On the Curvature of the Central Path of Linear Programming Theory.^{†}^{†}thanks: Mathematics Subject Classification (MSC2000): 90C51 (Primary), 90C60, 68Q25, 65H10 (Secondary).
Abstract
We prove a linear bound on the average total curvature of the central path of linear programming theory in terms on the number of independent variables of the primal problem, and independent of the number of constraints.
We dedicate this paper with great admiration and affection
to our friend and teacher Steve Smale for his seventy fifth birthday.
1 Introduction.
Consider a linear programming problem in the following primal/dual form:
Here and is an real matrix assumed to have rank , and are given vectors and is nonzero, and are unknown vectors ( is the vector of slack variables).
Our principal result bounds the total curvature of the union of all the central paths associated with all the feasible regions obtained by considering all the possible sign conditions
where is either or
Formal definitions will be given in subsequent sections. The rest of the results in the introduction follow from the next theorem which requires the rest of the paper.
Theorem 1.1.
Let . Let be an matrix of rank , and let and , nonzero. The sum over all sign conditions of the total curvature of the primal/dual central paths (resp. primal central paths, dual central paths) is less than or equal to (resp. , )
Theorem 1.1 allows us to conclude various results on the average curvature of the central paths corresponding to various probability measures on the space of problems. We begin with our main motivating example.
Central paths are numerically followed to the optimal solution of linear programming problems by interior point methods. For relevant background material on interior point methods see Renegar [21]. Our point in studying the total curvature is that curves with small total curvature may be easy to approximate with straight lines. So, small total curvature may contribute to the understanding of why long step interior point methods are seen to be efficient in practice. In DedieuShub [9] we studied the central paths of linear programming problems defined on strictly feasible compact polyhedra (polytopes)^{1}^{1}1A feasible region for a linear programming problem is a polyhedron, a compact polyhedron is a polytope. from a dynamical systems perspective. In this paper we optimistically conjectured that the worst case total curvature of a central path is . Our first average result and main theorem lends some credence to this conjecture, proving it on the average.
If we assume that the primal polyhedron is compact and strictly feasible (i.e., has nonempty interior), then the primal and dual problems have central paths which are each the projection of a primal/dual central path and all these central paths lead to optimal solutions. So for our purposes we will get a meaningful number if we divide the total curvature of the central paths of the all the strictly feasible polytopes arising from all possible sign conditions by the number of distinct strictly feasible polytopes associated with the sign conditions:
where is either or . The cardinality of the set of these polytopes is and equality holds for almost all , see section 6. When equality holds we say is in .
We use Theorem 1.1 to give an upper bound on the sum of the curvatures.
We obtain the following average result.
Main Theorem.
Let . Let be an matrix of rank , and let and , nonzero such that (A,b) is in general position. Then the average total curvature of the primal/dual central paths (resp. primal central paths, dual central paths) of the strictly feasible polytopes defined by (A,b) is less than or equal to (resp. , )
We may also average over more general probability measures on the data , , defining the problem. First we more precisely define the space of problems P and measures we consider. Here I is the open set of , by real matrices,and we assume for convenience that no row of any element of I is identically zero. Let D be the group with elements consisting of those by diagonal matrices whose diagonal entries are all either or . So for , D acts on P by The set of problems defined by the orbit of under the action of D is the same as considering with all possible sign conditions, so each orbit has distinct elements. We say that a probability measure is sign invariant if it is invariant under the action of D, i.e. for all .
We now generalize Theorem 1 once again averaging over problems with a strictly feasible primal polytope.
Let be a signinvariant probability measure on the data , , . If the the set of in P such that are in general position has full measure we will say that is full (for general position). This is the case for example if is supported on a finite union of orbits of D through elements in general position or if is absolutely continuous with respect to the Lebesgue measure, see section 6. For instance, an independent Gaussian probability distribution with zero average and arbitrary variance for each coefficient of the data is sign invariant and full.
Corollary 1.2.
Let and let be a signinvariant and full (for general position) probability measure on P. Let be the set of data with a strictly feasible primal polytope. Let be the conditional probability measure (with respect to ) defined for any measurable by
Then, the average (w.r.t. ) total curvature of the primal/dual central path (resp. primal central path, dual central path) is less than or equal to (resp. , )
This corollary while almost immediate requires a little proof which we carry out in section 6. There is another version of Corollary 1.2 which is perhaps a little more natural form the point of view of regions which have central paths defined for all positive parameter values. We state it below but don’t prove it as the proof is the same as for Corollary 1.2. For a primal/dual central path to exist for all positive parameter values a necessary and sufficient condition is that both primal and dual problems are strictly feasible see [29],[32] . If this is the case we say that the primal/dual polyhedra are jointly strictly feasible. Every strictly feasible primal polytope gives rise to primal/dual jointly strictly feasible polyhedra, but there are more of the later generally among the polyhedra arising from the possible sign conditions in a linear programming problem. Generally the number of jointly strictly feasible primal/dual polyhedra is . We may see this simply since there are generally vertices to the primal polyhedra and at each vertex almost all nonzero select a unique primal polyhedron for which that vertex minimizes the optimization problem, see[1]. When the number is we say that is in joint general position. If we consider a sign invariant probability measure which is full (for joint general position) i.e. the set of problems which are in joint general position has full measure, we get a slight improvement of Corollary 1.2.
Corollary 1.3.
Let and let be a signinvariant and full (for joint general position) probability measure on P. Let be the set of data with joint strictly feasible primal/dual polyhedra. Let be the conditional probability measure (with respect to ) defined for any measurable by
Then, the average (w.r.t. ) total curvature of the primal/dual central path (resp. primal central path, dual central path) is less than or equal to (resp. , )
2 Description of the central path.
When the optimal condition is attained, the primal and dual problems have the same value and the optimality conditions may be written as
where denotes the componentwise product of these two vectors. The primal/dual central path of this problem is the curve , , given by
(2.1) 
where denotes the vector in of all ’s.
The primal central path is the curve , , defined as the curve of minimizers of the function restricted to the primal polyhedron. By the use of Lagrange multipliers one sees that this is the curve defined by the existence of a vector satisfying the equations (2.1). Thus the primal central path is the projection of the primal/dual central path into the subspace.
Similarly, the dual central path is the curve , , defined as the curve of maximizers of the function restricted to the dual polyhedron. By use of Lagrange multipliers one sees that this curve is defined by the existence of vectors satisfying (2.1). So the dual central path is the projection of the primal/dual central path on the subspace.
Note, as we have alluded to in the introduction, that when the primal polyhedron is compact and strictly feasible the primal central path is defined for all and then so are the primal/dual and dual central paths.
3 Curvature.
Let be a map with nonzero derivative: for any . We denote by the arc length:
To the curve is associated another curve on the unit sphere, called the Gauss curve, defined by
which may also be parameterized by the arc length of :
with the length of the curve . The curvature is
see Spivak [28, chapter 1]. In terms of the original parameter we have
(3.2) 
The total curvature is the integral of the norm of the curvature vector:
Thus, is equal to the length of the Gauss curve on the unit sphere To compute we use integral geometry, the next section is devoted to that.
4 An integral geometry formula.
Let , , be a parametric curve contained into the unit sphere with at most a countable number of singularities (i.e. ). The parameter interval is not necessarily finite: Let us denote by the Grassmannian manifold of hyperplanes through the origin contained in . We also denote by the unique probability measure on invariant under the action of the orthogonal group.
Theorem 4.1.
The length of is equal to
where denotes the number of parameters such that : is the number of intersections counted with multiplicity.
Proof.
If is an embedding then Theorem 4.1 follows from Santaló [23, chapter 18, section 6], or also see Shub and Smale [25, section 4], where a similar theorem is proved for projective spaces or Edelman and Kostlan [11]. Now the set of such that may be written as a countable union of intervals on each of which is an embedding. ∎
Definition 4.2.
The parametric curve is transversal to (we also say is transversal to ) when at the intersection points.
Corollary 4.3.
If the number of intersections counted with multiplicity satisfies for all transversal then
Proof.
By a usual application of Sard’s Theorem, see GolubitskyGuillemin [12], nontransversality is a zero measure event. Thus, the integral giving only needs to be evaluated on the set of such that is transversal to . Since is a probability measure we get
∎
In order to bound the number of transversal intersections of the Gauss curve with a hyperplane , we will need the following fact: let
be of class , and assume that we are in the conditions of the Implicit Function Theorem, namely and (the derivative of with respect to the variables) has full rank. Let be the associated implicit function, and , and let denote the derivative of with respect to .
Let denote a hyperplane, with normal vector :
Lemma 4.4.
In the conditions above, if the Gauss curve intersects transversally for , then is a zero of the function
(4.3) 
Moreover, has full rank at that point.
Proof.
Equation (4.31) and (4.32) are the Implicit Function Theorem, and equation (4.33) is the intersection hypothesis. We write as the block matrix:
(4.4) 
where is the linear map . By hypothesis, is invertible. Hence, the block LU factorization of the matrix in (4.4) is:
where using (4.3):
Note that by construction, . Differentiating once with respect to , we obtain (4.32). Differentiating once again,
Solving for and replacing into , we obtain:
We need to show that . Our hypothesis was that . Multiplying equation (3.2) by to the left, we obtain:
Hence, does not vanish and is nonsingular at . ∎
5 A Bézout bound for multihomogeneous systems.
According to Theorem 4.1 to estimate the length of a curve we have to count the number of points in a certain set. To give such an estimate we use the multihomogeneous Bézout Theorem. While this theorem is wellknown to algebraic geometers, topologists and homotopy method theorists, the computation of the Bézout number is usually only carried out in the bihomogeneous case in textbooks. Morgan and Sommese [19] prove the theorem and give a simple description of how to compute the number, which we repeat here.
Let be a system of complex polynomial equations in complex variables. These variables are partitioned into groups with variables into the th group. is said multihomogeneous if for any index there exists a degree such that, for any scalar ,
In this case the system is called multihomogeneous. The Bézout number associated with this system and this structure is defined as the coefficient of in the product
We say that is a zero for when . In that case, for any tuple of complex scalars . For this reason it is convenient to associate a zero to a point in the product of projective spaces . We use the same notation for a point in and for any representative .
We say that a zero is nonsingular when the derivative
is surjective. Notice that this definition is independent of the representative . We have
Theorem 5.1.
(Multihomogeneous Bézout Theorem) Let be a multihomogeneous system. Then the number of isolated zeros of in is less than or equal to . If all the zeros are nonsingular then has exactly zeros.
6 The total curvature of the central path on the average.
To the matrix and the vector we associate the set of admissible points of the primal problem via the set of equalitiesinequalities
We may also consider the other polyhedra contained in the subspace and defined by the inequalities
where is one of the vectors of sign conditions.
Let denote the set of such primal strictly feasible polyhedra contained in the subspace and the set of those which are compact.
Lemma 6.1.
For and almost everywhere,
Proof.
This statement was proved by Buck [8] for and in general position. In particular, and are in general position except in a set of measure zero, Lemma 6.1 holds for and almost everywhere. ∎
Proposition 6.2.
A probability measure on P which is absolutely continuous with respect to Lebesgue measure is full.
Proof.
The set of in P where is not in general position has zero Lebesgue measure by the above lemma and Fubini’s theorem, thus it has zero measure for any measure absolutely continuous with respect to Lebesgue.∎
Now we prove the corollary of the introduction assuming the Main Theorem.
Proof.
The group D acts freely on P, let denote the orbit space. Then we may decompose the measure on the orbits of D. Since is sign invariant each point in the orbit gets equal measure and the same is true for the conditional measure , ie each strictly feasible polytope in the orbit of D gets equal measure when the measure is decomposed on orbits. Now we average over the orbits of points in general position, apply the Main Theorem and then average over ∎
It remains to prove Theorem 1.1.
Lemma 6.3.
For each , with nonempty primal central paths the Gauss curves associated with the central paths and are welldefined.
Proof.
The primal/dual (resp. primal, resp. dual) central path associated with a polyhedron satisfies the system of polynomial equations
(6.5) 
with , and this system is the same for all those polyhedra.
Let denote the diagonal matrix with diagonal entries . Since (equation 6.53), is invertible. The derivative of is equal to
and it factors as:
Therefore, since has full column rank, and , this derivative is nonsingular and we are in the conditions of the Implicit Function Theorem. The speed vector
is the unique solution of the implicit equations:
(6.6) 
The Gauss curve for the primaldual problem is . Notice that because of (6.63), , and cannot be together equal to so that this curve is welldefined.
The Gauss curve associated to the primal (resp. dual) central path is (resp. ). Those curves are well defined, for suppose that . Then equations (6.53) and (6.63) combined give:
Hence, dividing componentwise by and then multiplying by , one obtains:
which contradict the hypothesis . Suppose now . Then, by the same reasoning one obtains . Hence, by (6.61), is in the image of . Then by (6.51), is in the image of , and hence the polyhedron is either one point or unbounded. Thus, we showed that the Gauss curves for the primaldual, primal and dual central paths are welldefined. ∎
A point of the curve is the image under the map
of a point satisfying the systems (6.5) and (6.6) for some . Similarly, a point of the curve (resp. ) is the image of such a point under the map
Lemma 6.4.
It is assumed as above that and that and are defined as above. Let , , be not all zero.

Each transversal intersection of the Gauss curve with the hyperplane
is the image of a nonsingular solution of the polynomial system
(6.7) such that .

Let . Each transversal intersection of the Gauss curve with the hyperplane
is the image of a nonsingular solution of the polynomial system (6.7).

Let and . Each transversal intersection of the Gauss curve with the hyperplane
is the image of a nonsingular solution of the polynomial system (6.7).
Proof.
Part 2 follows from the fact that any transversal intersection of with the hyperplane corresponds to a transversal intersection of with the hyperplane . Indeed, if we set . Then if and only if .
Now, assume that the intersection of with is transversal. Then,
and therefore the intersection of with is also transversal.
The proof of Part 3 is similar. ∎
Proposition 6.5.
Let . Let be an matrix of rank , and let and , nonzero. Then, for any transversal hyperplane , the polynomial system (6.7) has at most
nonsingular solutions with .
If furthermore we have , the number of nonsingular solutions is bounded above by
If instead we have and , the number of nonsingular solutions is still bounded above by
Proof of Theorem 1.1.
The total curvature is the sum of the lengths of the Gauss curves corresponding to strictly feasible regions . According to Corollary 4.3, a bound for the number of intersections (counted with multiplicity) of the associated Gauss curves with a transversal hyperplane gives the bound for the length. Finally, by lemma 6.4 and proposition 6.5 may be taken as is proposition 6.5. ∎
7 Proof of Proposition 6.5
The proof of Proposition 6.5 is quite long, and occupies all of this section. There are actually three cases, that are quite similar and will be treated in parallel. The symbol stands for , or . Each of these cases will be known as the primaldual, the primal and the dual case, respectively.
We proceed as follows:
7.1 Complexification of the equations
The first step is to complexify the equations, i.e. to keep the coefficients fixed and to consider the variables as complex instead of real.
Lemma 7.1.
Proof.
A real root is, in particular, a complex root. It is nondegenerate if and only if the determinant of the Jacobian matrix of the derivative does not vanish. The nonvanishing of this determinant does not depend on whether the matrix is considered as real or complex. ∎
Note that when we complexify the equations, the terms stand for the usual transpose.
A standard application of Bézout’s Theorem implies that:
Lemma 7.2.
The number of nonsingular solutions of (6.7) in in with is bounded above by .
This estimate, while ensuring finiteness, is not sharp enough for our theorem.
7.2 Continuation of nondegenerate roots
More formally, we denote by the set of all complex where has rank , and , , are not simultaneously zero. We also denote by (resp. ) the intersection of with the linear space (resp. and ).
Then, will denote the maximal number of nondegenerate complex roots of (6.7) with , where is one of , , and the maximum is taken over all parameters in . As in Remark 7.2, is finite. Hence this maximal number is attained, and at that point all the nondegenerate complex roots may be continued in a certain neighborhood. Thus,
Lemma 7.3.
The maximal number of nondegenerate complex roots is attained in a certain open set of .
Proof.
Lemma 7.2 implies that is attained for some parameter .
By the Implicit Function Theorem, the nondegenerate complex roots of with can be continued to nondegenerate complex roots with , in a certain neighborhood of the parameter . ∎
7.3 Nondegeneracy at the maximum
The following fact will be needed in the sequel:
Proposition 7.4.
The complex roots of (6.7) with are all nondegenerate, almost everywhere in .
Proof.
is a regular value of if and only if is onto when (See [12, Ch II S1]).
Lemma 7.5.
is a regular value for .
This Lemma guarantees that is a smooth manifold and .
Now we consider the natural projection . By Sard’s theorem, the regular values of have full measure in . Since , is a regular value if and only if is an isomorphism at every point such that . For such systems, all the roots with are nondegenerate. ∎
Proof of Lemma 7.5.
We first reorder the equations and the variables of (6.7) as follows:
(7.8) 
In order to show that has full rank , we will show that a certain submatrix has rank . Namely, we will consider only the derivatives with respect to variables to , and derivation with respect to , , and will be omitted. We obtain the block matrix
Recall that , hence no coordinate of or can vanish and the diagonal matrices and have full rank.
Performing row operations on the previous matrix, one obtains:
for an invertible lower triangular matrix . Since not all of , , can be zero (Lemma 6.3) and has full rank, it remains only to check that has also full rank. This follows from the identity:
and from the fact that has full rank.
Hence, has rank , and we are done. ∎
7.4 Genericity
In this section we show that it is sufficient to bound the number of nondegenerate roots of systems satisfying conditions 1 through 5 of proposition 7.6.
Let . We define as the linear space of all .
Proposition 7.6.
Let . There is a point such that:

The maximal number of nondegenerate complex solutions of (6.7) with is attained at this point.


All the solutions at that point are nondegenerate.

For any , the linear space and the affine space intersect if and only if . In that case, the intersection has dimension .

For any , the linear space and the affine space intersect if and only if . In that case, the intersection has dimension
Proof.
By Lemma 7.3, item 1 holds on an open set . Items 2,3 will fail only on zero measure set (Proposition 7.4). For items 4 and 5, notice that with probability one, and . On the other hand, and . Thus it is easy to see that systems points violating items 3 and 4 will fail in a finite union of zero measure sets.
Hence, items 2 to 5 will hold on a subset of of full measure which has a nonempty intersection with the open set of Lemma 7.3. ∎
This result has the following consequence: to give a bound for the number of nondegenerate solutions of the system (6.7) with , we can replace the initial data by the data of Proposition 7.6.
Also, for convenience, we will count the number of isolated roots of the corresponding system, which is the same.
7.5 Simplification of the equations
Lemma 7.7.
Proof.
This last system is obtained from (