it is not a priori known how F behaves under the action of an arbitrary element of Aut(K). Note that this is in marked contrast to the case where F is the universal barrier function U , which is known to change only by an additive constant under the action of any element of the symmetry group of K. This explains in a sense the main difficulty one faces when proving the results below.
The next result is key in resolving this difficulty and is just a slight reformulation of the conjecture raised in [8], according to which self-scaled barriers on irreducible symmetric cones are isotropic. Let us denote by Aut(K) 0 the connected component of the identity in Aut(K).
Theorem 5.1. Let K be a symmetric cone. If H ∈ Aut(K) 0 is orthogonal, then F (Hx) = F (x) for all x ∈ int(K).
Proof. Koecher [10] proves that if K is a symmetric cone, then Aut(K) 0 is generated by {Q(u) : u ∈ V} where V is a neighbourhood of the identity f , see [10], Theorem 4.9 (b), pp. 88-89. Koecher's proof exploits the fact that all derivations of the Jordan algebra associated with U (x) are inner. An accessible proof for the case where K is irreducible is given in [1], Lemma VI.1.2, pp. 101-102, based on certain non-trivial results from the theory of Jordan algebras. If H ∈ Aut(K) 0 is orthogonal, it follows from Koecher's result that for some u i , v i ∈ int(K), (i = 1, . . . , l). Here the second equality follows from Lemma 3.4. Therefore, it follows from Equation (2.3) that Since Hf = f , setting x = f above yields l 1 F (v i ) = 0 and settles the claim of the theorem.
The group Aut(K) 0 already acts transitively on K, see [1], page 5. Thus, the above result is significant.
Remark 5.2. Hauser's approach [9] to solving the isotropy conjecture for the cone of positive semidefinite symmetric matrices is based on similar ideas. Hauser essentially rediscovers Koecher's Theorem 4.9 (b) in this special case, see [9] Proposition 3.3 and Corollary 4.3. He then uses Proposition 3.3 in conjunction with the fundamental formula as the key mechanism in the proof of the conjecture. Y. Lim [11] generalised this approach to arbitrary irreducible symmetric cones while refereeing the paper [9], thus completing the classification of self-scaled barriers. However, since Lim was an anonymous referee, his result was not publicly announced. Now, assume that K is irreducible and let U be the universal barrier function for K. Let k be the rank of the Jordan algebra associated with U (x). Let x be an arbitrary point in int(K). Then there exists an orthogonal frame {e 1 , . . . , e k } such that x = k i=1 λ i e i , λ i > 0, i = 1, . . . , k, see [1], Theorem III.1.2, pp. 44-45. By C let us denote the cone generated by this frame, that is, Note that C is a direct sum of the half-lines {λ i e i : λ i ≥ 0} and thus a symmetric cone. Lemma 5.3. If F is a self-scaled barrier function defined on the interior of the irreducible symmetric cone K, and if C is as defined above, then Proof. Let σ be any permutation of {1, . . . , k}. Theorem IV.2.5 in [1] implies that there exists an orthogonal automorphism H ∈ Aut(K) 0 such that He i = e σ(i) (i = 1, . . . , k). Using Theorem 5.1, we then obtain α i e i ), ∀α i > 0 (i = 1, . . . , k).
Note that g is a symmetric function. Consider a point e + k i=1 β i e i = k i=1 α i e i ∈ ri(C), with arbitrary β i ≥ 0 and α i := 1 + β i . Applying Theorem 5.1 in [14] repeatedly, we obtain (F (e + β i e i ) − F (e)).
The following theorem classifies self-scaled barrier functions for irreducible symmetric cones.
Theorem 5.4. Let K be an irreducible symmetric cone, and let F be a self-scaled barrier function defined on int(K). Then there exist constants α > 0 and β such that where U (x) is the universal barrier function on int K.
Proof. Lemma 5.3 describes the restriction of F on ri(C). Since the universal barrier function is also self-scaled, the same considerations apply to U (x) − U (e). Thus, the functions F and U are homothetic transformations of each other on ri(C), that is, there exist α > 0, β such that F (x) = αU (x) + β.
Let y ∈ int(K) be an arbitrary point with the spectral decomposition y = k i=1 ν i d i . Corollary IV.2.7 in [1] implies that there exists A ∈ O(Aut(K) 0 ) such that Ae i = d i (i = 1, . . . , k). We have y = Ax where x = k i=1 ν i e i ∈ ri(C). Theorem 5.1 gives F (y) = F (x) and U (y) = U (x), and hence the Identity (5.2) extends to all of int(K).
We are now ready to give the final classification theorem for self-scaled barrier functions on arbitrary symmetric cones. This theorem shows that all self-scaled barrier functions are related to the standard logarithmic or the universal barrier via homothetic transformations.
Theorem 5.5. Let F be a self-scaled barrier function for a symmetric cone K with irreducible decomposition (4.1). Then there exist constants c 0 and c 1 ≥ 1, . . . , c m ≥ 1 such that see Equation (4.2). Here det Ki x i denotes the determinant of x i ∈ ri(K i ) in the Jordan algebraic sense, see [1], Chapter 2. Conversely, any function of this form is a self-scaled barrier for K.
Proof. Theorems 4.4 and 5.4 imply that there exist constants d 0 and d 1 > 0, . . . , d m > 0 such that where u i (x i ) = ln ϕ Ki (x i ). It is known that u i (x i ) = const − n i /r i ln det x i , where r i is the rank of the Jordan algebra associated with u i (x), and n i is the dimension of the cone K i , see [1], Proposition III.4.3, p. 53. Finally, Theorem 4.1 in [6] implies that the function −α ln det x i is self-concordant if and only if α ≥ 1.

Acknowledgements
R. Hauser conducted his research on this paper partially while writing his Ph.D. at the School of Operations Research and Industrial Engineering of Cornell University, and partially as a postdoctoral fellow at the University of Cambridge. He wishes to thank Mike Todd, Jim Renegar and Arieh Iserles for numerous valuable discussions and for their continuous support. This research was supported by a "bourse pour chercheur débutant" from the Swiss National Science Foundation and the Swiss Academy of Technical Sciences, and by the Engineering and Physical Sciences Research Council of the UK, EPSRC grant GR/M30975.