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Uniform bounds on the limiting and marginal derivatives of the
analytic center solution over a set of normalized weights
A.G. Holder
a;∗, R.J. Caron
baDepartment of Mathematics, Trinity University, 715 Stadium Drive, San Antonio, TX 78212-7200, USA
bDepartment of Economics, Mathematics and Statistics, University of Windsor, 401 Sunset Avenue, Windsor, Ont., Canada N9B 3P4
Received 1 June 1998; received in revised form 1 September 1999
Abstract
While properties of the weighted central path for linear and non-linear programs have been studied for years, the eect that weighting the central path has on its limiting derivatives has not been fully explored. We show that the limiting derivatives of the central path for a linear program are uniformly bounded over a set of normalized weights. Consequently, this allows a similar bound for the marginal derivatives of the analytic center solution with respect to changes in the right-hand side.
c
2000 Elsevier Science B.V. All rights reserved.
Keywords:Analytic central path; Computational economics; Sensitivity analysis
1. Introduction
Consider the standard form primal linear program-ming problem
(LP) min{cx:Ax=b; x¿0}
and its associated dual:
(LD) max{yb:yA+s=c; s¿0};
where A∈Rm×n; m6n, rank(A) =m; b∈Rm; c∈ Rn; x∈Rn; s∈Rn; y∈Rm, and c; y, and s are taken to be row vectors. Since linear changes in the right-hand side,b, are considered, the primal and dual feasible regions are denoted by Pb, and D,
respec-tively. Similarly, the primal and dual optimal faces areP∗
b andD∗b, respectively.
∗Corresponding author.
Both the primal and dual feasibility regions are as-sumed to satisfy Slater’s constraint qualication. That is, we assume the existence of anx∈Pb and a pair
(y; s)∈D, such thatx ¿0 ands ¿0. The right-hand
side vector,b, isadmissibleif the corresponding LP satises Slater’s constraint qualication.
The assumption thatbis admissible implies the ex-istence of the omega central path [15],
{(x(; b; !);(y(; b; !); s(; b; !))): ¿0};
where!∈Rn++and the elements are the unique solu-tion to
Ax(; b; !) =b;
y(; b; !)A+s(; b; !) =c;
S(; b; !)x(; b; !) =!:
In the last equality, S(; b; !) is the diagonal matrix formed by s(; b; !). These equations are the neces-sary and sucient Lagrange conditions for
min
is a strictly complementary solution to the linear pro-grams, called theomega analytic center solution. This solution induces theoptimal partition:
B(b) ={i:x∗i(b; !)¿0}
and
N(b) ={1;2;3; : : : ; n} \B(b):
These sets do not depend on any specic choice of
!∈Rn++. If no confusion results, the argument (b) is dropped. The setsB andN are used as subscripts to indicate the components of a vector, or columns of a matrix, that correspond to elements inBandN, respectively.
Sonnevend [16 –18] showed the mathematical pro-gramming community that analytic centers were im-portant because of their connection to interior point algorithms. In fact, a polynomial-time algorithm for linear programming problems seems to require a cen-tering component in its search direction [8]. Because of this, properties of the omega central path are impor-tant and have been investigated by many researchers [3,4,6,10 –12,14,21,22].
In this paper, the limiting derivatives of the! cen-tral path are of interest. For any ¿0, the derivative ofx( ; b; !) is denoted by limiting derivatives was shown by Witzgall et al. [20] and Adler and Monteiro [1]. Guler [13] established the existence of the higher-order limiting derivatives. This result was used in [9] to show that along any given direction b, the analytic center solution is one-sided innitely, continuously dierentiable.
In Section 2, the vector of right-sided derivatives of x∗(b; !) with respect to , is shown to be uni-formly bounded over {!∈Rn++:k!k= 1}. In Sec-tion 3, a similar result is presented for the right-sided marginal derivatives, denoted D+x∗(b; !), where
b=b+pb. Notice that for any admissibleb and
anyb; bis admissible for suciently small.
2. The limiting derivatives of the omega central path
The components ofD+x∗(b; !) are partitioned into
Lemma 1. For any admissibleb;there exists a con-stantM¿0such that
sup{kD+x∗N(b; !)k:!∈Rn++; k!k= 1}6M¡∞:
Proof In all of [1,13,20], it is shown that
D+x∗
sucient optimality conditions for this program are the existence ofu∗(b; !) such that
u∗(b; !)¿0; (2)
Au∗(b; !) = 0; (3)
y∗(b; !)AB=cB; (4)
SN∗(b; !)u∗N(b; !) =!N; (6)
s∗N(b; !)¿0: (7)
Let ( ˆu;( ˆy; sNˆ )) = (u∗(b; e); (y∗(b; e); s∗N(b; e))), whereeis the vector of ones. Then,
ˆ
Combining (6), (13), and (14) with the fact that!is normalized, gives
ˆ
siu∗i(b; !)6eTN!N + ˆsNuˆN6n+ ˆsNuˆN: (15)
The following completes the result and holds from (6), (7) and (15),
where the superscript “+” denotes the Moore–Penrose generalized inverse. Hence, Lemma 1 implies that to boundD+x∗
B(b; !), only
B−1=2XB∗(b; !)(ABB−1=2XB∗(b; !))+ (17)
needs to be bounded. Lemma 3 provides the needed bound, and depends on Lemma 2 and its subsequent
corollary. The absence of a rank requirement makes the statements of Lemma 2 and Corollary 1 slight generalizations of Theorem 1 in [19] due to Stewart. Stewart’s proof, without modication, continues to work for the relaxed statement found in the next lemma. However, the proof of the corollary does not follow from the proof found in [19] without an ad-ditional property of the Moore–Penrose generalized inverse. Dene D++ to be the set of positive diag-onal matrices. For any D∈D++, set MD=D1=2M
and then use this bound to complete the result. Suppose for the sake of attaining a contradiction that K ∩ C6=∅. Then, there exists a sequence
tradiction that the right-hand side of the last equality is positive. Hence, K∩C=∅. Dene
≡ inf
(u;v)∈C×Kkv−uk¿0:
For any xed D∈D++; kPDk= maxkwk= 1kPDwk.
v∈leftnull(DM). SinceD1=2P
Dxis the projection of D1=2xonto col(D1=2M), we have
D1=2PDw =D1=2PDu+D1=2PDv
=D1=2PDu:
Furthermore, becauseD1=2u∈col(D1=2M),D1=2PDu=
D1=2u. Hence PDw=u, and to show that kPDk is bounded we may show that kuk is bounded. With-out loss of generality we assume that u6= 0. Since (u=kuk)∈C and (v=kuk)∈leftnull(DM),
Corollary 1. There existsMsuch that
sup
Lemma 3. The matrix in(17)is uniformly bounded over{!∈Rn++}.
Proof LetD=−B1=2X∗
B(b; !) andMT=AB. Then,
kB−1=2XB∗(b; !)(ABB−1=2XB∗(b; !))+k
=kD(MTD)+k=k(DM)+Dk:
From Corollary 1 there exists a constantM¿0 such
that for all!∈Rn++,
kB−1=2XB∗(b; !)(AB−B1=2XB∗(b; !))+k
6kA+
BkM¡∞:
The following theorem is the main result of this section, and it shows that the limiting derivatives of the omega analytic center solution are uniformly bounded over a set of normalized weights.
Theorem 1. For any admissible right-hand side b; there exists a constantM¿0such that
sup{kD+x∗(b; !)k:k!k= 1; ! ¿0}6M¡∞:
Proof Partition the vector of derivatives to obtain
kD+x∗(b; !)k6kD+xB∗(b; !)k+kD+x∗N(b; !)k:
Lemma 1, Eq. (16), and Lemma 3, imply the existence of anM1andM2such that
sup{kD+x∗(b; !)k:!∈Rn++; k!k= 1}
6kAT
BkM1kANkM2+M2
6M¡∞:
3. The marginal derivatives of the omega central path
In this section, the boundedness of the rst-order limiting derivatives is shown to imply that the rst-order marginal derivatives are also bounded. This follows because Dx∗(b; !) is expressible as
a limiting derivative of an omega central path. The following subset relationships are needed.
Lemma 4(Adler and Monteiro [2]). For a given b and suciently small,
B(b)⊆B(b) and N(b)⊇N(b):
;B=B(b)=B(b). From [7] we know thatB(b) is
invariant on (0;), for suciently small. The es-tablishment of the fact that the marginal derivatives of x∗(b; !) are the limiting derivatives of an omega central path is found in Lemma 5. The main result follows.
Lemma 5(Holder et al. [9]). For a givenb;let {(zB(); zB(); ()):¿0} (18)
be the!central path for
min{:ABzB+ABzB−b=b;
Theorem 2. For anyb; there exists a constantM with
sup{kD+x∗(b0; !)k:k!k= 1; ! ¿0}6M¡∞:
Proof Letvbe the dual slack vector associated with the omega central path in (18). Since the optimal par-tition for (18) is (B|B∪ {}), Eq. (16) and Lemma
The authors thank Harvey Greenberg for his com-ments on early drafts of this article. The rst author
also wishes to thank Andrew Knyazev. The work of the second author is supported by the Natural Sciences and Engineering Research Council of Canada.
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