Barrier Functions in Interior Point Methods - Semantic Scholar

Barrier Functions in Interior Point Methods Osman Guler Technical Report 94{01, March 1994 (Revised May 1995) Abstract We show that the universal barrier function of a convex cone introduced by Nesterov and Nemirovskii is the logarithm of the characteristic function of the cone. This interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone. This provides a simple method for calculating the universal barrier for homogeneous cones. We identify some known barriers as the universal barrier scaled by an appropriate constant. We also calculate some new universal barrier functions. Our results connect the eld of interior point methods to several branches of mathematics such as Lie groups, Jordan algebras, Siegel domains, dierential 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, homogeneous cones, 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: [email protected]). Research partially supported by the National Science Foundation under grant DMS{9306318. 

1

1 Introduction Since Karmarkar [17] introduced his polynomial{time projective algorithm for linear programming, the eld of interior point methods has been developing at a rapid rate. There are at present close to 1500 papers written in the eld. Most of these papers deal with the important problems of linear programming, convex quadratic programming, and monotone linear complementarity. At the same time, some researchers, especially Nesterov and Nemirovskii have successfully developed a general theory of interior point methods for nonlinear convex programming problems and monotone variational inequalities. The details 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 and the Newton method. This paper is concerned with the construction of barrier functions. We now recall some relevant concepts. Let Q  R n be an open convex set. A function F : Q ! R is called an {self{concordant function if F is at least three times dierentiable, convex, and satis es the property jD3F (x)[h; h; h]j  p2 (D2F (x)[h; h])3=2 ; (1) where Dk F (x)[h; : : : ; h] is the kth directional of F at x along the direction h 2 R n . The function 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 is responsible for the polynomial{time convergence of the interior point methods. They also show that, in solving constrained convex programming problems, a key role is played by self{concordant barrier functions which are 1{self{concordant and satisfy the additional property jDF (x)[h]j2  #D2F (x)[h; h]: (2) The constant # is called the parameter of the barrier function, and determines the speed of the underlying interior point method. We also recall the relevant barrier function concepts for a pointed convex cone K with non{empty interior, that is, a convex cone containing no lines and having a non{empty interior. (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 barrier function for K and satis es the property

F (tx) = F (x) ? # log t; 2

(3)

that is, the function '(x) = eF (x) is ?# homogeneous:

'(tx) = 't(#x) :

The function F is called a #{normal barrier for K if it is #{logarithmically homogeneous and 1{self{concordant. Nesterov and Nemirovskii [20], Proposition 5.1.4, show that any self{concordant{barrier function on a convex set with non{empty interior can be extended to a logarithmically homogeneous self{concordant barrier function on the cone tted to Q (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 homogeneous self{concordant barrier functions for convex cones. Nesterov and Nemirovskii show that any open convex set Q admits a universal barrier function which is also logarithmically homogeneous if Q is a pointed convex cone. They describe the universal barrier function in terms of the volume of the polar set, see Section 4. One of the main contributions of the present paper is to show that there exists a simpler representation of this function 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 has its origins in the works of Poincare, E. Cartan [3], C. L. Siegel, Pyatetskii{Shapiro, and others, (see the book [24] by Pyatetskii{Shapiro for details). The characteristic function also has connections with the Bergman kernel function on tube domains [8], etc; it even has uses in algebraic geometry [23]. The characteristic function for a cone K has invariance properties under the action of the automorphism group of K . This will be discussed below in Section 2. These invariance properties will help greatly in calculating the characteristic function of homogeneous cones, see Section 7. It is remarkable that homogeneous self{dual cones (\domains of positivity" in Koecher's terminology [18]) can be completely classi ed in terms of certain Jordan algebras, see Koecher [19], Hertneck [14], Vinberg [34], Satake [28], Faraut and Koranyi [8]. There exist 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{dual cones. 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]. These cones 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 metric in K . Thus, the characteristic function has intimate connections with Lie groups and 3

dierential geometry [18, 19, 26, 32, 4, 5, 28]. The characteristic function also has uses in carrying out Fourier analysis on K [15, 10, 30, 8]. The paper is organized as follows. In Section 2 we present some concepts and results from the theory of convex cones, especially concepts related to the automorphism group of the cone. In Section 3, we introduce the characteristic function of a cone K  R n and discuss its invariance properties. In Section 4, we prove the important result that the universal barrier function of K is essentially the logarithm of the characteristic function of K . In Section 5, we introduce the duality mapping which is an analytic bijection between K 0 and (K )0 , the interiors of K and its dual K  . We also show that if K is a homogeneous cone, then the (slightly modi ed) Fenchel dual of the universal barrier for K is the same as the universal barrier for K . In Section 6, we describe the important concept 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. Concluding remarks are made in Section 8.

2 Convex Cones In this section we present some elementary concepts and some relevant results from the theory of convex cones.

De nition 2.1 A subset K  R n is called a cone if for x 2 K and a scalar   0, we have x 2 K . A cone K is called convex if x; y 2 K implies x + y 2 K . If in addition K0 = 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 to these simply as cones. Two cones K1 and K2 are called isomorphic if there exists an invertible linear mapping A 2 R nn such that A(K1 ) = K2. Isomorphic cones can be considered equivalent. Let R n be endowed with an inner product hx; yi = xT Sy where S is a symmetric, positive de nite matrix. The dual of cone K is de ned as

K =

\

x2K

fy 2 R n : 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 ed that if hx; yiI = xT y is the standard inner product on R n , then the dual KI is related to K  by the equation K  = S ?1(KI): (5) 4

A cone is called self{dual if there exists an inner product such that K  = K . Note that a self{dual cone in one inner product may not be so in another one. The self{duality is a useful property, and this is the main reason why we allow inner products other than the standard one, see for example Section 7.3. Note that a self{dual cone K in the inner product h
;
iS is isomorphic to the cone S 1=2 (K ) which is self{dual in the standard inner product. Thus, if one is willing to work with transformed coordinates, one can always work with the standard inner product. However, this may not be convenient, since it might make it harder to describe the cone in the transformed coordinates, see again Section 7.3. Let Ki  R ni , i = 1; : : : ; k be cones. The direct sum of the cones Ki is the cone

K1  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. Otherwise, it is called indecomposable or irreducible. It is well known that a decomposable cone can be written as a direct sum of irreducible cones essentially in a unique way, see for 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  R n be a cone. The set of non-singular linear maps A : R n ! R n

leaving K invariant, that is satisfying A(K ) = K , is called the automorphism group of K and is denoted by Aut(K ). K is called homogeneous if Aut(K ) is transitive on K 0 , that is, given arbitrary points x; y in K 0 , 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 R n . It is also easy to see that Aut(K ) is a closed subgroup of GL(n; R ). Thus, by a theorem of von Neumann (or a more general result of E. Cartan), Aut(K ) is a Lie group. From (4) it follows that if A 2 Aut(K ), and x 2 K 0 , y 2 (K  )0, we have 0 < xT AT Sy = (Ax)T Sy = hAx; yi  hx; Ayi = xT SAy:

(6)

This shows that the conjugate map A = S ?1AT S 2 Aut(K  ). Similarly, if B 2 Aut(K ), then B  2 Aut(K ). It follows that

Aut(K  ) = fA : A = S ?1AT S; A 2 Aut(K )g; that is, the groups Aut(K ) and Aut(K ) are isomorphic. It follows from (6) that the mapping (A) = (A)?1 = S ?1(AT )?1S is a group isomorphism between Aut(K ) and Aut(K ). If K is homogeneous, then so is the dual cone K  , see (16). The automorphism group of a decomposable group is related 5

to the automorphism groups of its summands in the following way (see Vinberg [32]): if K = ki=1 Ki, then Qki=1 Aut(Ki) is a normal subgroup of nite index in Aut(K ). It is clear that if the cones Ki are homogeneous, then so is K . Irreducible homogeneous self{dual cones can be characterized completely in terms of formally real Jordan algebras [19, 14, 34, 28, 8], etc. These algebras, invented by P. Jordan in connection with quantum mechanics, are essentially classi ed in the very rst paper on Jordan algebras, the paper [16] by P. Jordan, J. von Neumann, and E. Wigner. Any homogeneous, 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 tool in this classi cation. Vinberg [34] gives an example of a homogeneous cone that is not isomorphic to a self{dual cone. In his seminal paper [32], he classi es the homogeneous cones 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 homogeneous cones. As mentioned above, it is possible to build up homogeneous cones in a recursive manner. This is discussed in some detail in Section 7.7.

3 Characteristic Function of a Cone In this section we state the de nition of the characteristic function of a cone and present its most important properties.

De nition 3.1 Let K  R n be a cone equipped with an inner product hx; yi = xT Sy, where S 2 R nn is a symmetric, positive de nite matrix. The characteristic function 'K : K 0 ! R of the cone K is the function 'K (x) =

Z

K

6

e?hx;yi dy:

(7)

We shall write ' when the cone under consideration is obvious. The function ' is essentially independent of the inner product. Consider the standard product in hx; yiI = xT y. Equation (5) implies KI = S (K  ) and we have

'I (x) =

Z

?xT y dy = e

KI

Z

S (K )

e?hx;S?1yidy =

Z

K

?hx;y0 i (det S )dy 0 = '(x) det S: e 

(8)

Consequently, the two characteristic functions dier by a multiplicative constant. The characteristic function has been introduced in connection with the classi cation of bounded homogeneous domains in complex analysis of several variables. Its main properties can be found in Koecher [18, 19], Rothaus [26], Vinberg [32], and Faraut and Koranyi [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 function F (x) = log('(x)) is strictly convex, (P3) If A is an automorphism of K , then

'(x) : '(Ax) = j det Aj

(9)

We note that since tI 2 Aut(K ) for any t > 0, we have

'(tx) = 't(nx) :

(10)

The properties (P1) and (P2) show that ' and F above are smooth barrier functions for K . These two functions, especially F will be important for interior point methods; we will 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 det Aj), we have

F (Ax) = F (x) ? log(j det Aj); which implies the important identity

Dk F (Ax)[Ah; Ah; : : : ; Ah] = Dk F (x)[h; h; : : : ; h]; 8h 2 R n ; k  1: 7

(11)

Property (P3) is important in calculating the barrier function of homogeneous cones. Let K be a homogeneous cone. Fix a point e 2 K 0 . Let x 2 K 0 be an arbitrary point, and suppose that Ax 2 Aut(K ) satis es Axe = x. Then (P3) implies '(e) = const : '(x) = '(Axe) = j det (12) Axj j det Axj Consequently, F (x) = const ? log(j det Axj): (13) We conclude this section by noting that D2F (x) de nes an invariant Riemannian metric on K . In fact, (11) implies that each derivative Dk F (x), k  1, is invariant under the action of Aut(K ). Moreover, (9) and the change of variables formula imply that the measure '(x)dx is invariant under Aut(K ), that is, if A 2 Aut(K ), then

Z

K

h(Ax)'(x)dx =

whenever the integral on the right exists.

Z

K

h(x)'(x)dx

4 Self{Concordance of the Characteristic Function In this section we prove the important result that the universal barrier function of Nesterov and Nemirovskii is essentially the logarithm of the characteristic function. This representation of the universal barrier function will make it easier to calculate barrier functions for cones. Let Q  R n be a convex set. Nesterov and Nemirovskii [20] de ne the universal barrier function 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 of Q centered at x, that is, Q(x) = fy 2 R n : 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  R n 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). Suppose  that 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 es hz0 ? x; ?yi  1; 8z0 2 K: Setting z0 = 0 above gives hx; yi  1. Also, setting z0 = x + z,  > 0, z 2 K gives hz; 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. ?hx;yi ?jjxjj The function e

=e

'(x) =

e?hx;yi dy =

Z

is constant on this hyperplane, and we can write '(x) as

Z1

Z

 d y d n ? 1 0 K K \H (x;) Z 1 ?jjxjj n?1  Z  voln?1(K  \ H (x; 1)): = e  d
 dn?1y = (njjx?jj1)! n 0 K \H (x;1) Here the second equality can be obtained by elementary methods, for example by transforming variables. It is also a direct application of the co{area formula, see [7], Theorem 2, Section 3.4.3, pp. 117, or [9], Theorem 3.2.3, pp. 243. Since  voln(K  \ H ?(x; 1)) = voln?1(K n\ H (x; 1)) ; we have '(x) = 1 vol (K  \ H ?(x; 1)) n! jjxjjn n = voln(K  \ H ?(x; 1=jjxjj)) = voln(fy 2 K  : hx; yi  1g) = voln(K (x)) = eu(x); e?jjxjj

9

where the fourth equality follows from Lemma 4.1. Let Q  R n be a closed convex set with non{empty interior. Endow Rn+1 with the inner product h(x; t); (y;  )i = hx; yi + t , and consider the tted cone K (Q)  R n+1 de 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 R n g, 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) = tnn+1! voln((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  1 '(x; t) = vol K (Q) \ H ?((x; t); 1)) n +1 (n + 1)! jj(x; t)jjn+1 1 vol (K (Q) \ H ((x; t); 1)) = (n + 1)jj(x; t)jjn+1 n 1 = vol (K (Q) \ H ((x; t); 1=jj(x; t)jj)): (n + 1)jj(x; t)jj n Now K (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; yi ) : hz ? x; ?yi  1; 8z 2 tQg t 1 ? hx; yi ) : y 2 ?(tQ) (x)g: = f(y; t Therefore,   '(x; t) = jj(x;n!t)jj voln f(y; 1 ? htx; yi ) : y 2 ?(tQ) (x)g : 10

The set in the above formula is the graph of the function  (y) = (1 ? hx; yi)=t over the domain ?(tQ) (x). By the surface area formula in calculus, it has volume

This shows that

jj(x=t; 1)jjvoln((tQ)(x)) = jj(tx;n+1t)jj voln(Q (x=t)):

and proves the theorem.

'(x; t) = tnn+1! voln(Q (x=t));

Corollary 4.1 De ne FQ(x) = log('K (Q)(x; 1)). Then FQ(x) = uQ(x) + log n!; where uQ is the universal barrier function for Q. In other words, FQ(x) and the universal barrier function for Q dier only by an additive constant.

Nesterov and Nemirovskii prove in their book, [20] (Theorem 2.5.1, pp. 50), that the universal barrier function is self{concordant with a parameter # = O(n). Their derivation of the bound on # is long and involves delicate moment inequalities. At least for homogeneous cones, one can give a simple proof that F is self{concordant, although the proof does not give any bound on the important self{concordance parameter #. (We know # = O(n) as mentioned above.)

Theorem 4.3 Let K  R n be a homogeneous cone. Then F (x) = log 'K (x) is a self{ concordant barrier function.

Proof.

Fix a point e 2 K 0 and let x 2 K 0 be an arbitrary point. Since K is homogeneous, there exists Ax 2 Aut(K ) such that Axe = x. It follows from (11) that

Dk F (x)[Axh; Axh; : : : ; Axh] = Dk F (e)[h; h; : : : ; h]; k  1:

(15)

Since ' satis es (10), it suces to show (see [20]) that there exists a constant c > 0 satisfying jD3F (x)[h0 ; h0; h0]j  c(D2F (x)[h0; h0])3=2 ; 8h0 2 R n : It follows from (15) that it is sucient to prove this inequality only at e, that is, it is enough to show that

jD3F (e)[h; h; h]j  c(D2F (e)[h; h])3=2 ; 8h 2 R n : 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 the whole set K 0 to calculating it at a single point x 2 K 0. In a number of papers in interior point methods, the expression log(det D2F (x)) appears in the barrier function, for example in the volumetric barrier, see Nesterov and Nemirovskii [20]. We close this section by showing that in the case where K is a homogeneous cone, the characteristic function can be written using the same expression.

Theorem 4.4 Let K  R n be a homogeneous cone and x a point e 2 K 0. We have q det D2F (x); '(x) = q '(e2) det D F (e) F (x) = const + 21 log(det D2F (x)):

Proof. Since K is homogeneous, there exists Ax 2 Aut(K ) satisfying Axe = x. Equation

(11) gives ATx D2F (x)Ax = D2F (e). This implies (det Ax)2 det D2F (x) = det D2 F (e), and

q ' ( e ) ' ( e ) q det D2 F (x): '(x) = j det A j = x det D2 F (e)

This proves the rst equality; the second one follows from the rst.

5 Duality Mapping In this section we de ne the duality mapping and present its main properties. It will be useful in determining barrier functions on dual cones. Let K  R n be a cone. Consider the characteristic function ' of K and its logarithm F (x) = log('(x)) both de ned in K 0 . Now DF (x) is a linear functional on R n , which in the standard inner product uT v on R n , is identi ed with the vector of the partial derivatives of F at x. If we endow R n with a new inner product hx; yi = xT Sy where S is a symmetric, positive de nite matrix S , then DF (x) can be written in the form

DF (x)[u] = h?x ; ui; 8u 2 R n : Thus, in this inner product, the linear functional ?DF (x) is identi ed with the vector x 2 R n . 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 homogeneous self{dual cones and in [32] when K is a homogeneous cone. Many of these properties also hold when K is an arbitrary cone. The following fundamental result can be found in Vinberg [32].

Theorem 5.1 Let K 

R n be a cone. The duality mapping is an analytic bijection

between K 0 and (K  )0 . We have

hx; xi = n:

In fact, x is characterized by the condition

x = arg minf'(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 theorem

can be found in [32]. The proof of the claim that x is the center of gravity of the above cross section follows, since we have

h?x ; hi = DF (x)[h] = implying

R D'(x)[h] = ? KR  hy; hie?hx;yi dy ; ?hx;yi dy '(x) K e

R ye?hx;yi dy  x = RK  e?hx;yidy :  K

Writing the integrals above in the form of iterated integrals as in Theorem 4.1 proves the result. Also, equation (16) follows from the fact that h 2 R n implies

hx ; 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)?1 x = (A)x . 13

It is known that the duality mapping has a unique xed point x = x which we denote by e. This can be seen as follows. Consider the convex minimization problem minf 1 hx; xi : FK (x)  0g; 2 where the constraint set is nonempty since x 2 K 0 and t ! 1 imply FK (tx) ! ?1. A solution to this problem exists and is unique, as the objective function is strictly convex and coercive. The constraint set can be shown to be bounded away from the origin, so that thep solution x satis es the condition x = (x) for some  > 0. Then the point x = x=  satis es x = x. The point e plays an important role in the classi cation of both homogeneous self{dual cones 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 point of K 0 of a homogeneous cone can be made into a center by choosing an appropriate inner product on R n . The existence of e implies immediately the following result in Ochiai [22], which can also be proved by an elementary separation argument.

Corollary 5.1 Let K  R n be a cone. Then K 0 \ (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 its easy proof.

Theorem 5.2 Let K  R n 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  (x )   'K (Ax)'K  ((Ax) ) = 'K (Ax)'K  ((A)?1x ) = j det'KA(jx
)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 its proof for completeness. Our proof follows Rothaus [26]. 14

Theorem 5.3 Let K  R n 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 ! R n , 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 . Dierentiating 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, dierentiating the above equation with respect 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{dual cone. The proofs can be found in Rothaus [26].

Theorem 5.4 Let K  R n be a homogeneous self{dual cone. The set of linear maps fHK (x) : x 2 K g forms a simply transitive subset of Aut(K ), that is, given any two a; b 2 K 0 , there exists a unique x 2 K 0 such that HK (x)a = b. Moreover, any A 2 Aut(K ) can be written uniquely as

A = B HK (x)

where B 2 Aut(K ) leaves the point e xed, that is, Be=e.

We remark that, in the inner product ha; bi = aT Sb, the linear mappings B and HK (x) are orthogonal and symmetric positive de nite, respectively; i.e., we have BB  = I and hHK (x)a; bi = ha; HK (x)bi > 0 for all a; b 2 R n . In other words, A = BHK (x) is the polar decomposition of A. Primal{dual interior point methods need a barrier function on both K and K , see Nesterov and Nemirovskii [20]. Thus it becomes important to calculate the barrier on 15

K  eciently. For self{dual cones, there is clearly no problem, as we can take the same barrier 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 K 0g;

(17)

is a barrier for K  and has the same parameter # as FK . (In the ordinary Fenchel dual, one has y instead of ?y in (17).) We use the properties of the duality mapping to determine the 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  R n 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 K such 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 K 0 such that x = y. Since hx; x i = n, we can rewrite (17), (FK )(y) = supfhx; ?yi ? FK (x) : x 2 K 0 ; hx; yi = ng = ?n ? log inf f'K (x) : x 2 K 0; hx; yi = ng: We show in the proof of Theorem 4.1 that 'K (x) is proportional to

V oln(fz 2 K  : hx; z i  ng); that is, the volume of the truncated cone after K  is cut by the hyperplane having normal x and passing through the point y 2 K  . Let  = fz 2 K  : hx; z i  hx; yig Kx;y

16

be such a cone. Combining the above results, we see that (FK ) is, up to an additive constant, equal to the function FK+ de ned by  ) : x 2 K g: FK+ (y) = ? log inf fV oln(Kx;y

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 FK+ given by   ) : y 2 K  g ; FK+(x) = ? log inf fV oln(Ky;x where Ky;x is truncated cone after K is cut by the hyperplane with normal y and passing through the point x 2 K . The description of the \characteristic function" related to FK+ can already be found in [31]. We show in Theorem 5.5 above that these two universal barrier functions coincide if K is a homogeneous cone.

6 Homogeneous Cones and Siegel Domains Homogeneous cones form an attractive class among cones because of their invariance properties. As we note in Section 2, Vinberg [32] is the rst to give an algebraic classi cation of these cones. Siegel domains described below play an essential role in the classi cation and 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 only the basic elements of this theory and those aspects of it that we need in order to calculate the universal barrier functions of homogeneous cones.

De nition 6.1 Let K be a cone in R k . A K {bilinear form B (u; v) in R p is a mapping from R p  R p to R k satisfying the following properties 1. 2. 3. 4.

B (1u1 + 2 u2; v) = 1 B (u1; v) + 2 B (u2; v) for 1 ; 2 2 R , B (u; v) = B (v; u), B (u; u) 2 K , 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 Siegel domain corresponding to K and B is the set

S (K; B ) = f(x; u) 2 R k  R p : x ? B (u; u) 2 K g: 17

De nition 6.3 A K {bilinear form B is called homogeneous if K is a homogeneous cone and there exists a transitive subgroup G  Aut(K ) such that for every g 2 G, there exists a linear transformation of g of R p such that g B (u; v) = B (gu; gv); that is, the following diagram p p p gg Rp  ?? R ?! R ?? R

B?y

Rk

g ?!

?yB

Rk

commutes.

The Siegel domain S (K; B ) corresponding to a homogeneous K {bilinear form B is ane homogeneous. This can be seen by checking that the following ane transformations form a transitive subgroup of S (K; B ): A1(x; u) = (x + 2B (u; a) + B (a; a); u + a); a 2 R p ; 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 ane{homogeneous domain D  Rn is ane equivalent to a Siegel domain.

The cone tted to a Siegel domain S (K; B ) is given by SC (K; B ) = f(x; u; t) 2 R k  Rp  R : t  0; tx ? B (u; u) 2 K g: We remark that a more general construction than SC (K; B ) can be found in Nesterov and Nemirovskii [20], Example 5, pp. 165, except that the homogeneity condition described 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, then the cone SC (K; B ) is homogeneous, and the following linear mappings form a transitive subgroup of Aut(SC (K; B )), p T1(x; u; t) = (x; u; t);  > 0; T2(x; u; t) = (x + 2B (u; a) + tB (a; a); u + ta; t); a 2 R p ; 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,0 using De nitions 6.1{6.3, that each map Ti 2 Aut(SC (K; B )). Let (x0 ; 0; 1) 2 SC (K; B ) be a xed point, where x0 2 K 0 . Let (x ; u; t ) 2 SC (K; B )0 be an arbitrary point, where x 2 K 0 , t > 0, and u 2 Rp . Consider the linear map T = T3 T2T1 , Ti 2 Aut(SC (K; B )), T1 (x; u; t) = (x; pu  ; tt ); t

T2 (x; u; t) = (x ? p2  B (u; u) + tt B (u; u); u ? pt  u; t) t t T3 (x; u; t) = (gx; gu; t); where g 2 G satis es g(x ? B (u ; u)=t ) = x0 . We note that  T1(x ; u; t) = (x ; pu  ; 1); t    T2 (x ; pu  ; 1) = (x ? B (ut; u ) ; 0; 1); t   T3(x ? B (ut; u ) ; 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 homogeneous 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 homogeneous K {bilinear form B such that K is linearly equivalent to SC (K; B ), see for example Gindikin [11], pp. 75. Consequently, it is possible to recursively construct an arbitrary homogeneous cone out of lower dimensional homogeneous cones, starting from the real half{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 positive semi{de nite matrices, see Section 7.3. This construction process yields the algebraic classi cation of homogeneous cones. The number of recursive steps necessary to build up a 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 and the 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 ) are given by B (u; u) )
det g; '(x; u; t) = const p +1
'K (x ? t t2 B ( F (x; u; t) = const + FK (x ? u;t u) ) + log(det g) ? ( 2p + 1) log t:

19

Proof. It follows from (12) that '(x; u; t) = '(x0 ; 0; 1)
det T; where x0 2 K 0 is a xed point, and T 2 Aut(SC (K; B )) satis es T (x; u; t) = (x0 ; 0; 1). It is thus sucient to calculate the determinants of the linear maps at the end of the proof p of the above lemma. It is easy to see that det(T1) = 1=t 2 +1 , det(T2) = 1, and det(T3 ) = det g
det g = const
'K (x ? B (u; u)=t)
det g; where the last equality follows from K (x0 ) 'K (x ? B (u; u)=t) = 'K (g?1x0 ) = 'det g?1 = const
det g:

The corollary is proved.

7 Characteristic Function of Some Cones In this section we calculate the characteristic function ' of some cones and the corresponding barrier function F . We demonstrate that, although the universal barrier function is usually very hard to calculate, it can be calculated in some important cases. It is 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 in these sections can be scaled to either agree with the optimal barrier functions or to have comparable parameter #. As we mentioned in the previous section, the calculation of the universal barrier functions in these sections bears a strong resemblance to the calculations carried out in Chapter 5 of [20]. (However, homogeneity is not considered in [20].) Using their classi cation theory, we determine in [13] the optimal parameter # for homogeneous cones. It is not known at present whether the universal barrier function of an arbitrary irreducible homogeneous cone can be scaled so that it has parameter # comparable to the optimal one. The barrier in Section 7.1 is the familiar logarithmic function. The barrier in Section 7.5 can be obtained by the methods in [20] and has # = n. We show here that it is the universal barrier function for the underlying cone. The barrier calculated in Section 7.4 seems to be new, and has # = O(n). However, it seems useless for interior point methods, since it would take eort exponential in n to calculate it and its derivatives. We note that, since the cones in Sections 7.4 and 7.5 are dual, the Fenchel dual of the universal barrier for one cone is the second \universal" barrier function for the other cone. We do not calculate explicit barriers in Section 7.6, but some barriers can be calculated using the formula in Lemma 7.5. In Section 7.7 we obtain the universal barrier function of some 20

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 homogeneous cone (using the classi cation of these cones). Finally, in Section 7.8, we show that the calculation of the universal barrier function of a polyhedral cone reduces, in theory, to the triangulation of the dual polyhedral cone. This shows, in particular, that the universal function of such a cone is the logarithm of a rational function. It also shows that it would be hard in general to calculate the universal barrier function of polyhedral cones.

7.1 The Non{Negative Orthant The non{negative orthant R n+ = R +  : : :  R + is the direct sum of n copies of R + . Thus,

'(x1 ; : : : ; xn) =

n Y

i=1

'R+ (xi):

R Since 'R+ (xi ) = 01 e?xiyi dyi = 1=xi, we have n X '(x1; : : : ; xn) = Qn1 x ; F (x) = ? log xi : i=1 i i=1 F (x) is the familiar self{concordant barrier function for R n +.

7.2 The Lorentz Cone q Kn = fx 2 R n : x21 + x22 + : : : x2n?1  xng: It is variously known as the spherical cone, light cone, ice cream cone, etc. The cone K4 plays a prominent role in special relativity. Note that Kn+1 is the cone tted to the unit ball Bn = fx 2 R n : jjxjj  1g. If we endow R n with the usual inner product, then this cone is self{dual. This can be inferred from Section 15 of Rockafellar [25]. We include a short proof. If the point (y;  ) 2 R n+1 is in Kn+1, then hx; yi + t  0 for all (x; t) satisfying jjxjj  t. This implies hu; yi +   0 for all u such that jjujj  1. Thus, supjjujj1hu; yi   and jjyjj   . Since the implications can be reversed, we have Kn+1 = Kn+1. Consider the cone SC (R+ ; Bn) = f(x; u; t) 2 R  R n  R : x  0; t  0; tx ? juj2  0g; where Bn(u; v) = uT v can be easily shown to be a homogeneous R + {bilinear form. After a rotation of the variables (t; x) the term tx becomes t2 ? x2 , so that SC (R + ; Bn) is linearly

This is the cone

21

isomorphic to Kn+2. Since the former cone is homogeneous by Lemma 6.1, the cone Sk is homogeneous for all k  3. The cone S1 = R + is obviously homogeneous, and it is easy to show that S2 is linearly isomorphic to R 2+ 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 Siegel domain construction in Section 6. De ne T 2 Aut(R + ), where p  > 0 and Tx = x. The corresponding linear transformation T  on R n is T u = u. By Corollary 6.1 2 F (x; u; t) = const ? log(x ? jutj ) + log(det T  ) ? ( n2 + 1) log t; where (x ? juj2=t) = 1. This gives det T  = (x ? juj2=t)?n=2, and F (x; u; t) = const ? n +2 2 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 corresponding barrier function are given by

'(x; t) = (t2 ? jjconst xjj2)(n+1)=2 ; F (x; t) = const ? n +2 1 log(t2 ? jjxjj2):

The barrier function F has parameter # = n + 1 which is much worse than the parameter # = 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 be obtained by scaling the universal barrier function.

7.3 Symmetric Positive Semi{De nite Matrices Consider the vector space n of n  n symmetric matrices endowed with the inner product

hx; yi = tr(xy): This is the same as the inner product on R n(n+1)=2 obtained as follows. Let x~; y~ be the vectors obtained by putting in some order the diagonal and strict upper diagonal elements of x and y into vectors in R n(n+1)=2 , respectively. Then

hx; yi = x~T Dy~; 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 we denote by +n . It is well known that +n is a self{dual cone, that is, (+n ) = +n . This can 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=2 x1=2 y) = tr(x1=2 yx1=2 )  0; where the inequality follows as x1=2 yx1=2 2 +n . This shows +n  (+n ) . Conversely, let y 2 (+n ) . Then tr(xy)  0 for all x 2 +n . If u 2 R n , then uT yu = tr(yuuT ) = hy; uuT i  0. Thus, y 2 +n , and consequently (+n ) = +n . The interior (+n )0 corresponds to the set 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 : R n  R n ! n is given by B (u; v) = (uvT + vuT )=2. If g 2 GL(n; R ), then the linear map Tg : n ! n given by Tg x = gxgT is evidently an automorphism of the cone +n . The corresponding linear map T g : R n ! R n is given by T g u = 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) matrix

1 0 t u x = @ T A; u x

where t 2 R , u 2 R n , and x 2 n is positive semide nite if and only if t  0, x 2 +n , and tx ? uuT 2 +n . The above claim follows easily from these. We now calculate the universal barrier function of the cones n . Using Corollary 6.1 and 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 obtain T Fn+1 (x) = const + Fn (x ? uut ) ? 21 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 matrices is the function

F (x) = const ? n +2 1 log(det x): 23

Note that the cone is one of the ve irreducible homogeneous self{dual cones listed in Section 2. The universal barrier function of the cone of positive semi de nite complex matrices and the cone of positive semi de nite quaternion matrices can be calculated similarly. The barrier function F has parameter # = n(n + 1)=2 which is much worse than the 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 from the universal barrier by scaling. Examples of convex programming problems which involve the cone of symmetric positive 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, combinatorial optimization, and control theory.

7.4 The 1 Unit Ball l

Here we calculate the characteristic function of the convex set Q = fx 2 R n : jjxjj1  1g. The tted cone is K (Q) = f(x; t) : jjxjj1  tg. It is easy to show that the dual cone is given by K (Q) = f(y;  ) : jjyjj1   g, see Rockafellar [25]. We calculate

'(x; t) = =

Z

Z

e?hx;yi dy)d = jjyjj1 0 jjyjj1 Z 1 Y n Z n  jxi j ? jxi j  Z1 Y e?t e?xi yi dyi = e?t e j?x ej d: 0 0 i i=1 ? i=1

It is easy to verify that

n Y

i=1

Thus, we have

Z1

e?t e?hx;yidyd

(e jxij ? e? jxij) =

e?t (

n X Y

( "i)e

"i =1 i=1

Pn

i=1 "i jxi j :

Z1 n Pn X Y '(x; t) = Qn 1 jx j ( "i) e? (t? i=1 "ijxij) d 0 i=1 i "i =1 i=1 Q n " X i=1 i ; = Qn 1 P n i=1 jxi j "i =1 t ? i=1 "i jxi j

the barrier function is

F (x; t) = log(

X "i =1

Qn " n X i=1 i log(jxi j); ) ? P t ? n " jx j i=1 i i

24

i=1

and the induced barrier on Q is

F (x) = F (x; 1) = log(

X "i =1

Qn " n X i=1 i ) ? log(jxij): P 1 ? n " jx j i=1 i i

i=1

The barrier function F has parameter # = O(n). Since l1 cone in Section 7.3 is dual to the l1 cone here, and the optimal barrier function for the former cone is at least n by Proposition 2.3.6 in [20], we see that F has parameter of optimal order. However, F is practically useless for interior point calculations for large n, since the eort to calculate it and its derivatives is exponential in n. For n = 2, the barrier of Q is F (x) = const ? log(1 ? (jx1 j + jx2j)2) ? log(1 ? (jx1 j ? jx2 j)2):

7.5 The

l1

Unit Ball

Here we calculate the characteristic function of the convex set Q = fx 2 R n : 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 R n is

n n?1 'n(x; t) = Qn2(t2t ? x2 ) ; i 1 n X Fn(x; t) = const ? log(t2 ? x2i ) + (n ? 1) log t: 1

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 Z 'n+1(x; t) = e?t e?hx;yi dyd Zjj1yjj1 Z = e?xn+1yn+1 ( 0 0 e?t e?hx0 ;y0i dy0d )dyn+1 f(y ; ):jjy jj1 ?jyn+1 jg Z?1 Z 1 ?x y = e n+1 n+1 e?tjyn+1j( 0 0 e?t( ?jyn+1 j)e?hx0 ;y0i dy0d )dyn+1 Zf(y ; ):jjy jj1 ?jyn+1jg0 0 0 Z?1 1 ?x y e n+1 n+1 e?tjyn+1j( 0 0 0 0 e?t e?hx ;y i dy0d 0)dyn+1 = ?1

f(y ; ):jjy jj1  g

25

Z1

 n n?1  e?xn+1yn+1 e?tjyn+1j Qn 2(tt2 ? x2 ) dyn+1 ?1 i i=1 Z1 n tn?1 Z 0 ?(xn+1 ?t)yn+1 2 ?(xn+1 +t)yn+1 dy  e dy + e = Qn 2 2 n+1 n+1 0 i=1 (t ? xi ) ?1 n n?1 = Qn 2 t2 2 ( 1 + 1 ): t+x (t ? x ) t ? x

=

i

i=1

n+1

n+1

Since 1=(t ? xn+1 ) + 1=(t + xn+1) = 2t=(t2 ? x2n+1), we have n+1 n 'n+1(x; t) = Qn+12 (t2 t? x2 ) : 1

i

This proves the lemma. The barrier function F has parameter # = n which is optimal, see [20], Proposition 2.3.6.

7.6 Epigraph of Convex Functions The following result is essentially contained in Rockafellar [25], Theorem 14.4, pp. 124.

Lemma 7.4 Let f : R n ! 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 the cone tted to Q. We have

K (Q) = cl(f (u; 1; ) : > 0; f  (?u)  g):

Lemma 7.5 Suppose f : R n ! 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) = tnn+1!

Z

du : D(f  ) [?hx=t; ui + =t + f  (u)]n

Proof. It is sucient to prove the lemma for t = 1. We have '(x; ; 1) =

Z

K (Q)

eh(x;;1);(y; ; )idyd d:

26

We use the description of K (O) in the above lemma and change the variables of integration 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) = = = =

Z

ZG

Z1 n +1 ? ( ?h x;u i +  ) e (  e?  d )d du f(u; ): >0;u2D(f  )g f (u) Z ? f  (u) e n +1 ? ( ?h x;u i +  ) d du e f(u; ): >0;u2D(f  )g Z1 Z ( ne? (?hx;ui++f  (u)) d )du  D(f ) 0

= n! The lemma is proved.

n+1e? (hx;?ui++ ) dd du

Z

du : D(f  ) [?hx; ui +  + f  (u)]n

7.7 Epigraph of Matrix Norms We consider the vector space m of symmetric m  m matrices, the cone K = +m of symmetric p.s.d. matrices in m , and the vector space R nm of n  m matrices, where m  n. We endow m with the inner product hx; yi = tr(xy) and R nm with the inner product hu; vi = tr(uT v). Note that Bn;m : Rnm  R nm ! n given by Bn;m(u; v) = (uT v + vT u)=2 is a +m {bilinear form. The subgroup G = fTg : g 2 (+m )0 g, where Tg is de ned in Section 7.3, is obviously a transitive subgroup of Aut(+m ). De ning T g : R nm ! R nm such that T g u = ug we see that Tg (Bn;m(u; v)) = Bn;m(T g u; T g v); that is, Bn;m is a homogeneous +m {bilinear form. Thus, the Siegel Domain S (+m; Bn;m) = f(x; u) 2 +m  R nm : x ? uT u 2 +m g is an ane homogeneous convex set, and the Siegel cone SC (+m ; Bn;m) = f(x; u; t) 2 +m  R nm  R : tx ? uT u 2 +m ; t  0g is homogeneous. We now calculate the characteristic function of the cone SC (K; B ) = SC (+m ; Bn;m). It is easy to verify that det T g = (det g)n. If T T Tg (x ? u t u ) = g(x ? u t u )gT = I; 27

then g = (x ? uTt u )?1=2 and det T g = det(x ? uTt u )?n=2. This and Corollary 6.1 give the following result.

Lemma 7.6 The characteristic function and the universal barrier function of the cone

SC (+m ; Bn;m) are given by

T '(x; u; t) = t? mn2 ?1(det(x ? u t u ))? m+2n+1 ; T + 1) log t: F (x; u; t) = const ? m + 2n + 1 log(det(x ? u t u )) ? ( mn 2

The cone SC (K; B ) 2 R l where l = 1 + mn + m(m + 1)=2. Hence by Theorem 2.5.1 in Nesterov and Nemirovskii [20], F a self{concordant barrier for SC (K; B ) 2 R l with parameter # = O(l). This parameter is much larger than the parameter # = m + 1 for the following barrier function for the same cone given in [20], pp. 200, Tu u H (x; u; t) = ? log(det(x ? t )) ? log t: The barrier function H can be shown to be optimal, see [13]. However, if we multiply F with 2=(m + n + 1), we obtain a scaled barrier function T +2 log t: G = ? log(det(x ? u t u )) ? mmn +n+1 Now, the function G is obtained by applying Proposition 5.1.8 in [20] to the barrier function G1 (x) = ? log det x for the cone +m and the barrier function G2(t) = ? log t for the non{negative real line R + with  = (mn +2)=(m + n +1). Since   1, it is easy to verify that G2 is a self{concordant barrier function for R + with parameter # =  . It then follows from Proposition 5.1.8 in [20] that G is a self{concordant barrier for SC (K; B ) with parameter + 2  2m: # = m +  = m + mmn +n+1 Thus, we see that the barrier function G and the optimal barrier function H have comparable parameters #, although H has a slightly better parameter. It a routine matter to calculate the dual barrier function G, since we know from Theorem 5.5 that it coincides with a multiple of the universal barrier function of the dual cone SC (K; B ). We do not calculate 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 the vector space m  R nm  R with the inner product h(x; u; t); (y; v; s)i = tr(xy) + tr(uT v) + ts: The method can be extended to calculate the dual of any Siegel cone SC (K; B ), and in fact to give a recursive \dual" procedure to build up any homogeneous cone, see [27]. 28

Lemma 7.7 We have

SC (+m ; Bn;m) = f(y; v; s) 2 +m  R nm  R : s  41 tr(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 all x; u such that x ? uT u 2 +m . This is equivalent to the requirement that tr(xy) + tr(uT v) + s  0; for all x; u satisfying x ? uT u 2 +m . We must have y 2 +m , since otherwise there exists  2 R m such that T y < 0. Choosing u = 0, and x(t) = t2 T , we obtain 2 T lim tr(x(t)T y) + s = tlim t!1 !1 t  y + s = ?1; a contradiction. Since tr(ab)  0 for all a; b 2 +n , we can rewrite the above inequality as

tr(uT uy) + tr(uT v) + s  0; 8u: De ning u = uy1=2, we can, in turn, write this last inequality as 0  tr(uT u) + tr(uT vy?1=2) + s ?1=2 ?1=2 vy vy 1 T = tr((u + ) (u + )) ? tr((vy?1=2)T vy?1=2) + s 2 2 4 for all u. The lemma follows immediately.

7.8 Polyhedral Cones Consider a polyhedral cone m \

K = fx 2 R n : hai ; xi  0g; 1

where ai 2 R n are given vectors. In this section, we give a formula for the characteristic function and the universal barrier function of K . Here we follow some ideas in Barvinok [2]. Note that the dual cone

m X K  = conv(a1 ; : : : ; am) = f iai : i  0g 1

29

(19)

is the convex conical hull of the vectors faigm1 . Suppose that we decompose K , that is, we write [ K  = Ki ; where each Ki = conv(d1; : : : ; dn) is a simplicial cone with linearly independent vectors fdign1 , and the intersection Ki1 \ Ki2 of two dierent cones has less than full dimension. We have Ki = D(R n+ ) where D is the matrix with columns dj . Thus, we can explicitly calculate the integral (7) over the cone Ki ,

Z

e?hx;yi dy = 

Ki

Consequently, we have

Z

D(Rn+ )

e?hx;yi dy =

Z

?hDT x;ui j det Djdu = Qj det Dj : e n hd ; xi n

R+

1

k

X

Qnhdci ; xi ; 1 k;i X ci ! FK (x) = log Qnhd ; xi ; 1 k;i

'K (x) =

where each summation is taken over the dierent simplicial cones in the decomposition of K  . By Theorem 2.5.1 in [20], FK is an O(n) self{concordant barrier function for K . We note 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 special cones in this way. At present, the above formula for FK should probably be considered a theoretical result, since it is not clear how one would decompose an arbitrary cone in (19) in an ecient manner.

8 Concluding Remarks In this paper, we have shown that the logarithm of the characteristic function is essentially equivalent to the universal barrier function of Nesterov and Nemirovskii. Our results connect the eld of interior point methods with several branches of mathematics, such as Lie groups, dierential geometry, symmetric spaces, complex variables, Jordan algebras, etc. It is hoped that the results obtained in the sizable literature in the mentioned elds will have applications to interior point methods. In [13], we make use of the classi cation of homogeneous cones in [32, 11], etc. to determine the best self{concordance parameter for such a cone. In particular, we show that the rank, Caratheodory number, and self{concordance parameter of a homogeneous cone are all equal. 30

The use of the automorphism group brings our approach closer to the one envisioned by Karmarkar. For example, Karmarkar [17] uses the automorphism group to bring an arbitrary point to the \center". We have shown for the rst time that the automorphism group 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 optimal self{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 function are either equal or have comparable barrier parameter. It is not known at present whether this is true for an arbitrary irreducible homogeneous cone K . (The corresponding result for 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 irreducible cones Ki individually.) Even if this is true, it is not clear whether the (scaled) universal barrier function F for K has any advantages over the optimal barrier function G for the same cone K , especially since both functions have the invariance properties

F (A1x) = F (x) + const1 ; G(A2 x) = G(x) + const2; where A1 2 Aut(K ) and A2 is in a transitive subset of Aut(K ). Here the rst equality follows from (9), and in the second equality the transitive subset can be taken to be the collection 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{step interior point methods on what they call self{scaled cones. These algorithms are strongly dependent on the special properties of the \self{scaled" cones and the special barriers on them. It is seen that these cones coincide with the homogeneous self{dual cones. Using the results of this paper, we see that the class of self{scaled cones are made up of direct products 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 are not addressed, such as how to obtain various interior point methods using these barrier functions. Many of these issues are discussed in Nesterov and Nemirovskii [20], but it may 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 special barrier functions and extend some of the algorithms in [21] to more general cones. We also have not addressed the dierential geometric issues involving the Riemannian geometry de ned on K 0 by the bilinear form D2F (x). A sizable literature exists on these 31

issues when K is homogeneous or homogeneous self{dual. For example, it is known that if 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 Riemannian curvature is everywhere non{positive, see for example [26]. The structure of the geodesics have also been studied, for example in [19, 26], etc. It seems reasonable that these should have a bearing on analyzing the behavior of interior point methods.

Acknowledgements. The author thanks Yurii Nesterov, Michael Todd, and Yinyu Ye for useful discussions during the preparation of the paper. The author also thanks Farid Alizadeh, Alexander Barvinok, Joseph Dorfmeister, Arkadii Nemirovskii, James Renegar, and Levent Tuncel for helpful comments and suggestions on the rst version of the paper. The research of this paper is partially supported by the National Science Foundation under grant DMS{9306318.

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