Barrier Functions in Interior Point MethodsOsman G�uler�Technical Report 94{01, March 1994(Revised May 1995)AbstractWe show that the universal barrier function of a convex cone introduced byNesterov and Nemirovskii is the logarithm of the characteristic function of the cone.This interpretation demonstrates the invariance of the universal barrier under theautomorphism group of the underlying cone. This provides a simple method forcalculating the universal barrier for homogeneous cones. We identify some knownbarriers as the universal barrier scaled by an appropriate constant. We also calculatesome new universal barrier functions. Our results connect the �eld of interior pointmethods to several branches of mathematics such as Lie groups, Jordan algebras,Siegel domains, di�erential geometry, complex analysis of several variables, etc.Key words. Barrier functions, interior point methods, self{concordance, convex cones,characteristic function, duality mapping, automorphism group of a cone, homogeneouscones, homogeneous self{dual cones.Abbreviated title. Barrier Functions in Interior Point Methods.AMS(MOS) subject classi�cations: primary 90C25, 90C60, 52A41; secondary 90C06,90C15, 90C20, 90C33.�Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore,Maryland 21228, USA. (e-mail: guler@math.umbc.edu). Research partially supported by the NationalScience Foundation under grant DMS{9306318.1

1 IntroductionSince Karmarkar [17] introduced his polynomial{time projective algorithm for linear pro-gramming, the �eld of interior point methods has been developing at a rapid rate. Thereare at present close to 1500 papers written in the �eld. Most of these papers deal with theimportant problems of linear programming, convex quadratic programming, and mono-tone linear complementarity. At the same time, some researchers, especially Nesterovand Nemirovskii have successfully developed a general theory of interior point methodsfor nonlinear convex programming problems and monotone variational inequalities. Thedetails of this theory can be found in the recent book by Nesterov and Nemirovskii [20].The two main components of this theory are the self{concordant barrier functions andthe Newton method. This paper is concerned with the construction of barrier functions.We now recall some relevant concepts. Let Q � Rn be an open convex set. A functionF : Q! R is called an �{self{concordant function if F is at least three times di�erentiable,convex, and satis�es the propertyjD3F (x)[h; h; h]j � 2p�(D2F (x)[h; h])3=2; (1)where DkF (x)[h; : : : ; h] is the kth directional of F at x along the direction h 2 Rn . Thefunction F is called strongly self{concordant if it is also a barrier function of Q, that is,F (x)!1 as x! @Q:One of the main contributions of Nesterov and Nemirovskii is to show that the (damped)Newton method performs well in minimizing a self{concordant function, and that this isresponsible for the polynomial{time convergence of the interior point methods. They alsoshow that, in solving constrained convex programming problems, a key role is played byself{concordant barrier functions which are 1{self{concordant and satisfy the additionalproperty jDF (x)[h]j2 � #D2F (x)[h; h]: (2)The constant # is called the parameter of the barrier function, and determines the speedof the underlying interior point method.We also recall the relevant barrier function concepts for a pointed convex cone K withnon{empty interior, that is, a convex cone containing no lines and having a non{emptyinterior. (There is no essential loss of generality in restricting attention to pointed cones.)A function F is called a #{logarithmically homogeneous barrier for K if it is a barrierfunction for K and satis�es the propertyF (tx) = F (x)� # log t; (3)2

that is, the function '(x) = eF (x) is �# homogeneous:'(tx) = '(x)t# :The function F is called a #{normal barrier for K if it is #{logarithmically homogeneousand 1{self{concordant. Nesterov and Nemirovskii [20], Proposition 5.1.4, show that anyself{concordant{barrier function on a convex set with non{empty interior can be extendedto a logarithmically homogeneous self{concordant barrier function on the cone �tted toQ (conic hull in the terminology of [20]). Thus we can restrict our attention to cones.In this paper, we shall be concerned with the construction of logarithmically homoge-neous self{concordant barrier functions for convex cones. Nesterov and Nemirovskii showthat any open convex set Q admits a universal barrier function which is also logarith-mically homogeneous if Q is a pointed convex cone. They describe the universal barrierfunction in terms of the volume of the polar set, see Section 4. One of the main contri-butions of the present paper is to show that there exists a simpler representation of thisfunction in terms of the characteristic function of a cone described below in Section 3.The characteristic function, introduced by Koecher [18] in 1957, is useful in the classi�-cation of homogeneous bounded domains in several complex variables. This subject hasits origins in the works of Poincar�e, E. Cartan [3], C. L. Siegel, Pyatetskii{Shapiro, andothers, (see the book [24] by Pyatetskii{Shapiro for details). The characteristic functionalso has connections with the Bergman kernel function on tube domains [8], etc; it evenhas uses in algebraic geometry [23].The characteristic function for a cone K has invariance properties under the action ofthe automorphism group of K. This will be discussed below in Section 2. These invari-ance properties will help greatly in calculating the characteristic function of homogeneouscones, see Section 7.It is remarkable that homogeneous self{dual cones (\domains of positivity" in Koecher'sterminology [18]) can be completely classi�ed in terms of certain Jordan algebras, seeKoecher [19], Hertneck [14], Vinberg [34], Satake [28], Faraut and Koranyi [8]. Thereexist only �ve classes of irreducible self{dual cones which will be mentioned in Section 2.Vinberg [34] is the �rst to give an example of a homogeneous cone that is not self{dual.The class of homogeneous cones is much larger than the class of homogeneous self{dualcones. However, homogeneous cones can also be classi�ed in terms of a class of non{associative matrix algebras, called T{algebras by Vinberg, see Vinberg [32, 33]. Thesecones can be constructed recursively, see Vinberg [32, 33], Gindikin [10], Rothaus [27],Dorfmeister [4, 5, 6], etc.The Hessian of the characteristic function de�nes an invariant Riemannian metricin K. Thus, the characteristic function has intimate connections with Lie groups and3

di�erential geometry [18, 19, 26, 32, 4, 5, 28]. The characteristic function also has uses incarrying out Fourier analysis on K [15, 10, 30, 8].The paper is organized as follows. In Section 2 we present some concepts and resultsfrom the theory of convex cones, especially concepts related to the automorphism groupof the cone. In Section 3, we introduce the characteristic function of a cone K � Rn anddiscuss its invariance properties. In Section 4, we prove the important result that theuniversal barrier function of K is essentially the logarithm of the characteristic functionof K. In Section 5, we introduce the duality mapping which is an analytic bijectionbetween K0 and (K�)0, the interiors of K and its dual K�. We also show that if K is ahomogeneous cone, then the (slightly modi�ed) Fenchel dual of the universal barrier forK is the same as the universal barrier for K�. In Section 6, we describe the importantconcept of a Siegel cone and how it relates to homogeneous cones and their classi�cation.In Section 7, we calculate the universal barrier functions for some cones. Concludingremarks are made in Section 8.2 Convex ConesIn this section we present some elementary concepts and some relevant results from thetheory of convex cones.De�nition 2.1 A subset K � Rn is called a cone if for x 2 K and a scalar � � 0, wehave �x 2 K. A cone K is called convex if x; y 2 K implies x + y 2 K. If in additionK0 6= ; and K contains no lines, then K is called a regular convex cone.In this paper, we shall always be concerned with regular convex cones, and will refer tothese simply as cones. Two cones K1 and K2 are called isomorphic if there exists aninvertible linear mapping A 2 Rn�n such that A(K1) = K2. Isomorphic cones can beconsidered equivalent.Let Rn be endowed with an inner product hx; yi = xTSy where S is a symmetric,positive de�nite matrix. The dual of cone K is de�ned asK� = \x2Kfy 2 Rn : hx; yi � 0g: (4)It is well{known that if K� is any closed convex cone, then K�� = K, and if K is regular,then so is K�. Note that the dual cone depends on the inner product; it can be veri�edthat if hx; yiI = xTy is the standard inner product on Rn , then the dual K�I is related toK� by the equation K� = S�1(K�I ): (5)4

A cone is called self{dual if there exists an inner product such that K� = K. Note thata self{dual cone in one inner product may not be so in another one. The self{duality isa useful property, and this is the main reason why we allow inner products other thanthe standard one, see for example Section 7.3. Note that a self{dual cone K in the innerproduct h�; �iS is isomorphic to the cone S1=2(K) which is self{dual in the standard innerproduct. Thus, if one is willing to work with transformed coordinates, one can always workwith the standard inner product. However, this may not be convenient, since it mightmake it harder to describe the cone in the transformed coordinates, see again Section 7.3.Let Ki � Rni , i = 1; : : : ; k be cones. The direct sum of the cones Ki is the coneK1 �K2 � : : :�Kk = f(x1; : : : ; xk) : xi 2 Kig:A cone K is called decomposable if it can be written as a direct product of cones. Oth-erwise, it is called indecomposable or irreducible. It is well known that a decomposablecone can be written as a direct sum of irreducible cones essentially in a unique way, seefor example Schneider [29], Theorem 3.2.1, pp. 142.The following de�nition formalizes the symmetries of a cone.De�nition 2.2 Let K � Rn be a cone. The set of non-singular linear maps A : Rn ! Rnleaving K invariant, that is satisfying A(K) = K, is called the automorphism group of Kand is denoted by Aut(K). K is called homogeneous if Aut(K) is transitive on K0, thatis, given arbitrary points x; y in K0, there exists A 2 Aut(K) such that Ax = y.It is easy to verify that Aut(K) forms a subgroup of the general linear group GL(n;R)of all non{singular linear transformations of Rn . It is also easy to see that Aut(K) is aclosed subgroup of GL(n;R). Thus, by a theorem of von Neumann (or a more generalresult of E. Cartan), Aut(K) is a Lie group. From (4) it follows that if A 2 Aut(K), andx 2 K0, y 2 (K�)0, we have0 < xTATSy = (Ax)TSy = hAx; yi � hx;A�yi = xTSA�y: (6)This shows that the conjugate map A� = S�1ATS 2 Aut(K�). Similarly, if B 2 Aut(K�),then B� 2 Aut(K). It follows thatAut(K�) = fA� : A� = S�1ATS;A 2 Aut(K)g;that is, the groups Aut(K) and Aut(K�) are isomorphic. It follows from (6) that themapping �(A) = (A�)�1 = S�1(AT )�1Sis a group isomorphism between Aut(K) and Aut(K�). If K is homogeneous, then so isthe dual cone K�, see (16). The automorphism group of a decomposable group is related5

to the automorphism groups of its summands in the following way (see Vinberg [32]): ifK = �ki=1Ki, then Qki=1Aut(Ki) is a normal subgroup of �nite index in Aut(K). It isclear that if the cones Ki are homogeneous, then so is K.Irreducible homogeneous self{dual cones can be characterized completely in terms offormally real Jordan algebras [19, 14, 34, 28, 8], etc. These algebras, invented by P. Jordanin connection with quantum mechanics, are essentially classi�ed in the very �rst paperon Jordan algebras, the paper [16] by P. Jordan, J. von Neumann, and E. Wigner. Anyhomogeneous, irreducible self{dual cone is isomorphic to one of the following �ve cones:(i) the cone of positive semi{de�nite symmetric matrices (see Section 7.3),(ii) the Lorentz cone (see Section 7.2),(iii) the cone of positive semi{de�nite Hermitian matrices,(iv) the cone of positive semi{de�nite Hermitian quaternion matrices,(v) a 27 dimensional exceptional cone.The characteristic function of a cone discussed in Section 3 below is an important toolin this classi�cation. Vinberg [34] gives an example of a homogeneous cone that is notisomorphic to a self{dual cone. In his seminal paper [32], he classi�es the homogeneouscones in term of T{algebras, a class of matrix algebras that he invents for this purpose.Again, the characteristic function plays a central role in the classi�cation of homogeneouscones. As mentioned above, it is possible to build up homogeneous cones in a recursivemanner. This is discussed in some detail in Section 7.7.3 Characteristic Function of a ConeIn this section we state the de�nition of the characteristic function of a cone and presentits most important properties.De�nition 3.1 Let K � Rn be a cone equipped with an inner product hx; yi = xTSy,where S 2 Rn�n is a symmetric, positive de�nite matrix. The characteristic function'K : K0 ! R of the cone K is the function'K(x) = ZK� e�hx;yidy: (7)6

We shall write ' when the cone under consideration is obvious. The function ' is essen-tially independent of the inner product. Consider the standard product in hx; yiI = xTy.Equation (5) implies K�I = S(K�) and we have'I(x) = ZK�I e�xT ydy = ZS(K�) e�hx;S�1yidy = ZK� e�hx;y0i(detS)dy0 = '(x) detS: (8)Consequently, the two characteristic functions di�er by a multiplicative constant.The characteristic function has been introduced in connection with the classi�cationof bounded homogeneous domains in complex analysis of several variables. Its mainproperties can be found in Koecher [18, 19], Rothaus [26], Vinberg [32], and Faraut andKoranyi [8], etc. The most important properties of ' are(P1) ' is an analytic function de�ned on the interior of K and '(x)!1 as x! @K.(P2) ' is logarithmically strictly convex, that is, the functionF (x) = log('(x))is strictly convex,(P3) If A is an automorphism of K, then'(Ax) = '(x)j detAj : (9)We note that since tI 2 Aut(K) for any t > 0, we have'(tx) = '(x)tn : (10)The properties (P1) and (P2) show that ' and F above are smooth barrier functionsfor K. These two functions, especially F will be important for interior point methods; wewill show in Section 4 that F is essentially the universal barrier function of K.Property (P3) is the important invariance property of '. Note that it is obtained from(7) by the change of variables formula. Since '(Ax) = '(x)=(j detAj), we haveF (Ax) = F (x)� log(j detAj);which implies the important identityDkF (Ax)[Ah;Ah; : : : ; Ah] = DkF (x)[h; h; : : : ; h]; 8h 2 Rn ; k � 1: (11)7

Property (P3) is important in calculating the barrier function of homogeneous cones. LetK be a homogeneous cone. Fix a point e 2 K0. Let x 2 K0 be an arbitrary point, andsuppose that Ax 2 Aut(K) satis�es Axe = x. Then (P3) implies'(x) = '(Axe) = '(e)j detAxj = constj detAxj : (12)Consequently, F (x) = const� log(j detAxj): (13)We conclude this section by noting that D2F (x) de�nes an invariant Riemannianmetric on K. In fact, (11) implies that each derivative DkF (x), k � 1, is invariant underthe action of Aut(K). Moreover, (9) and the change of variables formula imply that themeasure '(x)dxis invariant under Aut(K), that is, if A 2 Aut(K), thenZK h(Ax)'(x)dx = ZK h(x)'(x)dxwhenever the integral on the right exists.4 Self{Concordance of the Characteristic FunctionIn this section we prove the important result that the universal barrier function of Nes-terov and Nemirovskii is essentially the logarithm of the characteristic function. Thisrepresentation of the universal barrier function will make it easier to calculate barrierfunctions for cones.Let Q � Rn be a convex set. Nesterov and Nemirovskii [20] de�ne the universal barrierfunction for Q as u(x) = log(voln(Q�(x));where voln stands for the n{dimensional Lebesgue measure, and Q�(x) is the polar set ofQ centered at x, that is,Q�(x) = fy 2 Rn : hz � x; yi � 1; 8z 2 Qg: (14)We need the following result in the proof of Theorem 4.1.Lemma 4.1 Let K � Rn be a cone. Then for x 2 K,K�(x) = �fy 2 K� : hy; xi � 1g:8

Proof. Denote the set on the right by U . First, we show that U � K�(x). Supposethat u 2 U . Then h�x; ui � 1 and since u 2 �K�, we also have hz; ui � 0 for all z 2 K.Adding these two inequalities gives hz � x; ui � 1 for all z 2 K, implying u 2 K�(x).Conversely, suppose that u 2 K�(x). Then y = �u satis�eshz0 � x;�yi � 1; 8z0 2 K:Setting z0 = 0 above gives hx; yi � 1. Also, setting z0 = x + �z, � > 0, z 2 K giveshz; yi � �1=�. Letting �!1 shows hz; yi � 0, that is, y 2 K�. Thus, u 2 U .We de�ne the sets H(x; �) = fy : hx=jjxjj; yi = �g;H�(x; �) = fy : hx=jjxjj; yi � �g:The following theorem is one of the main results of this paper.Theorem 4.1 Let ' be the characteristic function of a cone K, and de�ne F (x) =log('(x)). Then F (x) = u(x) + log n!:Proof. The hyperplane H(x; �) is orthogonal to x and has distance � from the origin.The function e�hx;yi = e��jjxjj is constant on this hyperplane, and we can write '(x) as'(x) = ZK� e�hx;yidy = Z 10 e��jjxjj�ZK�\H(x;�) dn�1y�d�= �Z 10 e��jjxjj�n�1d�� � �ZK�\H(x;1) dn�1y� = (n� 1)!jjxjjn voln�1(K� \H(x; 1)):Here the second equality can be obtained by elementary methods, for example by trans-forming variables. It is also a direct application of the co{area formula, see [7], Theorem2, Section 3.4.3, pp. 117, or [9], Theorem 3.2.3, pp. 243. Sincevoln(K� \H�(x; 1)) = voln�1(K� \H(x; 1))n ;we have '(x)n! = 1jjxjjn voln(K� \H�(x; 1))= voln(K� \H�(x; 1=jjxjj)) = voln(fy 2 K� : hx; yi � 1g)= voln(K�(x)) = eu(x); 9

where the fourth equality follows from Lemma 4.1.Let Q � Rn be a closed convex set with non{empty interior. Endow Rn+1 with theinner product h(x; t); (y; �)i = hx; yi + t� , and consider the �tted cone K(Q) � Rn+1de�ned by K(Q) = cl(f(x; t) : x 2 tQ; t > 0g = ft(z; 1) : z 2 Q; t > 0g):Since Q is identi�ed with the cross section K(Q) \ f(x; 1) : x 2 Rng, the restriction of'K(Q) to the cross section gives a \characteristic function" for Q.We begin by calculating 'K(Q).Theorem 4.2 Let '(x; t) be the characteristic function of the cone K(Q). Then'(x; t) = n!tn+1voln((Q)�(x=t)):Proof. Theorem 4.1 gives'(x; t) = (n + 1)! voln+1�K(Q)� \ f(y; �) : h(x; t); (y; �)i � 1g�= (n + 1)! voln+1�K(Q)� \H�((x; t); 1=jj(x; t)jj�:Thus, we have'(x; t)(n + 1)! = 1jj(x; t)jjn+1voln+1�K(Q)� \H�((x; t); 1))= 1(n + 1)jj(x; t)jjn+1voln(K(Q)� \H((x; t); 1))= 1(n + 1)jj(x; t)jjvoln(K(Q)� \H((x; t); 1=jj(x; t)jj)):NowK(Q)� \H((x; t); 1=jj(x; t)jj) = f(y; �) : hz; yi+ t� � 0; 8z 2 tQ; and hx; yi+ t� = 1g= f(y; 1� hx; yit ) : hz � x;�yi � 1; 8z 2 tQg= f(y; 1� hx; yit ) : y 2 �(tQ)�(x)g:Therefore, '(x; t) = n!jj(x; t)jjvoln�f(y; 1� hx; yit ) : y 2 �(tQ)�(x)g�:10

The set in the above formula is the graph of the function �(y) = (1 � hx; yi)=t over thedomain �(tQ)�(x). By the surface area formula in calculus, it has volumejj(x=t; 1)jjvoln((tQ)�(x)) = jj(x; t)jjtn+1 voln(Q�(x=t)):This shows that '(x; t) = n!tn+1 voln(Q�(x=t));and proves the theorem.Corollary 4.1 De�ne FQ(x) = log('K(Q)(x; 1)). ThenFQ(x) = uQ(x) + log n!;where uQ is the universal barrier function for Q. In other words, FQ(x) and the universalbarrier function for Q di�er only by an additive constant.Nesterov and Nemirovskii prove in their book, [20] (Theorem 2.5.1, pp. 50), thatthe universal barrier function is self{concordant with a parameter # = O(n). Theirderivation of the bound on # is long and involves delicate moment inequalities. At leastfor homogeneous cones, one can give a simple proof that F is self{concordant, althoughthe proof does not give any bound on the important self{concordance parameter #. (Weknow # = O(n) as mentioned above.)Theorem 4.3 Let K � Rn be a homogeneous cone. Then F (x) = log'K(x) is a self{concordant barrier function.Proof. Fix a point e 2 K0 and let x 2 K0 be an arbitrary point. Since K ishomogeneous, there exists Ax 2 Aut(K) such that Axe = x. It follows from (11) thatDkF (x)[Axh;Axh; : : : ; Axh] = DkF (e)[h; h; : : : ; h]; k � 1: (15)Since ' satis�es (10), it su�ces to show (see [20]) that there exists a constant c > 0satisfying jD3F (x)[h0; h0; h0]j � c(D2F (x)[h0; h0])3=2; 8h0 2 Rn :It follows from (15) that it is su�cient to prove this inequality only at e, that is, it isenough to show thatjD3F (e)[h; h; h]j � c(D2F (e)[h; h])3=2; 8h 2 Rn :11

This is obvious, since D2F (e) is a symmetric positive de�nite matrix.We remark that the above proof reduces the calculation of the parameter # on thewhole set K0 to calculating it at a single point x 2 K0.In a number of papers in interior point methods, the expression log(detD2F (x)) ap-pears in the barrier function, for example in the volumetric barrier, see Nesterov andNemirovskii [20]. We close this section by showing that in the case where K is a homo-geneous cone, the characteristic function can be written using the same expression.Theorem 4.4 Let K � Rn be a homogeneous cone and �x a point e 2 K0. We have'(x) = '(e)qdetD2F (e)qdetD2F (x);F (x) = const+ 12 log(detD2F (x)):Proof. Since K is homogeneous, there exists Ax 2 Aut(K) satisfying Axe = x. Equation(11) gives ATxD2F (x)Ax = D2F (e). This implies (detAx)2 detD2F (x) = detD2F (e), and'(x) = '(e)j detAxj = '(e)qdetD2F (e)qdetD2F (x):This proves the �rst equality; the second one follows from the �rst.5 Duality MappingIn this section we de�ne the duality mapping and present its main properties. It will beuseful in determining barrier functions on dual cones. Let K � Rn be a cone. Considerthe characteristic function ' of K and its logarithm F (x) = log('(x)) both de�ned inK0. Now DF (x) is a linear functional on Rn , which in the standard inner product uTvon Rn , is identi�ed with the vector of the partial derivatives of F at x. If we endow Rnwith a new inner product hx; yi = xTSy where S is a symmetric, positive de�nite matrixS, then DF (x) can be written in the formDF (x)[u] = h�x�; ui; 8u 2 Rn :Thus, in this inner product, the linear functional �DF (x) is identi�ed with the vectorx� 2 Rn . 12

De�nition 5.1 The mapping x 7! x� is called the duality mapping.The basic properties of the duality mapping can be found in [19, 26, 8] for homogeneousself{dual cones and in [32] when K is a homogeneous cone. Many of these properties alsohold when K is an arbitrary cone.The following fundamental result can be found in Vinberg [32].Theorem 5.1 Let K � Rn be a cone. The duality mapping is an analytic bijectionbetween K0 and (K�)0. We have hx; x�i = n:In fact, x� is characterized by the conditionx� = argminf'(y) : y 2 K�; hx; yi = ng:Moreover, x� is the center of gravity of the cross section fy 2 K� : hx; yi = ng of K�.If A 2 Aut(K), then (Ax)� = �(A)x� = S�1(AT )�1Sx�: (16)In particular, if t > 0, then (tx)� = x�t :Proof. We give here only a sketch of the proof; a more detailed proof of the theoremcan be found in [32]. The proof of the claim that x� is the center of gravity of the abovecross section follows, since we haveh�x�; hi = DF (x)[h] = D'(x)[h]'(x) = � RK�hy; hie�hx;yidyRK� e�hx;yidy ;implying x� = RK� ye�hx;yidyRK� e�hx;yidy :Writing the integrals above in the form of iterated integrals as in Theorem 4.1 proves theresult. Also, equation (16) follows from the fact that h 2 Rn implieshx�; hi = �DF (x)[h] = �DF (Ax)[Ah] = h(Ax)�; Ahi = hA�(Ax)�; hi;where the second equality follows from (11). This implies A�(Ax)� = x� or (Ax)� =(A�)�1x� = �(A)x�. 13

It is known that the duality mapping has a unique �xed point x� = x which we denoteby e. This can be seen as follows. Consider the convex minimization problemminf12hx; xi : FK(x) � 0g;where the constraint set is nonempty since x 2 K0 and t !1 imply FK(tx) ! �1. Asolution to this problem exists and is unique, as the objective function is strictly convexand coercive. The constraint set can be shown to be bounded away from the origin, sothat the solution x satis�es the condition x = �(x)� for some � > 0. Then the pointx = x=p� satis�es x� = x.The point e plays an important role in the classi�cation of both homogeneous self{dualcones and the homogeneous cones. The point e can be called the \center" of the cone K.However, we note that e has signi�cance only with respect to the given inner product;changing the inner product will change the center. In fact, it can be shown that any pointof K0 of a homogeneous cone can be made into a center by choosing an appropriate innerproduct on Rn .The existence of e implies immediately the following result in Ochiai [22], which canalso be proved by an elementary separation argument.Corollary 5.1 Let K � Rn be a cone. Then K0 \ (K�)0 6= ;.When K is a homogeneous cone, the duality mapping has further useful properties.For example, the following important result can be found in Vinberg [32]. We include itseasy proof.Theorem 5.2 Let K � Rn be a homogeneous cone. Then'K(x)'K�(x�) = const;FK(x) + FK�(x�) = const:Proof. Let A 2 Aut(K). Equation (16) gives'K(Ax)'K�((Ax)�) = 'K(Ax)'K�((A�)�1x�) = 'K(x)'K�(x�)j detAj � j det(A�)�1j = 'K(x)'K�(x�):Since K is homogeneous, the theorem is proved.The following result is also well{known, see for example [19, 26, 32, 8]. We include itsproof for completeness. Our proof follows Rothaus [26].14

Theorem 5.3 Let K � Rn be a homogeneous cone. Then(x�)� = x:Proof. It is well known that the bilinear form D2F (x) can be represented by a self{adjoint linear mapping HK(x) : Rn ! Rn ,D2F (x)[u; v] = hHK(x)u; vi:Since hx�; ui = �DF (x)[u], HK(x) is the Jacobian of the mapping x 7! �x�. Di�erenti-ating the equation hx�; xi = n gives HK(x)x = x�:See also [20], equation 2.3.12, pp. 41. In addition, Theorem 5.2 gives�FK(x)� FK�(x�) = const:Since (x+ h)� = x��HK(x)h+ o(jhj) as jhj ! 0, di�erentiating the above equation withrespect to x gives x� �HK(x)(x�)� = 0:Thus, HK(x)x = HK(x)(x�)� = x�:Since HK(x) is non{singular, we have (x�)� = x.We mention some other useful results in the case where K is a homogeneous self{dualcone. The proofs can be found in Rothaus [26].Theorem 5.4 Let K � Rn be a homogeneous self{dual cone. The set of linear mapsfHK(x) : x 2 Kg forms a simply transitive subset of Aut(K), that is, given any twoa; b 2 K0, there exists a unique x 2 K0 such that HK(x)a = b. Moreover, any A 2 Aut(K)can be written uniquely as A = BHK(x)where B 2 Aut(K) leaves the point e �xed, that is, Be=e.We remark that, in the inner product ha; bi = aTSb, the linear mappings B and HK(x)are orthogonal and symmetric positive de�nite, respectively; i.e., we have BB� = I andhHK(x)a; bi = ha;HK(x)bi > 0 for all a; b 2 Rn . In other words, A = BHK(x) is thepolar decomposition of A.Primal{dual interior point methods need a barrier function on both K and K�, seeNesterov and Nemirovskii [20]. Thus it becomes important to calculate the barrier on15

K� e�ciently. For self{dual cones, there is clearly no problem, as we can take the samebarrier both on K and K�. Nesterov and Nemirovskii [20] (Theorem 2.4.4) show that the(slightly modi�ed) Fenchel dual of the barrier function of FK,(FK)�(y) = supfhx;�yi � FK(x) : x 2 K0g; (17)is a barrier forK� and has the same parameter # as FK. (In the ordinary Fenchel dual, onehas y instead of �y in (17).) We use the properties of the duality mapping to determinethe properties of (FK)�. Note that if y 2 (K�)0, then the maximization problem in (17)has a unique solution x satisfying y = x�. Since hx; x�i = n by Theorem 5.1, we obtain(FK)�(x�) = �n� FK(x): (18)Thus, in the case where K is a homogeneous cone we have the following important result.Theorem 5.5 Let K � Rn be a homogeneous cone. If y 2 (K�)0, then(FK)�(y) = const + FK�(y):Proof. Since K is homogeneous, Theorem 5.3 implies that there exists a unique x 2 Ksuch that y = x�. Then(FK)�(x�) = �n� FK(x) = const+ FK�(y);where the �rst equality follows from (18) and the second one from Theorem 5.2.We end this section with a geometric description of the dual barrier function (FK)�.As mentioned above, the optimal value is achieved in (17) at a point x 2 K0 such thatx� = y. Since hx; x�i = n, we can rewrite (17),(FK)�(y) = supfhx;�yi � FK(x) : x 2 K0; hx; yi = ng= �n� log inff'K(x) : x 2 K0; hx; yi = ng:We show in the proof of Theorem 4.1 that 'K(x) is proportional toV oln(fz 2 K� : hx; zi � ng);that is, the volume of the truncated cone after K� is cut by the hyperplane having normalx and passing through the point y 2 K�. LetK�x;y = fz 2 K� : hx; zi � hx; yig16

be such a cone. Combining the above results, we see that (FK)� is, up to an additiveconstant, equal to the function F+K� de�ned byF+K�(y) = � log inffV oln(K�x;y) : x 2 Kg:We summarize these results for the cone K as follows. For a proper cone K, we have,in addition to the universal barrier function FK, a second \universal barrier" function F+Kgiven by F+K (x) = � log �inffV oln(K�y;x) : y 2 K�g� ;where Ky;x is truncated cone after K is cut by the hyperplane with normal y and passingthrough the point x 2 K. The description of the \characteristic function" related to F+Kcan already be found in [31]. We show in Theorem 5.5 above that these two universalbarrier functions coincide if K is a homogeneous cone.6 Homogeneous Cones and Siegel DomainsHomogeneous cones form an attractive class among cones because of their invariance prop-erties. As we note in Section 2, Vinberg [32] is the �rst to give an algebraic classi�cationof these cones. Siegel domains described below play an essential role in the classi�cationand in the construction of homogeneous cones. The literature on these two topics is large,see for example [32, 33, 10, 11, 27, 4, 5, 6], etc. Here we will be content to describe onlythe basic elements of this theory and those aspects of it that we need in order to calculatethe universal barrier functions of homogeneous cones.De�nition 6.1 Let K be a cone in Rk . A K{bilinear form B(u; v) in Rp is a mappingfrom Rp � Rp to Rk satisfying the following properties1. B(�1u1 + �2u2; v) = �1B(u1; v) + �2B(u2; v) for �1; �2 2 R,2. B(u; v) = B(v; u),3. B(u; u) 2 K,4. B(u; u) = 0 implies u = 0.De�nition 6.2 Let B and K satisfy the conditions (1)-(4) in De�nition 6.1. The Siegeldomain corresponding to K and B is the setS(K;B) = f(x; u) 2 Rk � Rp : x� B(u; u) 2 Kg:17

De�nition 6.3 A K{bilinear form B is called homogeneous if K is a homogeneous coneand there exists a transitive subgroup G � Aut(K) such that for every g 2 G, there existsa linear transformation of g of Rp such thatg B(u; v) = B(gu; gv);that is, the following diagram Rp � Rp g�g�! Rp � RpB???y ???yBRk g�! Rkcommutes.The Siegel domain S(K;B) corresponding to a homogeneous K{bilinear form B isa�ne homogeneous. This can be seen by checking that the following a�ne transformationsform a transitive subgroup of S(K;B):A1(x; u) = (x+ 2B(u; a) +B(a; a); u+ a); a 2 Rp ;A2(x; u) = (gx; gu); g 2 G � Aut(K):The following remarkable theorem is due to Vinberg, see [32, 10].Theorem 6.1 Any a�ne{homogeneous domain D � Rn is a�ne equivalent to a Siegeldomain.The cone �tted to a Siegel domain S(K;B) is given bySC(K;B) = f(x; u; t) 2 Rk � Rp � R : t � 0; tx�B(u; u) 2 Kg:We remark that a more general construction than SC(K;B) can be found in Nes-terov and Nemirovskii [20], Example 5, pp. 165, except that the homogeneity conditiondescribed in De�nition 6.3 is not considered there.Lemma 6.1 If K is a homogeneous cone and B is a homogeneous K{bilinear form, thenthe cone SC(K;B) is homogeneous, and the following linear mappings form a transitivesubgroup of Aut(SC(K;B)),T1(x; u; t) = (x;p�u; �t); � > 0;T2(x; u; t) = (x + 2B(u; a) + tB(a; a); u+ ta; t); a 2 Rp ;T3(x; u; t) = (gx; gu; t); g 2 G � Aut(K);where G is a transitive subgroup of Aut(K).18

Proof. It is easy to show, using De�nitions 6.1{6.3, that each map Ti 2 Aut(SC(K;B)).Let (x0; 0; 1) 2 SC(K;B)0 be a �xed point, where x0 2 K0. Let (x�; u�; t�) 2 SC(K;B)0be an arbitrary point, where x� 2 K0, t� > 0, and u� 2 Rp . Consider the linear mapT = T3T2T1, Ti 2 Aut(SC(K;B)),T1(x; u; t) = (x; upt� ; tt� );T2(x; u; t) = (x� 2pt�B(u; u�) + tt�B(u�; u�); u� tpt�u�; t)T3(x; u; t) = (gx; gu; t);where g 2 G satis�es g(x� �B(u�; u�)=t�) = x0. We note thatT1(x�; u�; t�) = (x�; u�pt� ; 1);T2(x�; u�pt� ; 1) = (x� � B(u�; u�)t� ; 0; 1);T3(x� � B(u�; u�)t� ; 0; 1) = (x0; 0; 1):This shows that T (x�; u�; t�) = (x0; 0; 1).The above lemma demonstrates that a homogeneous cone K gives rise to a homoge-neous cone SC(K;B) in a higher dimensional space. The converse is also true. That is,given a homogeneous cone K, there exists a lower dimensional cone K and a homoge-neous K{bilinear form B such that K is linearly equivalent to SC(K;B), see for exampleGindikin [11], pp. 75. Consequently, it is possible to recursively construct an arbitraryhomogeneous cone out of lower dimensional homogeneous cones, starting from the realhalf{line fx 2 R : x � 0g. This is a generalization of the familiar construction of the(n+1)�(n+1) symmetric positive semi{de�nite matrices from the n�n symmetric posi-tive semi{de�nite matrices, see Section 7.3. This construction process yields the algebraicclassi�cation of homogeneous cones. The number of recursive steps necessary to build upa homogeneous cone is invariant, and is called the rank of the cone, see Vinberg [32].We end this section by giving a recursive formula for the characteristic function andthe universal barrier function of a homogeneous cone.Corollary 6.1 Let K be a homogeneous cone and B a homogeneous K{bilinear form.The characteristic function and the universal barrier function of the cone SC(K;B) aregiven by '(x; u; t) = constt p2+1 � 'K(x� B(u; u)t ) � det g;F (x; u; t) = const + FK(x� B(u; u)t ) + log(det g)� (p2 + 1) log t:19

Proof. It follows from (12) that'(x; u; t) = '(x0; 0; 1) � detT;where x0 2 K0 is a �xed point, and T 2 Aut(SC(K;B)) satis�es T (x; u; t) = (x0; 0; 1). Itis thus su�cient to calculate the determinants of the linear maps at the end of the proofof the above lemma. It is easy to see that det(T1) = 1=t p2+1, det(T2) = 1, anddet(T3) = det g � det g = const � 'K(x�B(u; u)=t) � det g;where the last equality follows from'K(x� B(u; u)=t) = 'K(g�1x0) = 'K(x0)det g�1 = const � det g:The corollary is proved.7 Characteristic Function of Some ConesIn this section we calculate the characteristic function ' of some cones and the corre-sponding barrier function F . We demonstrate that, although the universal barrier func-tion is usually very hard to calculate, it can be calculated in some important cases. Itis known that the universal barrier function does not always have the optimal parameter#, see for example Sections 7.2, 7.3, and 7.7. However, the universal barrier functions inthese sections can be scaled to either agree with the optimal barrier functions or to havecomparable parameter #. As we mentioned in the previous section, the calculation of theuniversal barrier functions in these sections bears a strong resemblance to the calculationscarried out in Chapter 5 of [20]. (However, homogeneity is not considered in [20].) Usingtheir classi�cation theory, we determine in [13] the optimal parameter # for homogeneouscones. It is not known at present whether the universal barrier function of an arbitraryirreducible homogeneous cone can be scaled so that it has parameter # comparable to theoptimal one.The barrier in Section 7.1 is the familiar logarithmic function. The barrier in Section7.5 can be obtained by the methods in [20] and has # = n. We show here that it is theuniversal barrier function for the underlying cone. The barrier calculated in Section 7.4seems to be new, and has # = O(n). However, it seems useless for interior point methods,since it would take e�ort exponential in n to calculate it and its derivatives. We notethat, since the cones in Sections 7.4 and 7.5 are dual, the Fenchel dual of the universalbarrier for one cone is the second \universal" barrier function for the other cone. We donot calculate explicit barriers in Section 7.6, but some barriers can be calculated using theformula in Lemma 7.5. In Section 7.7 we obtain the universal barrier function of some20

cones related to matrix norms given in [20], Section 5.4.6. The method used here can,in principle, be applied to calculate the universal function of an arbitrary homogeneouscone (using the classi�cation of these cones). Finally, in Section 7.8, we show that thecalculation of the universal barrier function of a polyhedral cone reduces, in theory, to thetriangulation of the dual polyhedral cone. This shows, in particular, that the universalfunction of such a cone is the logarithm of a rational function. It also shows that it wouldbe hard in general to calculate the universal barrier function of polyhedral cones.7.1 The Non{Negative OrthantThe non{negative orthant Rn+ = R+ � : : :�R+ is the direct sum of n copies of R+ . Thus,'(x1; : : : ; xn) = nYi=1'R+(xi):Since 'R+(xi) = R10 e�xiyidyi = 1=xi, we have'(x1; : : : ; xn) = 1Qni=1 xi ; F (x) = � nXi=1 log xi:F (x) is the familiar self{concordant barrier function for Rn+.7.2 The Lorentz ConeThis is the cone Kn = fx 2 Rn : qx21 + x22 + : : : x2n�1 � xng:It is variously known as the spherical cone, light cone, ice cream cone, etc. The coneK4 plays a prominent role in special relativity. Note that Kn+1 is the cone �tted to theunit ball Bn = fx 2 Rn : jjxjj � 1g. If we endow Rn with the usual inner product,then this cone is self{dual. This can be inferred from Section 15 of Rockafellar [25]. Weinclude a short proof. If the point (y; �) 2 Rn+1 is in K�n+1, then hx; yi + t� � 0 forall (x; t) satisfying jjxjj � t. This implies hu; yi + � � 0 for all u such that jjujj � 1.Thus, supjjujj�1hu; yi � � and jjyjj � � . Since the implications can be reversed, we haveK�n+1 = Kn+1.Consider the coneSC(R+ ; Bn) = f(x; u; t) 2 R � Rn � R : x � 0; t � 0; tx� juj2 � 0g;where Bn(u; v) = uTv can be easily shown to be a homogeneous R+{bilinear form. After arotation of the variables (t; x) the term tx becomes t2�x2, so that SC(R+ ; Bn) is linearly21

isomorphic to Kn+2. Since the former cone is homogeneous by Lemma 6.1, the cone Sk ishomogeneous for all k � 3. The cone S1 = R+ is obviously homogeneous, and it is easyto show that S2 is linearly isomorphic to R2+ which is homogeneous. Thus all cones Sk,k � 1, are homogeneous.We now calculate the characteristic function of the cone SC(R+ ; Bn) using the Siegeldomain construction in Section 6. De�ne T� 2 Aut(R+), where � > 0 and T�x = �x.The corresponding linear transformation T � on Rn is T �u = p�u. By Corollary 6.1F (x; u; t) = const� log(x� juj2t ) + log(det T �)� (n2 + 1) log t;where �(x� juj2=t) = 1. This gives det T� = (x� juj2=t)�n=2, andF (x; u; t) = const� n+ 22 log(tx� juj2):After of a rotation of the variables (t; x), we obtain the following lemma.Lemma 7.1 The characteristic function of the Lorentz cone Kn+1 and the correspondingbarrier function are given by'(x; t) = const(t2 � jjxjj2)(n+1)=2 ;F (x; t) = const� n+ 12 log(t2 � jjxjj2):The barrier function F has parameter # = n + 1 which is much worse than theparameter # = 2 of the optimal barrier function G(x; t) = � log(t2 � jjxjj2). However,note that G = (2=(n+1))F up to a constant, so that the optimal barrier function can beobtained by scaling the universal barrier function.7.3 Symmetric Positive Semi{De�nite MatricesConsider the vector space �n of n�n symmetric matrices endowed with the inner producthx; yi = tr(xy):This is the same as the inner product on Rn(n+1)=2 obtained as follows. Let ~x; ~y be thevectors obtained by putting in some order the diagonal and strict upper diagonal elementsof x and y into vectors in Rn(n+1)=2 , respectively. Thenhx; yi = ~xTD~y;22

where D is a diagonal matrix with Dii = 1 and Dij = 2 for 1 � i < j � n.It is easy to see that the set of positive de�nite matrices form a cone in �n which wedenote by �+n . It is well known that �+n is a self{dual cone, that is, (�+n )� = �+n . Thiscan be shown as follows. First, let x; y 2 �+n , and let x1=2 2 �+n be the square root of x.Then tr(xy) = tr(x1=2x1=2y) = tr(x1=2yx1=2) � 0;where the inequality follows as x1=2yx1=2 2 �+n . This shows �+n � (�+n )�. Conversely, lety 2 (�+n )�. Then tr(xy) � 0 for all x 2 �+n . If u 2 Rn , then uTyu = tr(yuuT ) = hy; uuT i �0. Thus, y 2 �+n , and consequently (�+n )� = �+n . The interior (�+n )0 corresponds to theset of symmetric positive de�nite matrices.The cone �+n+1 can be realized as a Siegel domain cone over �+n . In fact, we have�+n+1 = SC(�+n ; Bn);where the bilinear form Bn : Rn � Rn ! �n is given by B(u; v) = (uvT + vuT )=2. Ifg 2 GL(n;R), then the linear map Tg : �n ! �n given by Tgx = gxgT is evidently anautomorphism of the cone �+n . The corresponding linear map T g : Rn ! Rn is given byT gu = gu and has determinant det g. Thus, Bn is a homogeneous �+n {bilinear form. Also,it is well known that a symmetric (n+ 1)� (n+ 1) matrixx = 0@ t uuT x 1A ;where t 2 R, u 2 Rn , and x 2 �n is positive semide�nite if and only if t � 0, x 2 �+n , andtx� uuT 2 �+n . The above claim follows easily from these.We now calculate the universal barrier function of the cones �n. Using Corollary 6.1and the fact g(x � uuT=t)gT = I, or g = (x � uuT=t)�1=2 implies det T g = det(x �uuT=t)�1=2, we obtainF�n+1(x) = const+ F�n(x� uuTt )� 12 log det(tx� uuT )� n2 log t:Since det x = det(tx� uuT ), we can easily prove the following result by induction.Lemma 7.2 The universal barrier for the cone of symmetric positive semi{de�nite ma-trices is the function F (x) = const� n + 12 log(det x):23

Note that the cone is one of the �ve irreducible homogeneous self{dual cones listed inSection 2. The universal barrier function of the cone of positive semi de�nite complexmatrices and the cone of positive semi de�nite quaternion matrices can be calculatedsimilarly.The barrier function F has parameter # = n(n + 1)=2 which is much worse thanthe parameter # = n of the optimal barrier function G(x) = � log det x. Since G =(2=(n + 1))F up to a constant, the optimal barrier function can again be obtained fromthe universal barrier by scaling.Examples of convex programming problems which involve the cone of symmetric pos-itive de�nite matrices can be found in Nesterov and Nemirovskii [20], Alizadeh's Ph.D.thesis [1], etc. Some of these problems naturally occur in matrix analysis, combinatorialoptimization, and control theory.7.4 The l1 Unit BallHere we calculate the characteristic function of the convex set Q = fx 2 Rn : jjxjj1 � 1g.The �tted cone is K(Q) = f(x; t) : jjxjj1 � tg. It is easy to show that the dual cone isgiven by K(Q)� = f(y; �) : jjyjj1 � �g, see Rockafellar [25]. We calculate'(x; t) = Zjjyjj1�� e�t�e�hx;yidyd� = Z 10 e�t� (Zjjyjj1�� e�hx;yidy)d�= Z 10 e�t�� nYi=1 Z ��� e�xiyidyi� = Z 10 e�t� nYi=1 e� jxij � e�� jxijjxij d�:It is easy to verify thatnYi=1(e� jxij � e�� jxij) = X"i=�1( nYi=1 "i)ePni=1 �"ijxij:Thus, we have '(x; t) = 1Qni=1 jxij X"i=�1( nYi=1 "i) Z 10 e��(t�Pni=1 "ijxij)d�= 1Qni=1 jxij X"i=�1 Qni=1 "it�Pni=1 "ijxij ;the barrier function isF (x; t) = log( X"i=�1 Qni=1 "it�Pni=1 "ijxij)� nXi=1 log(jxij);24

and the induced barrier on Q isF (x) = F (x; 1) = log( X"i=�1 Qni=1 "i1�Pni=1 "ijxij)� nXi=1 log(jxij):The barrier function F has parameter # = O(n). Since l1 cone in Section 7.3 is dualto the l1 cone here, and the optimal barrier function for the former cone is at least n byProposition 2.3.6 in [20], we see that F has parameter of optimal order. However, F ispractically useless for interior point calculations for large n, since the e�ort to calculateit and its derivatives is exponential in n.For n = 2, the barrier of Q isF (x) = const� log(1� (jx1j+ jx2j)2)� log(1� (jx1j � jx2j)2):7.5 The l1 Unit BallHere we calculate the characteristic function of the convex set Q = fx 2 Rn : jjxjj1 � 1g.The �tted cone is K(Q) = f(x; t) : jjxjj1 � tg. The dual cone is K(Q)� = f(y; �) :jjyjj1 � �g.Lemma 7.3 The characteristic function of the unit l1 ball in Rn is'n(x; t) = 2ntn�1Qn1 (t2 � x2i ) ;Fn(x; t) = const� nX1 log(t2 � x2i ) + (n� 1) log t:Proof. We prove the lemma by induction. For n = 1, it is a routine task to verify that'1(x; t) = 2=(t2�x2). Suppose that the lemma holds true for n; we will prove it for n+1.We denote x = (x1; : : : ; xn; xn+1) = (x0; xn+1). Similarly, we write y = (y0; yn+1). We have'n+1(x; t) = Zjjyjj1�� e�t�e�hx;yidyd�= Z 1�1 e�xn+1yn+1(Zf(y0;�):jjy0jj1���jyn+1jg e�t�e�hx0;y0idy0d�)dyn+1= Z 1�1 e�xn+1yn+1e�tjyn+1j(Zf(y0;�):jjy0jj1���jyn+1jg e�t(��jyn+1j)e�hx0;y0idy0d�)dyn+1= Z 1�1 e�xn+1yn+1e�tjyn+1j(Zf(y0;� 0):jjy0jj1�� 0g e�t� 0e�hx0;y0idy0d� 0)dyn+125

= Z 1�1 e�xn+1yn+1e�tjyn+1j� 2ntn�1Qni=1(t2 � x2i )�dyn+1= 2ntn�1Qni=1(t2 � x2i )�Z 0�1 e�(xn+1�t)yn+1dyn+1 + Z 10 e�(xn+1+t)yn+1dyn+1�= 2ntn�1Qni=1(t2 � x2i )( 1t� xn+1 + 1t+ xn+1 ):Since 1=(t� xn+1) + 1=(t+ xn+1) = 2t=(t2 � x2n+1), we have'n+1(x; t) = 2n+1tnQn+11 (t2 � x2i ) :This proves the lemma.The barrier function F has parameter # = n which is optimal, see [20], Proposition2.3.6.7.6 Epigraph of Convex FunctionsThe following result is essentially contained in Rockafellar [25], Theorem 14.4, pp. 124.Lemma 7.4 Let f : Rn ! R[f1g be a proper closed convex function. Let Q = f(x; �) :f(x) � �g be the epigraph of f and let K(Q) = ft(x; �; 1) : f(x) � �; t � 0g denote thecone �tted to Q. We haveK(Q)� = cl(f�(u; 1; ) : � > 0; f �(�u) � g):Lemma 7.5 Suppose f : Rn ! R [ f1g, Q, and K(Q) are as in the above lemma. If(x; �; t) 2 K(Q)0, we have'K(Q)(x; �; t) = n!tn+1 ZD(f�) du[�hx=t; ui+ �=t+ f �(u)]n :Proof. It is su�cient to prove the lemma for t = 1. We have'(x; �; 1) = ZK(Q)� eh(x;�;1);(y;�;�)idyd�d�:26

We use the description of K(O)� in the above lemma and change the variables of inte-gration from (y; �; �) to (u; �; ), where y = ��u, and � = � . Notice that the Jacobian@(y; �; �)=@(u; �; ) = �n+1. Thus, de�ning G = f(u; �; �) : � � 0; f �(u) � g, we have'K(Q)(x; �; 1) = ZG �n+1e��(hx;�ui+�+�)d�d�du= Zf(u;�):�>0;u2D(f�)g �n+1e��(�hx;ui+�)(Z 1f�(u) e���d�)d�du= Zf(u;�):�>0;u2D(f�)g �n+1e��(�hx;ui+�) e��f�(u)� d�du= ZD(f�)(Z 10 �ne��(�hx;ui+�+f�(u))d�)du= n! ZD(f�) du[�hx; ui+ � + f �(u)]n :The lemma is proved.7.7 Epigraph of Matrix NormsWe consider the vector space �m of symmetric m � m matrices, the cone K = �+mof symmetric p.s.d. matrices in �m, and the vector space Rn�m of n � m matrices,where m � n. We endow �m with the inner product hx; yi = tr(xy) and Rn�m withthe inner product hu; vi = tr(uTv). Note that Bn;m : Rn�m � Rn�m ! �n given byBn;m(u; v) = (uTv + vTu)=2 is a �+m{bilinear form. The subgroup G = fTg : g 2 (�+m)0g,where Tg is de�ned in Section 7.3, is obviously a transitive subgroup of Aut(�+m). De�ningT g : Rn�m ! Rn�m such that T gu = ugwe see that Tg(Bn;m(u; v)) = Bn;m(T gu; T gv);that is, Bn;m is a homogeneous �+m{bilinear form. Thus, the Siegel DomainS(�+m; Bn;m) = f(x; u) 2 �+m � Rn�m : x� uTu 2 �+mgis an a�ne homogeneous convex set, and the Siegel coneSC(�+m; Bn;m) = f(x; u; t) 2 �+m � Rn�m � R : tx� uTu 2 �+m; t � 0gis homogeneous.We now calculate the characteristic function of the cone SC(K;B) = SC(�+m; Bn;m).It is easy to verify that detT g = (det g)n. IfTg(x� uTut ) = g(x� uTut )gT = I;27

then g = (x � uT ut )�1=2 and detT g = det(x � uT ut )�n=2. This and Corollary 6.1 give thefollowing result.Lemma 7.6 The characteristic function and the universal barrier function of the coneSC(�+m; Bn;m) are given by'(x; u; t) = t�mn2 �1(det(x� uTut ))�m+n+12 ;F (x; u; t) = const� m + n+ 12 log(det(x� uTut ))� (mn2 + 1) log t:The cone SC(K;B) 2 Rl where l = 1 +mn +m(m + 1)=2. Hence by Theorem 2.5.1in Nesterov and Nemirovskii [20], F a self{concordant barrier for SC(K;B) 2 Rl withparameter # = O(l). This parameter is much larger than the parameter # = m + 1 forthe following barrier function for the same cone given in [20], pp. 200,H(x; u; t) = � log(det(x� uTut ))� log t:The barrier function H can be shown to be optimal, see [13]. However, if we multiply Fwith 2=(m+ n + 1), we obtain a scaled barrier functionG = � log(det(x� uTut ))� mn + 2m+ n + 1 log t:Now, the function G is obtained by applying Proposition 5.1.8 in [20] to the barrierfunction G1(x) = � log det x for the cone �+m and the barrier function G2(t) = �� log tfor the non{negative real line R+ with � = (mn+2)=(m+n+1). Since � � 1, it is easy toverify that G2 is a self{concordant barrier function for R+ with parameter # = �. It thenfollows from Proposition 5.1.8 in [20] that G is a self{concordant barrier for SC(K;B)with parameter # = m + � = m+ mn + 2m + n+ 1 � 2m:Thus, we see that the barrier function G and the optimal barrier function H have com-parable parameters #, although H has a slightly better parameter. It a routine matter tocalculate the dual barrier function G�, since we know from Theorem 5.5 that it coincideswith a multiple of the universal barrier function of the dual cone SC(K;B)�. We do notcalculate G� here as it would take us far a�eld.We end this subsection by describing the dual cone SC(�m; Bn;m). Here we endow thevector space �m � Rn�m � R with the inner producth(x; u; t); (y; v; s)i = tr(xy) + tr(uTv) + ts:The method can be extended to calculate the dual of any Siegel cone SC(K;B), and infact to give a recursive \dual" procedure to build up any homogeneous cone, see [27].28

Lemma 7.7 We haveSC(�+m; Bn;m)� = f(y; v; s) 2 �+m � Rn�m � R : s � 14tr(vy�1vT )g:Proof. A point (y; v; s) is in the dual cone if and only if ht(x; u; 1); (y; v; s)i � 0 for allx; u such that x� uTu 2 �+m. This is equivalent to the requirement thattr(xy) + tr(uTv) + s � 0;for all x; u satisfying x � uTu 2 �+m. We must have y 2 �+m, since otherwise there exists� 2 Rm such that �Ty� < 0. Choosing u = 0, and x(t) = t2��T , we obtainlimt!1 tr(x(t)Ty) + s = limt!1 t2�Ty�+ s = �1;a contradiction. Since tr(ab) � 0 for all a; b 2 �+n , we can rewrite the above inequality astr(uTuy) + tr(uTv) + s � 0; 8u:De�ning u = uy1=2, we can, in turn, write this last inequality as0 � tr(uTu) + tr(uTvy�1=2) + s= tr((u+ vy�1=22 )T (u+ vy�1=22 ))� 14tr((vy�1=2)Tvy�1=2) + sfor all u. The lemma follows immediately.7.8 Polyhedral ConesConsider a polyhedral cone K = m\1 fx 2 Rn : hai; xi � 0g;where ai 2 Rn are given vectors. In this section, we give a formula for the characteristicfunction and the universal barrier function of K. Here we follow some ideas in Barvi-nok [2].Note that the dual coneK� = conv(a1; : : : ; am) = f mX1 �iai : �i � 0g (19)29

is the convex conical hull of the vectors faigm1 . Suppose that we decompose K�, that is,we write K� =[K�i ;where each Ki = conv(d1; : : : ; dn) is a simplicial cone with linearly independent vectorsfdign1 , and the intersection Ki1 \Ki2 of two di�erent cones has less than full dimension.We have Ki = D(Rn+) where D is the matrix with columns dj. Thus, we can explicitlycalculate the integral (7) over the cone K�i ,ZK�i e�hx;yidy = ZD(Rn+) e�hx;yidy = ZRn+ e�hDT x;uij detDjdu = j detDjQn1 hdk; xi :Consequently, we have 'K(x) = X ciQn1 hdk;i; xi ;FK(x) = log X ciQn1 hdk;i; xi! ;where each summation is taken over the di�erent simplicial cones in the decompositionof K�.By Theorem 2.5.1 in [20], FK is an O(n) self{concordant barrier function for K. Wenote that we can calculate the barrier functions in Sections 7.4 and 7.5 using this approach.It might also be possible to calculate the universal barrier function of some other specialcones in this way. At present, the above formula for FK should probably be considereda theoretical result, since it is not clear how one would decompose an arbitrary cone in(19) in an e�cient manner.8 Concluding RemarksIn this paper, we have shown that the logarithm of the characteristic function is essentiallyequivalent to the universal barrier function of Nesterov and Nemirovskii. Our resultsconnect the �eld of interior point methods with several branches of mathematics, such asLie groups, di�erential geometry, symmetric spaces, complex variables, Jordan algebras,etc. It is hoped that the results obtained in the sizable literature in the mentioned �eldswill have applications to interior point methods.In [13], we make use of the classi�cation of homogeneous cones in [32, 11], etc. todetermine the best self{concordance parameter for such a cone. In particular, we showthat the rank, Carath�eodory number, and self{concordance parameter of a homogeneouscone are all equal. 30

The use of the automorphism group brings our approach closer to the one envisionedby Karmarkar. For example, Karmarkar [17] uses the automorphism group to bring anarbitrary point to the \center". We have shown for the �rst time that the automorphismgroup helps a great deal in calculating the universal barrier function of certain cones,such as a homogeneous cone K. In fact, using the classi�cation of homogeneous cones,one can explicitly calculate the universal barrier function of any such cone K. Moreover,this classi�cation theory and the results of [13] helps us to calculate explicitly an optimalself{concordant barrier for K.In the examples of irreducible homogeneous cones given in Sections 7.2, 7.3, and 7.7,we see that the optimal barrier function and a suitably scaled universal barrier functionare either equal or have comparable barrier parameter. It is not known at present whetherthis is true for an arbitrary irreducible homogeneous cone K. (The corresponding resultfor reducible homogeneous cones K = K1 � K2 � : : : � Kk is easily shown to be false.However, in this case one would scale the universal barrier functions of the irreduciblecones Ki individually.) Even if this is true, it is not clear whether the (scaled) universalbarrier function F for K has any advantages over the optimal barrier function G for thesame cone K, especially since both functions have the invariance propertiesF (A1x) = F (x) + const1;G(A2x) = G(x) + const2;where A1 2 Aut(K) and A2 is in a transitive subset of Aut(K). Here the �rst equalityfollows from (9), and in the second equality the transitive subset can be taken to be thecollection of the operators T1, T2, and T3, in Lemma 6.1, see also [13].After this paper was written, Nesterov and Todd [21] have obtained some long{stepinterior point methods on what they call self{scaled cones. These algorithms are stronglydependent on the special properties of the \self{scaled" cones and the special barriers onthem. It is seen that these cones coincide with the homogeneous self{dual cones. Usingthe results of this paper, we see that the class of self{scaled cones are made up of directproducts of the �ve irreducible homogeneous self{dual cones listed in Section 2.In this paper, we address only the most basic results. Some important issues arenot addressed, such as how to obtain various interior point methods using these barrierfunctions. Many of these issues are discussed in Nesterov and Nemirovskii [20], but itmay be possible to improve on their results. As mentioned above, using \self{scaled"barrier functions, some long step interior point algorithms have been obtained in [21]for homogeneous self{dual cones. In a forthcoming paper [12], we propose some specialbarrier functions and extend some of the algorithms in [21] to more general cones.We also have not addressed the di�erential geometric issues involving the Riemanniangeometry de�ned on K0 by the bilinear form D2F (x). A sizable literature exists on these31

issues when K is homogeneous or homogeneous self{dual. For example, it is known thatif K is homogeneous, then the Riemannian space is symmetric if and only if K is self{dual. It is also known that if K is a homogeneous self{dual cone, then the Riemanniancurvature is everywhere non{positive, see for example [26]. The structure of the geodesicshave also been studied, for example in [19, 26], etc. It seems reasonable that these shouldhave a bearing on analyzing the behavior of interior point methods.Acknowledgements. The author thanks Yurii Nesterov, Michael Todd, and Yinyu Yefor useful discussions during the preparation of the paper. The author also thanks FaridAlizadeh, Alexander Barvinok, Joseph Dorfmeister, Arkadii Nemirovskii, James Renegar,and Levent Tun�cel for helpful comments and suggestions on the �rst version of the paper.The research of this paper is partially supported by the National Science Foundationunder grant DMS{9306318.References[1] Alizadeh, F. (1991) Combinatorial Optimization with Interior Point Methods and Semi{De�nite Matrices. Ph.D. Thesis, Univ. of Minnesota.[2] Barvinok, A. B. (1992) Partition functions in optimization and computational problems.St. Petersburg Math. Jour. 4 1{49.[3] Cartan, E. (1935) Domaines born�es homog�enes de l'espace de n variables complexes. Abh.Math. Sem. Univ. Hamburg 11 116{162.[4] Dorfmeister, J. (1979) Inductive construction of homogeneous cones. Trans. Amer. Math.Soc. 252 321{349.[5] Dorfmeister, J. (1979) Algebraic description of homogeneous cones. Trans. Amer. Math.Soc. 255 61{89.[6] Dorfmeister, J. (1982) Homogeneous Siegel domains. Nagoya Math. Jour. 86 39{83.[7] Evans, L. C. and R. F. Gariepy (1992) Measure Theory and Fine Properties of Functions.CRC Press, Boca Raton, Florida.[8] Faraut, J. and A. Koranyi (1994) Analysis on Symmetric Cones. Oxford Univ. Press, NewYork.[9] Federer, H. (1969) Geometric Measure Theory. Springer{Verlag, New York, Berlin.[10] Gindikin, S. G. (1964) Analysis in homogeneous domains. Russian Math. Surveys 19 1{89.[11] Gindikin, S. G. (1992) Tube Domains and the Cauchy Problem Transl. Math. Monog., Vol.111, Amer. Math. Soc., Providence, Rhode Island.32

[12] G�uler, O. (1995) Hyperbolic polynomials and interior{point methods for convex program-ming. In preparation.[13] G�uler, O. and L. Tun�cel (1995) Characterizations of the barrier parameter of homogeneousconvex cones. Research Report CORR 95{14, Department of Combinatorics and Optimiza-tion, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada.[14] Hertneck, C. (1962) Positivit�atbereiche und Jordan{Strukturen. Math. Ann. 146 433{455.[15] Hua, L. K. (1963) Harmonic Analysis of Functions of Several Complex Variables in theClassical Domains. Transl. Math. Monog., Amer. Math. Soc., Providence, Rhode Island.[16] Jordan, P., J. von Neumann, and E. Wigner (1934) On an algebraic generalization of thequantum mechanical formalism. Ann. Math. 35 29{64.[17] Karmarkar, N. (1984). A new polynomial{time algorithm for linear programming. Combi-natorica 4 373{395.[18] Koecher, M. (1957) Positivit�atbereiche im Rn . Amer. Jour. Math. 79 575{596.[19] Koecher, M. (1962) Jordan Algebras and their Applications. Lecture Notes, Department ofMathematics, University of Minnesota, Minneapolis, Minnesota.[20] Nesterov, Yu. E. and A. S. Nemirovskii (1993) Interior Point Polynomial Methods in ConvexProgramming. SIAM Publications, Philadelphia, Pennsylvania.[21] Nesterov, Yu. E. and M. J. Todd (1994), revised May 1995. Self{scaled barriers and interior{point methods for convex programming. Technical Report 1091, Cornell University, Ithaca,New York.[22] Ochiai, T. (1966) A lemma on open convex cones. Jour. Fac. Sci. Univ. Tokyo 12 231{234.[23] Oda, T. (1985) Convex Bodies and Algebraic Geometry. Springer{Verlag, Berlin, New York.[24] Pyatetskii{Shapiro, I. I. (1969) Geometry of Classical Domains and the Theory of Auto-morphic Functions. Gordon and Breach, New York.[25] Rockefellar, R. T. (1970) Convex Analysis. Princeton Univ. Press, Princeton, New Jersey.[26] Rothaus, O. S. (1960) Domains of Positivity. Abh. Math. Sem. Univ. Hamburg 24 189{235.[27] Rothaus, O. S. (1966) The construction of homogeneous convex cones. Ann. Math. 83358{376.[28] Satake, I. (1980) Algebraic Structures of Symmetric Domains. Princeton Univ. Press,Princeton, New Jersey.[29] Schneider, R. (1993) Convex Bodies: The Brunn{Minkowski Theory. Cambridge Univ.Press, Cambridge, United Kingdom. 33

[30] Stein, E. M. and G. Weiss (1971) Introduction to Fourier Analysis on Euclidean Spaces.Princeton Univ. Press, Princeton, New Jersey.[31] Venkov B. A. and B. B. Venkov (1962) On the automorphism of a convex cone. (In Russian,English summary.) Vestnik Leningrad. Univ. Ser. Mat. Mek. Astronom. 17 42{57.[32] Vinberg, �E. B. (1963) The theory of homogeneous convex cones. Trans. Moscow Math. Soc.12 340{403.[33] Vinberg, �E. B. (1965) The structure of the groups of automorphisms of a homogeneousconvex cone. Trans. Moscow Math. Soc. 13 63{93.[34] Vinberg, �E. B. (1960) Homogeneous cones. Soviet Math. Doklady 1 787{790.

34