BARRIERS ON PROJECTIVE CONVEX SETS

. Modern interior-point methods used for optimization on convex sets in aﬃne space are based on the notion of a barrier function. Projective space lacks crucial properties inherent to aﬃne space, and the concept of a barrier function cannot be directly carried over. We present a self-contained theory of barriers on convex sets in projective space which is build upon the projective cross-ratio. Such a projective barrier equips the set with a Codazzi structure, which is a generalization of the Hessian structure induced by a barrier in the aﬃne case. The results provide a new interpretation of the aﬃne theory and serve as a base for constructing a theory of interior-point methods for projective convex optimization.

1. Introduction. Self-concordant barriers are one of the main tools in convex optimization. Modern interior point methods for linear, conic quadratic and semidefinite programming are based on the use of logarithmically homogeneous self-concordant barriers on the corresponding convex cones [1]. A barrier F : C → R on an open convex subset C of a real vector space V is a smooth locally strongly convex function which tends to infinity as the argument tends to the boundary of the set. A barrier defines a Hessian structure on the convex set, i.e., a pair (∇, g), consisting of a flat affine connection ∇ and a Riemannian metric g, such that the covariant derivative ∇g is symmetric in all three indices [2]. Here the role of the metric g is played by the second derivative F , while the affine connection is the canonical connection of the ambient vector space V . Thus ∇g is simply given by the third derivative F . T! he function F is called self-concordant if F is uniformly bounded with respect to the metric. Any Hessian structure defines a dual Hessian structure (∇ , g), which shares the metric with the primal one, but whose flat affine connection is in general different. This dual connection has a simple interpretation by the gradient map x → F (x), which maps C bijectively to a subset C * of the dual space V * . Namely, ∇ is generated on C by the canonical flat affine connection of V * . This duality of Hessian structures is vital to the theory of interior-point methods in convex optimization and reflects the well-known convex duality between a barrier F and its Legendre transform F * . Namely, F * generates the dual Hessian structure in the same way as F generates the primal one. Hessian structures can be generalized to Codazzi structures by dropping the requirement that the affine connection ∇ is flat.
In projective space, several key properties used for the construction of a theory of barriers, and ultimately, for a theory of interior-point methods, are not available. The projective space does not possess a canonical affine connection with respect BARRIERS ON PROJECTIVE CONVEX SETS 673 to which a covariant derivative could be defined. The notion of convexity of a function, contrary to the notion of convexity of a set, is therefore not well-defined, and the concept of a barrier cannot be directly carried over from the affine case. Any projective space P, however, still comes along with a dual projective space P * , and a convex set C ⊂ P defines a dual convex set C * ⊂ P * . We exploit this duality by defining a barrier on a convex open subset C of projective space as a bijection between C and its dual C * . We then construct a quantitative theory of projective barriers from a basic numerical invariant, the projective cross-ratio. In particular, the interpl! ay between the barrier and the cross-ratio equips the set with a Codazzi structure. The connection of this Codazzi structure is projectively flat, i.e., projectively equivalent to flat connection [3,Def. 3.3,p.17]. This allows to define projective counterparts for most objects appearing in the affine theory, including the notion of self-concordance.
It turns out that projective barriers on a given convex set C in projective space are in close relation with logarithmically homogeneous barriers on the convex cone K corresponding to the set C. This furnishes a new interpretation of the logarithmically homogeneous barriers and reveals parts of their structure that remained hidden in the affine setting. Our results can serve as a base for a reformulation of the theory of conic programming and for an improvement of existing interior-point methods in conic optimization.
The present paper is close in spirit to affine differential geometry, and we use notations and a presentation style that is common in the corresponding literature. An excellent textbook on affine differential geometry is [3]. For a more specific introduction to Hessian structures, see [2].
2. Barrier functions on affine convex sets. In this section we consider the classical case of barrier functions defined on convex subsets of a real vector space. As a special case, we consider logarithmically homogeneous barriers on convex cones.
Let V be an n-dimensional real vector space and ∇ the canonical flat affine connection on V . In any affine coordinate system on V the Christoffel symbols of ∇ vanish identically, such that the covariant derivative equals the partial derivative.
Let C ⊂ V be an open convex set. A barrier on C is a smooth function F : C → R with everywhere positive definite Hessian such that lim x→∂C F (x) = +∞. A special case of convex sets is given by regular convex cones, i.e., closed convex cones containing no lines and with nonempty interior. Definition 2.1. Let C ⊂ V be the interior of a regular convex cone. A logarithmically homogeneous barrier with homogeneity parameter ν is a barrier F on C satisfying the relation The Hessian F induces a Riemannian metric g on C. Together with the affine connection ∇ this metric constitutes a Hessian structure (∇, g) on C. This follows from the fact that in every affine coordinate system the covariant derivative ∇g equals the third partial derivative F , which is symmetric in all three indices. Thus ∇g is symmetric in every coordinate system.
Since F is positive definite, the gradient map x → F (x) is a smooth injective mapping from C into the dual vector space. It defines a bijection between C and a subset C * ⊂ V * . Now the Legendre transform F * defines on C * a Hessian structure (∇ , g ) in the same way as F on C * . Here ∇ is the flat affine connection on V * , and g is given by the second covariant derivative ∇ µ ∇ ν F * , which coincides with the Hessian of F * in any affine coordinate system on V * .
To account for this duality, we from now on do not work with the sets C or C * , but define a submanifold By the above, M carries two different Hessian structures (∇, g) and (∇ , g ). The theory of Hessian structures now states that these two structures are dual to each other [2, p.26]. In particular, the metrics g, g coincide and∇ = 1 2 (∇ + ∇ ) is the Levi-Civita connection of g [2, p.25]. Since∇g = 0, we see that (∇ −∇)g = F in every affine coordinate system on V , and (∇ −∇)g = (F * ) in every affine coordinate system on V * . By the above, we then get F + (F * ) = 0, but the derivatives of F and F * are here with respect to different coordinate systems, see also [1, p.45].
For a barrier to be useful for practical purposes, it must be self-concordant [1, Def.
] A convex function F on a convex set C is said to be ϑ-self-concordant if in any affine coordinate system on C for every x ∈ C and every nonzero tangent vector h ∈ T x C the inequality holds.
In view of the above, this definition is equivalent to the condition that the infinity norm of the tensor ∇g with respect to the metric g is bounded by 2ϑ −1/2 . Let now C be the interior of a regular convex cone and let F be a logarithmically homogeneous barrier on C with homogeneity parameter ν. Then the functioñ 3. Projective space. In this section we first consider some elementary properties of projective space. This part of the material in this section is classical. Then we introduce a symmetric 2-point function and an invariant bilinear form on a dense subset of the product of projective space with its dual. These tools will allow us to perform quantitative analysis in projective space. Finally, we consider convex sets in projective space. Let X = RP n be the n-dimensional real projective space. It is defined as the set of 1-dimensional subspaces of the real vector space R n+1 . Its dual, which we denote by P , is defined as the set of 1-dimensional subspaces of R n+1 , the dual space to R n+1 . For an element p ∈ P , all nonzero linear functionals on the line p ⊂ R n+1 share the same kernel. This kernel is a hyperplane in R n+1 and hence defines a projective hyperplane in the projective space X. Thus points in P can be identified with hyperplanes in X. In the same manner, points in X can be identified with hyperplanes in P . A pair (x, p) ∈ X × P of points is said to be orthogonal, x ⊥ p, if x is contained in the projective hyperplane defined by p. Clearly this is equivalent to the condition that p is contained in the hyperplane defined by x.
On the product X × P there exists a natural atlas of affine charts. Any such affine chart can be constructed as follows. Let {e 0 , e 1 , . . . , e n } be a basis of R n+1 and {e 0 , . . . , e n } the corresponding dual basis of R n+1 . The linear hulls of the basis vectors e 0 , e 0 define pointsx ∈ X,p ∈ P , respectively. Let now z = (x, p) ∈ X × P be such that x ⊥p, p ⊥x. Then the line x ⊂ R n+1 intersects the affine hyperplane {e 0 + n k=1 x k e k | x 1 , . . . , x n ∈ R} ⊂ R n+1 in exactly one point, and we can associate the coordinate vector (x 1 , . . . , x n ) T of this point to x. Likewise, p ⊂ R n+1 intersects the affine hyperplane {e 0 + n k=1 p k e k | p 1 , . . . , p n ∈ R} ⊂ R n+1 in exactly one point, and we can associate the coordinate vector (p 1 , . . . , p n ) T of this poin! t to p. In this way affine charts on X and P are defined. The coordinate vector (x 1 , . . . , x n , p 1 , . . . , p n ) ∈ R 2n can then be associated to z and defines a chart on the set {z = (x, p) | x ⊥p, p ⊥x}. Note that the point (x,p) lies at the origin of this chart. On the other hand, for every point z = (x, p) such that x ⊥ p there exists an affine chart that has z as its origin. If z = (x, p) lies in some affine chart, then x ⊥ p if and only if 1 + x T p = 0 in this chart.
Linear coordinate transformations in R n+1 are given by nonsingular matrices being defined by a similar partition. Then the coordinate transformation induced on X × P is given by the linear fractional map The projective cross-ratio is a function that associates to 4 collinear points in projective space a number in R ∪ {∞}. Let l ⊂ X be a projective line, let x 1 , x 2 , x 3 , x 4 be points on l. Then the cross-ratio is defined as whenever this ratio makes sense, i.e., whenever the numerator and the denominator are not simultaneously equal to zero. Here the numbers x 1 , x 2 , x 3 , x 4 on the righthand side of the formula stand for the coordinates of the corresponding point in any affine chart on l RP 1 containing the four points.
We shall now use a generalization of the cross-ratio defined in [4, p.4] to define a numerical invariant function on a dense subset of (X × P ) 2 . Let x, x ∈ X and p, p ∈ P , such that no point is orthogonal to both points in the respective dual space simultaneously. Draw a projective line l through x, x and consider the intersection points u, u of l with the projective hyperplanes defined by p, p . Then x, x , u, u are collinear and the expression (u, x ; u , x) makes sense due to the above non-orthogonality assumption. The map taking the quadruple (x, p, x , p ) to the cross-ratio (u, x ; u , x) is well-defined and determines a R ∪ {∞}-valued function f on a dense subset of X × P × X × P . Indeed, if x = x , then (u, x ; u , x) = 0 independently of which projective line l is drawn through x in order to construct the points u, u . By [4,Theorem 2.10] this definition is equi! valent to the definition of the generalized cross-ratio mentioned above. For a pair of points z = (x, p), is well-defined, symmetric in the arguments, finite, and given by the explicit expression in any affine chart on X × P containing both z and z .
Proof. Symmetricity of (z; z ) follows from the symmetry (u, x ; u , x) = (u , x; u, x ) of the cross-ratio. If x = x , then (z; z ) = 0 and the other assertions of the theorem are evident. Suppose that x = x and choose an appropriate affine chart on X × P . Since x ⊥ p and x ⊥ p , we have 1 + p T x = 0, 1 + x T p = 0. Assume the notations in the paragraph preceding the theorem. Since the points u, u lie on the projective line l through x and x , their coordinates are given by u = λ u x + (1 − λ u )x and u = λ u x + (1 − λ u )x , respectively, for some real numbers λ u , λ u . Then l can be affinely parameterized such that the points x, x , u, u are assigned the coordinates 1, 0, λ u , λ u , respectively. Further we have u ⊥ p and u ⊥ p , which can be expressed as 1 + u T p = 1 + u T p = 0, yielding and finally which is now easily seen to be equal to expression (3).
If we now fix the first argument in (3) and consider (z; z ) as a scalar function f (z ) = f (x , p ) on a neighbourhood of z in X × P , then the gradients ∂f ∂x , ∂f ∂p vanish at z = z and hence the second derivative defines an invariant bilinear form Q : T x X × T p P → R. From (3) it follows that the matrix of this form is given by Since det Q = (1 + p T x) −(n+1) , we have that Q is non-degenerated for all z = (x, p) such that x ⊥ p. Let now π x : T (X ×P ) → T X, π p : T (X ×P ) → T P be the canonical projections of the tangent bundle of X × P onto the tangent bundles of the factor spaces. For every z = (x, p) such that x ⊥ p we can then define a bilinear form Q on T z (X × P ) by for all tangent vectors u, v ∈ T z (X × P ).
We now pass to the definition of convex sets and their duals in projective space.
To an open convex set C we can assign the interior K o C of a regular convex cone This cone is determined up to multiplication by −1. It is not hard to check that to C * then corresponds the interior of the dual cone K * C . Hence C * is also convex and (C * ) * = C.

4.
Barriers on projective convex sets. This section contains the main results. We define and investigate barriers on convex sets in projective space and uncover their relation to logarithmically homogeneous barriers on convex cones.

4.1.
Submanifolds of X × P . In this subsection we show that the bilinear form (6) defines a pair of affine connections on n-dimensional submanifolds M ⊂ X × P satisfying certain regularity assumptions. Of special interest will be the case when the form Q itself determines a metric on M . We show that then the connections and the metric form a Codazzi structure.
Let M ⊂ X ×P be a smooth n-dimensional submanifold such that the projections of M on X and P are injective and regular and for every z = (x, p) ∈ M we have x ⊥ p. From the regularity of (5) it then follows that the restriction of the bilinear form Q on M , which we will also denote by Q, is defined everywhere and nondegenerate.
Every affine chart on X defines a chart on M , and M can be represented by a smooth function p = p(x). In these coordinates, the form Q is given by where v = p 1+p T x is a vector-valued function on M . Note that v is not an invariant 1-form, since it does not transform as a 1-form when we pass to an affine coordinate system with different origin. However, we have the following result.
We shall now compare the accelerationsẍ , first computed from the parameters of the original geodesic by applying the coordinate transformation, and then from the geodesic equationẍ = 2 v ,ẋ ẋ in the new , and hencë with e k the k-th basis vector of R n . Thus we getẍ = 2a −2 Since every point z ∈ M can be put at the origin of an appropriate affine chart, the assertion of the theorem becomes equivalent to the condition (b 0 +a −1 0 b T 2 a 2 )a 1 = (−A T + a −1 0 a 1 a T 2 )b 2 , which is easily verified.
Likewise, we can let p be the independent variables on M . Then the form Q is given by In a similar manner, we get the following result. Theorem 4.2. Assume above notations. Then are the Christoffel symbols of an invariant affine connection ∇ on M . The geodesics of this connection project to straight lines in P .
Now suppose that the form Q is symmetric on M . In view of (7), the gradient ∂v ∂x is then also symmetric, and v must itself be the gradient of some scalar function f (x). Note again that f is not an invariant scalar field. For brevity of notation, we shall write f ,i for the partial derivative ∂f ∂x i , f ,ij for ∂ 2 f ∂x i ∂x j etc. In index notation, we then have Q ij = f ,ij + f ,i f ,j . The covariant derivative of Q with respect to the connection (8) is then given by If we put h = exp(2f ), then above expression simplifies to ∇ k Q ij = (2h) −1 h ,ijk . In particular, this covariant derivative is symmetric in all three indices, i.e., the pair (∇, Q) satisfies the Codazzi equation [3, p.21]. Note also that h ,ij = 2hQ ij . Likewise, if we assume p to be the independent variables on M , then w is the gradient of some scalar function f (p) and with the indices now standing for the partial derivative with respect to p. The pair (∇ , Q) then also satisfies the Codazzi equation. From the defining equation for v . We then get Hence we can normalize f such that f + f = log(1 + p T x). Proof. Consider a pointẑ = (x,p) ∈ M and pass to an affine coordinate system witĥ z at the origin. Then the gradients of f, f are zero at the origin, ∂p Denoting the partial derivatives of p with respect to x as above by indices and assuming the Einstein summation convention over repeating indices, we further get Letting x be the independent variables on M , the covariant derivatives of Q at the origin are given by ∇ k Q ij = f ,ijk , ∇ k Q ij = f ,abc p a ,i p b ,j p c ,k . Inserting the above formulas, we finally get, after some computations, that ∇ k Q ij + ∇ k Q ij = 0. This proves the assertion of the theorem at the origin. Sinceẑ was chosen arbitrarily, the identity∇ = 1 2 (∇ + ∇ ) holds everywhere on M . 4.2. Barriers for projective convex sets. Definition 4.4. A barrier for an open convex set C ⊂ X is a smooth n-dimensional submanifold M ⊂ X × P such that the projections on X and P map M diffeomorphically onto C and C * , respectively, and such that the restriction of the bilinear form Q on M is negative definite.
Let M be a barrier for a convex set C. The form g = −Q is then a Riemannian metric on M , and (∇, g), (∇ , g) are mutually dual Codazzi structures. Note that the function f defined in the previous subsection has a negative definite Hessian, since f ,ij = Q ij − f ,i f ,j . The function −f cannot, however, serve as a real-valued barrier, because, as already mentioned, it is not invariant against coordinate transformations. It does not need to tend to ∞ as the argument tends to the boundary either. We can, however, define an invariant function D : M × M → R by D(z, z ) = −(z; z ).
(10) Theorem 4.5. Let M be a barrier for a convex set C. Then the function D is symmetric, non-negative, compatible with the Riemannian metric g if regarded as a distance function, and D(z, z ) = 0 if and only if z = z .
We also have that D tends to +∞ if one of the arguments tends to the boundary of M in X × P , and every such boundary point z = (x, p) satisfies x ⊥ p. However, D is not a true distance function on M , since it violates the triangle inequality. We will not prove these assertions here for reasons of space limitation.
Self-concordance of a projective barrier can be defined similar to the affine case.

4.3.
Relation to logarithmically homogeneous barriers. In this subsection we show that every logarithmically homogeneous barrier F on the interior of a regular convex cone defines a projective barrier M for the corresponding convex set C in projective space, and study the relations of the corresponding notions of self-concordance. In particular, this proves that for every projective convex set C there exists a self-concordant projective barrier, because self-concordant barriers exist for every regular convex cone [1, Section 2.5].
Let C ⊂ X be a convex set and M a barrier for C. Pass to an affine chart with x being the independent variables on M and define a functionF on the cone where f (x) is the function defined in Subsection 4.1. ClearlyF is a logarithmically homogeneous function on K C with homogeneity parameter 1. Proof. Assume above notations. The gradient ofF is given by

BARRIERS ON PROJECTIVE CONVEX SETS 681
We have to prove that the gradient remains invariant under coordinate changes, in which caseF can differ only by an additive constant in different coordinate systems. We shall proceed as in the proof of Theorem 4.1. Choose a point z ∈ M and pass without restriction of generality to an affine chart such that this point lies at the origin. Then we have x = 0, p = 0, v = 0 at this point and we can putx = e 0 . It follows that ∂F ∂x = −e 0 . Consider now a change of coordinates given by (2) and put We shall now compute the gradient of the functionF in the new coordinate system in two different ways, first from the gradient ofF by applying the transformation matrix A, and then by the above formula, using the expressions On the other hand, we have which by the relation a 0 b 0 + b T 2 a 2 = 1 leads to the same expression as in the first case. This proves the invariance of the functionF .
Let us now compute the Hessian ofF . We have But − ∂v ∂x − vv T is the matrix of the metric g and hence positive definite. Therefore ∂ 2F ∂x 2 is also positive definite. Note, however, thatF does not need to be a barrier.

ROLAND HILDEBRAND
Now note that the relation γ = ν−2 √ ν−1 is equivalent to The assertion of the theorem now easily follows.
It is not hard to see that the chain of arguments can be reversed, and every selfconcordant logarithmically homogeneous barrier on the interior of a regular convex cone with homogeneity parameter ν generates a γ-self-concordant projective barrier on the projective convex set corresponding to the cone, where γ and ν are related as in the previous theorem. In particular, since γ ≥ 0, there cannot exist ν-selfconcordant barriers with ν < 2 on cones in dimension 2 or higher. Proof. Let z ∈ M be an arbitrary point and pass to an affine chart with z at the origin. Normalize the function f such that h(0) = 1 2 , where h = exp(2f ). The condition γ = 0 implies ∇ k Q ij = (2h) −1 h ,ijk = 0, and h must be quadratic and hence equal to h(x) = 1 2 (1 − x T Ax) with A = −Q(0) positive definite. Let now B = (−A) 1/2 and perform the coordinate transformation x → Bx, p → B −1 p. Note that f and thus h are invariant with respect to this transformation. Then in the new coordinates h = 1 − ||x || 2 2 , and the assertion of the theorem easily follows.
In particular, it follows that if a regular convex cone in dimension 2 or higher possesses a self-concordant logarithmically homogeneous barrier with parameter 2, then it must be the Lorentz cone, up to linear coordinate changes.

5.
Conclusions. We developed a concept of barriers for convex sets in projective space. The barrier is not defined as a real-valued function on the convex set C, it is rather a bijection between C and its dual C * satisfying certain integrability and positivity properties. As in the affine case, the barrier defines two affine connections ∇, ∇ and a Riemannian metric on the convex set. The connections are not flat, however, and these objects define a Codazzi structure rather than a Hessian structure as in the affine case. The differential-geometric structures are generated by a macroscopic object, the distance-like function (10). This function is in turn generated by the projective cross-ratio. Its affine analog is the function x − x, F (x ) − F (x) , which has similar properties. The theory of projective barriers outlined in this paper can serve as a base for the design of interior-point methods for the solution of optimization problems over convex projective sets. In particular, the function D can serve both as a measure of distance from the central path and as a measure of progress from one iteration to the next. The details will be developed in a subsequent publication. Since the projective barriers introduced in this paper are closely related to logarithmically homogeneous barriers on convex cones, it is to be expected that new developments can be initiated also in this area.