Two simple proofs for analyticity of the central path in linear programming

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Abstract

Several papers have appeared recently establishing the analyticity of the central path at the boundary point for both linear programming (LP) and linear complementarity problems (LCP). While the proofs for LP are long, proceeding from limiting properties of the corresponding derivatives, the proofs for LCP are very simple, consisting of an application of the implicit function theorem to a certain system of equations. Inspired by the approach for LCP, this paper gives two simple ways of proving the analyticity of the central path for LP. One follows the idea for LCP, the other is based on a proper partition of the system defining the central path.

Introduction

Consider the following pair of dual linear programming (LP) problems:(P)min{cTx|Ax=b,x⩾0},(D)max{bTy|ATy+s=c,s⩾0},where A is an m×n matrix, rank(A)=m<n;c,x,s∈Rn;b,y∈Rm. We restrict our attention to problems satisfying the assumption(AS)thereexistx>0,s>0suchthatAx=b,ATy+s=c.

In the context of interior point methods for solving (P) and (D), it is important to study the properties of certain interior point paths. We recall here the definition and some properties of these paths, called ω-weighted central paths.

For any ω>0,ω∈Rn, define the following μ parameterized systemAx=b,x⩾0,ATy+s=c,s⩾0,Xs=μω,where Xdiag(x) and μ∈R,μ⩾0. Note that for μ=0 system (1) gives necessary and sufficient conditions for optimality of both (P) and (D). Under assumption (AS), for any μ>0 there exists the unique solution (x(μ),y(μ),s(μ)) of (1) such that x(μ)>0,s(μ)>0. The set of points {x(μ),y(μ),s(μ)}μ>0 is called an (ω-weighted) central path. Due to the rank condition for A, we have a one-to-one correspondence between y and s in (1). This enables us to omit y(μ) from the definition of an (ω-weighted) central path.

The usual central path is the ω-weighted central path corresponding to the weight vector ω=(1,…,1)T. In this paper we consider the (ω-weighted) central path corresponding to the arbitrarily chosen weight vector ω>0. We study the properties of this central path under the natural parameterization given in (1). That is, we are interested in the properties of (x(μ),s(μ)) as a function of μ>0.

It is easy to see that the central path has nice analytical properties. Actually, the functionG(x,y,s;μ)=Ax−bATy+s−cXs−μωis real analytic and its Jacobian with respect to (x,y,s) is nonsingular at those points where all components of both x and s are non-zero. Moreover, the central path satisfiesG(x(μ),y(μ),s(μ);μ)=0for each μ>0. Thus, by the implicit function theorem, the central path is analytic in μ for μ>0. That is, it is infinitely differentiable and the Taylor series of (x(μ),s(μ)) for any μ0>0 converges to (x(μ),s(μ)) at a neighborhood of μ0. (More analytical properties of the central path for μ>0 were studied in [17], [15].)

The limiting property of the ω-weighted central path is that there exists a finite limit of (x(μ),s(μ)) as μ↓0, and this limit value forms a strictly complementary solution of (P) and (D). For the proof of this assertion see [9], [6], [8]. Therefore, we can extend the domain of the central path (as a function of μ) to the closed interval [0,∞) by(x(0),s(0))limμ↓0(x(μ),s(μ)).

Now, Eq. (2) holds even for μ=0, but the study of the analytical properties of the central path becomes more complicated since some information contained in system (1) vanishes at μ=0. In fact, system (1) can have many solutions at μ=0, and the Jacobian, G′(x,y,s;0), at these solutions is singular. Consequently, the implicit function theorem does not apply to Eq. (2) at μ=0. Thus effort was concentrated on the analysis of limiting properties of the kth derivatives, (x(k)(μ),s(k)(μ)), as μ↓0. The existence of such limits was established in [1], [16] for k=1 and, in [4] for any k⩾1.

Recently, almost simultaneously, several papers have appeared proving the analyticity of the central path at the boundary [7], [12], [13], [14], [15]. While two of them deal with LP problems [7], [15], the others treat linear complementarity problems (LCP) where the linear programming problem is regarded as a special case. The proofs for LP by Halická [7] and Wechs [15] proceed from the investigation of the limits of explicit formulas for high-order derivatives for μ>0 and also establish the geometric growth of the derivatives at the boundary. Although these proofs for LP are constructive and insightful, they are rather technical and quite long. On the contrary, the proof by Stoer and Wechs [12] (see also [13]) for the monotone LCP is, surprisingly, very simple. It consists of an application of the implicit function theorem to a system of equations, the existence of which was deduced from the system of equations defining the central path. Let us note that the construction of that system is described in the follow-up paper by Stoer et al. [14].

Inspired by the approach for LCP, this paper gives two simple ways of proving the analyticity of the central path for LP. The main idea of both proof techniques is common: we rewrite the system of equations describing the central path, so that the corresponding Jacobian remains nonsingular as μ↓0. One proof technique (Section 3) follows the ideas for LCP from [12], [13], [14] and allows geometric interpretation. The other (Section 4) uses a partition of A, first described in [7]. Here, the system is rewritten in a different form and the resulting Jacobian exhibits some (skew-)symmetric properties. A common conclusion of both these proof techniques is given in Section 5.

Section snippets

Preliminaries

The main tool in developing limiting properties of the central path is the concept of an optimal partition (B,N) of (P) and (D). Recall that the optimal partition is a partition of the index set {1,…,n} such that every strictly complementary optimal solution (x,s) of (P) and (D) has the following property: xi>0,si=0 for iB and xi=0,si>0 for iN. For further details of the optimal partition concept see [2], or [11].

We use the notation xB and xN to refer to the restriction of any vector x∈Rn to

The first approach

As mentioned above, Stoer and Wechs proved the analyticity of the central path for the monotone LCP in [12] and for the sufficient LCP in [13]. The main idea of their proof resides in adding some equations to the original system of equations describing the central path. Then, the enlarged (overdetermined) system of equations has a Jacobian with full column rank, and the existence of a subsystem with a nonsingular Jacobian is ensured. Thus, the analyticity follows by an application of the

The second approach

We now present an alternative procedure that yields a different algebraic description for the central path with non-vanishing Jacobian. First we note that, prior to this section, the symbol was used to define the object on its left-hand side. In this section we also use the symbol =: to define the object on the right.

Let rank(AB)=:r. From the rank conditions on A and AB one has that A can be partitioned in the form (by reordering the columns in AB or in AN, if necessary)A=[ABAN]=:AB11AB12A

Conclusion

The results of 3 The first approach, 4 The second approach imply the analyticity of the central path. In fact, the variables xB,x̃N,y,s̃B,sN and μ enter into both systems (described in Theorem 1, Theorem 2) analytically and thus the application of the analytic version of the implicit function theorem yields the analyticity of the “tilde” path as the function of μ at any μ⩾0. Since xN(μ)=μx̃N(μ), sB(μ)=μs̃B(μ), the analyticity of xN(μ) and sB(μ) is evident.

For some applications it is often

Acknowledgements

The author wishes to thank Milan Hamala for many stimulating discussions on the subject of this paper. Thanks also to Pavol Brunovský, Joseph Gruendler and two anonymous referees for their comments and suggestions which resulted in the improvement of the readability of this paper. This work was supported in part by VEGA grants 1/4302/97 and 1/7675/20.

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