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### Augmented self-concordant barriers and nonlinear optimization problems with finite complexity

NESTOROV, Yurii, VIAL, Jean-Philippe

**Abstract**

In this paper we study special barrier functions for the convex cones, which are the sum of a self-concordant barrier for the cone and a positive-semidefinite quadratric form. We show that the central path of these augmented barrier functions can be traced with linear speed. We also study the complexity of finding the analytic center of the augmented barrier. This problem itself has some interesting applications. We show that for some special classes of quadratic forms and some convex cones, the computation of the analytic center requires an amount of operations independent on the particular data set. We argue that these problems form a class that is endoweed with a property which we call finite polynomial complexity.

NESTOROV, Yurii, VIAL, Jean-Philippe. *Augmented self-concordant barriers and*
*nonlinear optimization problems with finite complexity*. 2000

Available at:

http://archive-ouverte.unige.ch/unige:5851

Disclaimer: layout of this document may differ from the published version.

## Augmented self-concordant barriers and nonlinear optimization problems

## with finite complexity

### Yu. Nesterov

^{∗}### J.-Ph. Vial

^{†}### October 9, 2000

**Abstract**

In this paper we study special barrier functions for the convex cones, which are the sum of a self-concordant barrier for the cone and a positive-semidefinite quadratic form.

We show that the central path of these augmented barrier functions can be traced with linear speed. We also study the complexity of finding the analytic center of the augmented barrier. This problem itself has some interesting applications. We show that for some special classes of quadratic forms and some convex cones, the computation of the analytic center requires an amount of operations independent on the particular data set. We argue that these problems form a class that is endowed with a property which we call finite polynomial complexity.

**Keywords**Augmented barrier, self-concordant functions, finite methods, nonlinear opti-
mization.

*∗*CORE, Catholic University of Louvain, 34 voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium;

e-mail: nesterov@core.ucl.ac.be

*†*HEC, Universit´e de Gen`eve, 40 Bd du Pont d’Arve, CH-1211 Geneva, Switzerland. This author acknowl-
edges the support of the grant 1214-057093-99 of the Swiss National Science Foundation.

**1** **Introduction**

In the recent years, homotopy methods have become a focal point of interest in interior
point methods for convex optimization. The basic idea of these methods is to trace
approximately the*central path trajectoryx(t). This path is defined by*

*x(t) = arg min*

*x**∈**Q* *thc, xi*+*F*(x), (1.1)
where *h·,·i* stands for the scalar product in *R** ^{n}*,

*c*is a fixed cost vector,

*Q*

*⊂*

*R*

*is a bounded convex set and*

^{n}*F*(x) is a convex barrier function for

*Q. The main issue for these*methods is the following:

*What is the class of barrier functions, for which we can follow the central path*
*with a high speed?*

The answer to that question is given by the theory of*self-concordant functions*[3]. More
specifically, the analysis relies on the concept of that *self-concordant barrier.* *F*(x) is a
self-concordant barrier if the following conditions are satisfied. Firstly, *F*(x) must be a
self-concordant function:

*D*^{3}*F(x)[h, h, h]≤*2hF* ^{00}*(x)h, hi

^{3/2}

*∀x∈*int

*Q, h∈R*

^{n}*.*

Secondly, the local norm of the gradient of *F*(x) must be uniformly bounded by some
positive constant*ν*:

*hF** ^{0}*(x), hi

^{2}

*≤νhF*

*(x)h, hi,*

^{00}*∀x∈*int

*Q, h∈R*

^{n}*.*

The term *ν* is called the *parameter* of the barrier. In this framework, it can be proved
that the approximations to the points *x(t) can be updated with a constant complexity*
for the change

*t→*
µ

1*±O(1)*

*√ν*

¶

*t.* (1.2)

The theory of self-concordant functions naturally extends to conic sets [3]. If *Q≡K*
is a proper cone, the theory introduces the further assumption that the self-concordant
barrier *F(x) for this set is* *logarithmically homogeneous*^{1}:

*F(τ x) =F*(x)*−ν*ln*τ,* *x∈*int*K, τ >*0.

With such a barrier one can trace the central path trajectory on the intersection of the
cone*K* with some linear subspace [3].

Note that the problem of tracing the trajectory (1.1) for a standalone logarithmically
homogeneous barrier has no sense. (In this case, the trajectory *{x(t)}**t>0* either does not
exist or it is a straight line.) However, in this paper we show, that there exists a class
of barrier functions for convex cones, for which the tracing problem is meaningful and
not trivial, while its complexity remains on the level of (1.2). We call these functions the
*augmented self-concordant barriers. Such a function Φ(x) is formed as follows:*

Φ(x) = ^{1}_{2}*hQx, xi*+*F*(x),

1The barrier possessing all above mentioned properties is called*ν-normal barrier. The results on* *ν-normal*
barrier that we use in the paper can be found in [2]. A short summary of these results can also be found in [4].

where*Q*is a positive-semidefinite matrix and *F*(x) is a *ν-normal barrier for the coneK.*

Note that Φ(x) is a self-concordant function, but it is *not* a self-concordant barrier. So,
the complexity of tracing its central path cannot be obtained from the standard results
[3].

Augmented self-concordant barriers have already been used in the framework of cut-
ting plane algorithms, either in the main iterative scheme [4], or in an auxiliary problem
[1]. The present paper gives a more systematic treatment and discusses some additional
applications. The more striking result in this paper is perhaps the fact that, for some
nonlinear optimization problems, the complexity of a path-following method is indepen-
dent of the particular data set. This strong property is remindful of the situation in
linear algebra, for which interior point methods have *finite polynomial complexity. For*
this reason, we suggest to call the methods possessing the above property “finite methods
in nonlinear optimization”.

The paper is organized as follows. In Section 2 we define the augmented self-concordant
barriers and study their properties. In Section 3 we analyze the complexity of finding the
analytic center of the augmented barrier. Our complexity results are given in terms of
the quality of the starting point with respect to the global behavior of the quadratic form
*hQx, xi* on the cone *K. The main motivations for such approach are some applications*
(Section 5), for which the complexity of finding the analytic center appears to be inde-
pendent on a particular data set. We conclude the paper with a short discussion of the
complexity results (Section 6).

**2** **Augmented self-concordant barriers**

**Definition 1** *Let* *K* *be a proper cone endowed with a* *ν-normal barrier* *F(x)* *and let* *Q*
*be a positive semidefinite matrix. The function*

Φ(x) = ^{1}_{2}*hQx, xi*+*F(x)*

*is called an* augmented*ν-self-concordant barrier* *for the cone* *K.*

**Remark 1** *An augmented self-concordant barrier with* *Q* *6= 0* *is* not *a self-concordant*
*barrier itself. A self-concordant barrier has the property that the Newton decrement is*
*uniformly bounded on* int*K. An augmented self-concordant barrier is a self-concordant*
*function, but the quadratic augmentation term precludes the uniform boundedness of the*
*Newton decrement.*

Just as in the theory of self-concordant functions, we will use local norms for measuring
the size of the vectors. We need three types of local norms. The first one is defined by
the Hessian of the barrier*F*(x):

*θ**x*(u) =*hF** ^{00}*(x)u, ui

^{1/2}

*,*

*θ*

_{x}*(g) =*

^{∗}*hg,*[F

*(x)]*

^{00}

^{−}^{1}

*gi*

^{1/2}

*,*

*x∈*int

*K, u, g∈R*

^{n}*.*The second one is given by the Hessian of the augmented barrier:

*λ**x*(u) =*hΦ** ^{00}*(x)u, ui

^{1/2}

*,*

*λ*

^{∗}*(g) =*

_{x}*hg,*[Φ

*(x)]*

^{00}

^{−}^{1}

*gi*

^{1/2}

*,*

*x∈*int

*K, u, g∈R*

^{n}*.*

Finally, we need a mixed norm:

*σ**x*(u) =*hΦ** ^{00}*(x)u,[F

*(x)]*

^{00}

^{−}^{1}Φ

*(x)ui*

^{00}^{1/2}

*,*

*σ*

^{∗}*(g) =*

_{x}*hF*

*(x)[Φ*

^{00}*(x)]*

^{00}

^{−}^{1}

*g,*[Φ

*(x)]*

^{00}

^{−}^{1}

*gi*

^{1/2}

*,*

*x∈*int*K, u, g* *∈R*^{n}*.*

Note that Φ* ^{00}*(x) =

*F*

*(x) +*

^{00}*QºF*

*(x). Therefore the above norms are related as follows:*

^{00}*θ**x*(u)*≤λ**x*(u)*≤σ**x*(u), *x∈*int*K, u∈R*^{n}*.* (2.1)
*σ*^{∗}* _{x}*(g)

*≤λ*

^{∗}*(g)*

_{x}*≤θ*

_{x}*(g),*

^{∗}*x∈*int

*K, g∈R*

^{n}*.*(2.2) In particular,

*σ*^{∗}* _{x}*(Φ

*(x))*

^{0}*≤λ*

^{∗}*(Φ*

_{x}*(x))*

^{0}*≤θ*

^{∗}*(Φ*

_{x}*(x)),*

^{0}*x∈*int

*K.*

Note that *λ*^{∗}* _{x}*(Φ

*(x)) is a local norm of the gradient of function Φ(x) with respect to its Hessian. It is uniformly bounded by*

^{0}*ν*

^{1/2}only when the quadratic term is absent (Q= 0).

However, it possible to derive a uniform bound on *σ*_{x}* ^{∗}*(Φ

*(x)) and a weaker bound on*

^{0}*λ*

^{∗}*(Φ*

_{x}*(x)).*

^{0}**Lemma 1** *For any* *x∈*int*K* *we have*

*σ*_{x}* ^{∗}*(Φ

*(x))*

^{0}*≤ν*

^{1/2}

*,*(2.3)

*λ*^{∗}* _{x}*(Φ

*(x))*

^{0}*≤*[hQx, xi+

*ν]*

^{1/2}

*.*(2.4)

**Proof:**

Indeed, since the barrier *F(x) is logarithmically homogeneous, we have for all* *x∈*int*K*
Φ* ^{0}*(x) =

*Qx*+

*F*

*(x) =*

^{0}*Qx−F*

*(x)x= (Q*

^{00}*−F*

*(x))x.*

^{00}Therefore, for all*x∈*int*K*

¡*σ*^{∗}* _{x}*(Φ

*(x))*

^{0}^{¢}

^{2}=

*hF*

*(x)[Q+*

^{00}*F*

*(x)]*

^{00}

^{−}^{1}(Q

*−F*

*(x))x,[Q+*

^{00}*F*

*(x)]*

^{00}

^{−}^{1}(Q

*−F*

*(x))xi. (2.5) Let*

^{00}*H*= (Q*−F** ^{00}*(x))[Q+

*F*

*(x)]*

^{00}

^{−}^{1}

*F*

*(x)[Q+*

^{00}*F*

*(x)]*

^{00}

^{−}^{1}(Q

*−F*

*(x)).*

^{00}Let *V* be an orthogonal matrix of eigenvectors and Λ the associated diagonal matrix of
eigenvalues of [F* ^{00}*(x)]

^{−}^{1/2}

*Q[F*

*(x)]*

^{00}

^{−}^{1/2}. Then

*H*= [F* ^{00}*(x)]

^{1/2}

*V*(Λ

*−I)(Λ +I*)

^{−}^{2}(Λ

*−I)V*

*[F*

^{T}*(x)]*

^{00}^{1/2}

*¹F*

*(x),*

^{00}since for positive *λ**i*, one has (λ*i**−*1)^{2}(λ*i*+ 1)^{−}^{2} *≤*1. Therefore, from equation (2.5),

¡*σ*_{x}* ^{∗}*(Φ

*(x))*

^{0}^{¢}

^{2}

*≤ hF*

*(x)x, xi=*

^{00}*ν.*

On the other hand,

¡*λ*^{∗}* _{x}*(Φ

*(x))*

^{0}^{¢}

^{2}=

*h[Q*+

*F*

*(x)]*

^{00}

^{−}^{1}(Q

*−F*

*(x))x,(Q*

^{00}*−F*

*(x))xi.*

^{00}Using a similar argument as above, we have

(Q*−F** ^{00}*(x))[Q+

*F*

*(x)]*

^{00}

^{−}^{1}(Q

*−F*

*(x))*

^{00}*¹Q*+

*F*

*(x).*

^{00}Thus, (2.4) holds. *2*

The result of Lemma 2.2 is crucial for the analysis of a path-following scheme. We also need an additional result on the convergence of the Newton method on the augmented barrier.

Consider the problem

min*x* [ψ(x)*≡ hc, xi*+ Φ(x)]*.* (2.6)

Let us analyze the performance of the Newton method

*x*+=*x−*[ψ* ^{00}*(x)]

^{−}^{1}

*ψ*

*(x) (2.7) for finding the minimum of*

^{0}*ψ. Note thatψ*is a self-concordant function. Therefore the standard description of the region of quadratic convergence of the Newton scheme is as follows (see [3]):

*λ*^{∗}* _{x}*(ψ

*(x))*

^{0}*≡ hψ*

*(x),[ψ*

^{0}*(x)]*

^{00}

^{−}^{1}

*ψ*

*(x)i*

^{0}^{1/2}

*<*2 3 +

*√*

5 = 0.3819660*. . . .*

Let us show that, due to the quadratic augmentation term, the region of quadratic con- vergence of the Newton scheme for problem (2.6) is wider.

**Theorem 1** *Let* *x*+ *be defined by (2.7) andσ*^{∗}* _{x}*(ψ

*(x))*

^{0}*<*1. Then

*x*+

*∈*int

*K*

*and*

*σ*

_{x}

^{∗}_{+}(ψ

*(x+))*

^{0}*≤λ*

^{∗}

_{x}_{+}(ψ

*(x+))*

^{0}*≤θ*

^{∗}

_{x}_{+}(ψ

*(x+))*

^{0}*≤*

µ *σ*^{∗}* _{x}*(ψ

*(x)) 1*

^{0}*−σ*

^{∗}*(ψ*

_{x}*(x))*

^{0}¶2

*.* (2.8)

*The region of quadratic convergence of the Newton method is then*

*σ*_{x}* ^{∗}*(ψ

*(x))*

^{0}*<*2 3 +

*√*

5*.*
**Proof:**

We prove first that the process is well-defined. Denote
*r*^{2}*≡ hF** ^{00}*(x)(x+

*−x), x*+

*−xi.*

By definition,

(σ^{∗}* _{x}*(ψ

*(x)))*

^{0}^{2}=

*hF*

*(x)[Φ*

^{00}*(x)]*

^{00}

^{−}^{1}

*ψ*

*(x),[Φ*

^{0}*(x)]*

^{00}

^{−}^{1}

*ψ*

*(x)i,*

^{0}=*hF** ^{00}*(x)[ψ

*(x)]*

^{00}

^{−}^{1}

*ψ*

*(x),[ψ*

^{0}*(x)]*

^{00}

^{−}^{1}

*ψ*

*(x)i=*

^{0}*r*

^{2}

*.*

The statement *x*+*∈*int*K* follows from *r <*1. Thus, the scheme (2.8) is well defined.

The first two inequalities in (2.8) follow from (2.2). Let us prove the last one. From
the definition of*x*^{+}, one has *ψ** ^{0}*(x) +

*ψ*

*(x)(x+*

^{00}*−x) = 0. Thus,*

*ψ** ^{0}*(x+) =

*ψ*

*(x+)*

^{0}*−ψ*

*(x)*

^{0}*−ψ*

*(x)(x+*

^{00}*−x),*

=^{h}
Z1
0

[ψ* ^{00}*(x+

*τ*(x+

*−x))−ψ*

*(x)]dτ*

^{00}^{i}(x+

*−x),*

=^{h}
Z1
0

[F* ^{00}*(x+

*τ*(x+

*−x))−F*

*(x)]dτ*

^{00}^{i}(x+

*−x)≡G(x*+

*−x).*

In view of Corollary 4.1.4 [2] the matrix*G* is comparable to*F** ^{00}*(x):

*−*
Ã

*r−r*^{2}
3

!

*F** ^{00}*(x)

*¹G¹*

*r*

1*−rF** ^{00}*(x).

On the other hand, by Theorem 2.1.1 [3], we have *F** ^{00}*(x+)

*º*(1

*−r)*

^{2}

*F*

*(x). Thus,*

^{00}*G[F*

*(x+)]*

^{00}

^{−}^{1}

*G¹*1

(1*−r)*^{2}*G[F** ^{00}*(x)]

^{−}^{1}

*G¹*

*r*

^{2}

(1*−r)*^{4}*F** ^{00}*(x).

We conclude that

³

*θ*_{x}^{∗}_{+}(ψ* ^{0}*(x+))

^{´}

^{2}=

*h[F*

*(x+)]*

^{00}

^{−}^{1}

*ψ*

*(x+), ψ*

^{0}*(x+)i,*

^{0}=*h[F** ^{00}*(x+)]

^{−}^{1}

*G(x*+

*−x), G(x*+

*−x)i,*

*¹* *r*^{2}

(1*−r)*^{4}*hF** ^{00}*(x)(x+

*−x), x*+

*−xi*=

µ *σ*^{∗}* _{x}*(ψ

*(x)) 1*

^{0}*−σ*

^{∗}*(ψ*

_{x}*(x))*

^{0}¶4

*.*
*2*

**Remark 2** *In the above proof we did not use the fact thatF(x)is a self-concordant barrier*
*for a cone. Thus, the result of Theorem 1 is also valid for an arbitrary self-concordant*
*functionF*(x).

Theorem 1 makes it possible to construct a path-following scheme relative to the following central path trajectory:

*x(t) = arg min*

*t* *ψ**t*(x), *ψ**t*(x) =*thc, xi*+ Φ(x), t >0.

Let us analyze one iteration of the path-following scheme.

**Theorem 2** *Assume that for somet >*0 *we have a point* *x∈*int*K* *such that*
*σ*^{∗}* _{x}*(ψ

^{0}*(x))*

_{t}*≤β <*3

*−√*

5.

*Let us chooset*+= (1 + ∆)t *and*

*x*+=*x−*[ψ^{00}* _{t}*(x)]

^{−}^{1}

*ψ*

_{t}

^{0}_{+}(x). (2.9)

*If* ∆ *satisfies the condition*

*|∆|(β*+*√*
*ν)≤*

*√β*
1 +*√*

*β* *−β,*
*thenx*+*∈*int*K* *and* *σ*^{∗}_{x}_{+}(ψ^{0}_{t}_{+}(x+))*≤θ*^{∗}_{x}_{+}(ψ^{0}_{t}_{+}(x+))*≤β.*

**Proof:**

In view of the initial centering condition and Lemma 1, we have

*β≥σ*^{∗}* _{x}*(ψ

^{0}*(x)) =*

_{t}*σ*

^{∗}*(tc+ Φ*

_{x}*(x))*

^{0}*≥tσ*

_{x}*(c)*

^{∗}*−σ*

^{∗}*(Φ*

_{x}*(x))*

^{0}*≥tσ*

_{x}*(c)*

^{∗}*−√*

*ν.*

Thus,*tσ*_{x}* ^{∗}*(c)

*≤β*+

*√*

*ν*and

*σ*^{∗}* _{x}*(ψ

^{0}

_{t}_{+}(x)) =

*σ*

_{x}*(∆tc+*

^{∗}*ψ*

^{0}*(x))*

_{t}*≤ |∆|tσ*

*x*

*(c) +*

^{∗}*σ*

_{x}*(ψ*

^{∗}

_{t}*(x)),*

^{0}*≤ |∆|(β*+*√*

*ν) +β* *≤* *√*

*β*
1+*√*

*β**.*
Hence, in view of Theorem 1,

*θ*_{x}^{∗}_{+}(ψ^{0}_{t}_{+}(x+))*≤*

Ã *σ*_{x}* ^{∗}*(ψ

^{0}

_{t}_{+}(x)) 1

*−σ*

_{x}*(ψ*

^{∗}

_{t}

^{0}_{+}(x))

!_{2}

*≤β.*

*2*
In particular, for*β* = ^{1}_{9} we have the following bounds for ∆: *|∆| ≤* _{4+36}^{5}^{√}* _{ν}*.

**3** **Finding the analytic center**

Consider now the problem of finding the analytic center of the augmented barrier:

*φ** ^{∗}*= min

*x* Φ(x)*≡*min

*x* [^{1}_{2}*hQx, xi*+*F*(x)], (3.1)
where*Q* is a positive-semidefinite matrix. For the sake of simplicity, we assume that the
the solution*x** ^{∗}* of this problem always exists.

When *Q* is non-degenerate, problem to (3.1) has a nice symmetric dual.

**Lemma 2** *Let* *F** _{∗}*(s)

*be a barrier conjugate to*

*F*(x). Then the problem

*φ*

*= min*

_{∗}*s* [^{1}_{2}*hs, Q*^{−}^{1}*si*+*F** _{∗}*(s)] (3.2)

*is dual to (3.1) with zero duality gap:*

*φ*

*+*

^{∗}*φ*

*= 0.*

_{∗}**Proof:**

Indeed, in view of definition of the conjugate barrier we have
min*x* [^{1}_{2}*hQx, xi*+*F(x)] = min*

*x* max

*s* [^{1}_{2}*hQx, xi − hs, xi −F** _{∗}*(s)],

= max

*s* min

*x* [^{1}_{2}*hQx, xi − hs, xi −F** _{∗}*(s)],

= max

*s* [−^{1}_{2}*hs, Q*^{−}^{1}*si −F** _{∗}*(s)].

*2*
In the non-degenerate case, duality makes it possible to find the analytic center of the
augmented barrier either from the primal or from the dual problem. Surprisingly enough,
we shall see in Section 5 that the problems in this pair may have different complexity. Let
us also mention that, in Section 4, we shall discuss duality theory for problem (3.1) with
generically degenerate quadratic function.

Since the primal and dual problems have similar formulation, we focus without loss of generality on the primal problem (3.1). In the rest of this section, we derive complexity estimates for some basic minimization schemes.

Complexity estimates depend on characteristics of problem (3.1). Let us define the
characteristics as follows. Choose some point ˆ*x∈*int*K* and let the characteristics be:

*γ**u*(ˆ*x)* *≥* max

*x* [*hQx, xi*: *−hF** ^{0}*(ˆ

*x), xi*= 1, x

*∈K*],

*γ*

*l*(ˆ

*x)*

*≤*min

*x* [hQx, xi: *hF** ^{00}*(ˆ

*x)x, xi*= 1, x

*∈K*].

Thus, for any*x∈K* we have:

*γ**u*(ˆ*x)hF** ^{0}*(ˆ

*x), xi*

^{2}

*≥ hQx, xi ≥γ*

*l*(ˆ

*x)hF*

*(ˆ*

^{00}*x)x, xi.*(3.3) Note that

*γ*

*l*(ˆ

*x) can be positive even for a degenerate*

*Q. Since*

*F(x) is a self-concordant*barrier, we have

*hF** ^{0}*(¯

*x), xi*

^{2}

*≥ hF*

*(¯*

^{00}*x)x, xi ≥*1

*νhF** ^{0}*(¯

*x), xi*

^{2}

for any *x* *∈* *K. So, we can use different combinations of these quadratic forms in the*
two-side bounds for the variation of the form*hQx, xi* over the cone *K. All these bounds*
are equivalent up to a factor that is polynomial in *ν. Our choice (3.3) is motivated by*
some applications (see Section 5).

**3.1** **Damped Newton scheme**

Let us choose some scaling coefficient*ρ >*0. Consider the following Newton scheme:

*x*0 =*ρx,*ˆ *x**k+1* =*x**k**−* [Φ* ^{00}*(x

*k*)]

^{−}^{1}

1 +*λ**x** _{k}*(Φ

*(x*

^{0}*k*))

*,*

*k*= 0, . . . . (3.4)

**Theorem 3**

*1) If we chooseρ*=

^{√}^{1}

_{ν}*, then the process (3.4) enters the region of quadratic*

*convergence of the Newton method (2.7) at most after*

*O*(νγ*u*(ˆ*x)) +O*
µ

*ν*ln 1
*γ**l*(ˆ*x)*

¶

+*O*(νln*ν).*
*iterations.*

*2) If we choose* *ρ* = *√* ^{1}

*νγ**u*(ˆ*x)**, then the upper bound for the number of iterations is as*
*follows:*

*O*
µ

*ν*ln*νγ**u*(ˆ*x)*
*γ**l*(ˆ*x)*

¶
*.*

**Proof:**

Indeed, (3.4) is a standard damped Newton method for minimizing the self-concordant
function Φ(x). Therefore, it enters the region of the quadratic convergence after at most
*O(Φ(x*0)*−*Φ(x* ^{∗}*)) iterations. Note that in view of inequality (3.3) we have

Φ(x* ^{∗}*) = min

*x* [^{1}_{2}*hQx, xi*+*F*(x)]*≥*min

*x* [^{1}_{2}*γ**l*(ˆ*x)hF** ^{00}*(ˆ

*x)x, xi*+

*F*(x)].

The solution of this problem is uniquely defined by *γ**l*(ˆ*x)F** ^{00}*(ˆ

*x)x*+

*F*

*(x) = 0. Let ¯*

^{0}*x*= ˆ

*x/*^{p}*γ**l*(ˆ*x). Since* *F* is logarithmically homogeneous, *γ**l*(ˆ*x)F** ^{00}*(ˆ

*x)¯x*=

*−*

^{p}

*γ*

*l*(ˆ

*x)F*

*(ˆ*

^{0}*x) and*

*F*

*(¯*

^{0}*x) =*

^{p}

*γ*

*l*(ˆ

*x)F*

*(ˆ*

^{0}*x). Hence, ¯x*is the solution and

Φ^{∗}*≥* ^{1}_{2}*ν(1 + lnγ**l*(ˆ*x)) +F*(ˆ*x).*

On the other hand,

Φ(x0) = ^{1}_{2}*ρ*^{2}*hQˆx,x*ˆ*i*+*F*(ρ*x),*ˆ

*≤* ^{1}_{2}*ρ*^{2}*γ**u*(ˆ*x)hF** ^{0}*(ˆ

*x),xi*ˆ

^{2}+

*F*(ˆ

*x)−ν*ln

*ρ,*

= ^{1}_{2}*ρ*^{2}*γ**u*(ˆ*x)ν*^{2}+*F(ˆx)−ν*ln*ρ.*

Thus,

Φ(x0)*−*Φ(x* ^{∗}*)

*≤*

^{1}

_{2}

*ν*[ρ

^{2}

*γ*

*u*(ˆ

*x)ν−*2 ln

*ρ−*ln

*γ*

*l*(ˆ

*x)−*1]. (3.5) To conclude the proof it suffices to introduce the appropriate values for

*ρ*into (3.5).

*2*

**3.2** **Path-following scheme**

Let us look now at the complexity of a path-following scheme as applied to the problem
(3.1). Assume that the scaling coefficient*ρ >*0 has been fixed to some value (to be made
precise later). Set the tolerance parameter*β* = ^{1}_{9} and let ∆ be defined as in Theorem 2

∆ = 1

*β*+*√*
*ν*

Ã *√*
*β*
1 +*√*

*β* *−β*

!

= 5

4 + 36*√*

*ν.* (3.6)

Denote

*ψ**t*(x) =*−thΦ** ^{0}*(ρ

*x), xi*ˆ + Φ(x).

Consider the following method:

Initialization: *t*0= 1, *x*0=*ρˆx.*

Iteration #k: If*λ**x** _{k}*(Φ

*(x*

^{0}*k*))

*>*2β, then set

*t**k+1*= (1*−*∆)t*k**,* *x**k+1* =*x**k**−*[ψ_{t}^{00}* _{k}*(x

*k*)]

^{−}^{1}

*ψ*

^{0}

_{t}*(x*

_{k+1}*k*), else stop.

(3.7)

Prior to analyzing the complexity of this scheme, we need one auxiliary result on the local
norms defined by a*ν-normal barrier.*

**Lemma 3** *Let* *F(x)* *be a* *ν-normal barrier for the cone* *K. Then for any* *x* *and* *y* *from*
int*K* *and any* *s∈R*^{n}*we have:*

*θ*^{∗}* _{y}*(s)

*≤ −2hF*

*(x), yi ·*

^{0}*θ*

^{∗}*(s), (3.8)*

_{x}*and*

[F* ^{00}*(y)]

^{−}^{1}

*¹*4hF

*(x), yi*

^{0}^{2}[F

*(x)]*

^{00}

^{−}^{1}

*.*(3.9)

**Proof:**

Since *s*is arbitrary, it is straightforward to see that (3.8) implies (3.9).

Let us fix an arbitrary *x* and *y* in int*K. Note that* *x* *∈* int*K, implies that* *F** ^{0}*(x)

*∈*int

*K*

*, and thus*

_{∗}*hF*

*(x), y*

^{0}*i<*0. Let us scale

*y*to

¯

*y*= *νy*

*−hF** ^{0}*(x), yi

*,*

so that*−hF** ^{0}*(x),

*y*¯

*i*=

*ν. Let us fix an arbitrary directions∈R*

*. Denote*

^{n}*u*= [F

*(¯*

^{00}*y)]*

^{−}^{1}

*s/θ*

^{∗}

_{y}_{¯}(s)

and consider two points *y** _{±}* = ¯

*y±u. Both points belong to*

*K, since*

*θ*

*y*¯(u) = 1, and at least one of them belongs to the set

*S* =*{v∈K*: *−hF** ^{0}*(x), vi ≤

*ν}.*

Since *F* is a *ν-normal barrier,* *hF** ^{00}*(x)x, xi =

*ν*and

*F*

*(x)x =*

^{00}*F*

*(x); besides, for any*

^{0}*v∈K,hF*

*(x)v, v*

^{00}*i*=

*hF*

*(x), v*

^{0}*i*

^{2}. Therefore, if we take any

*v∈S*

*⊂K, we have*

(θ*x*(v*−x))*^{2} =*hF** ^{00}*(x)(v

*−x), v−xi,*

=*hF** ^{00}*(x)v, vi+ 2hF

*(x), vi+*

^{0}*ν,*

*≤ hF** ^{0}*(x), vi

^{2}+ 2hF

*(x), vi+*

^{0}*ν.*

Recall that for a*ν-normal barrier−hF** ^{0}*(x), vi ≥0, while, by definition of

*S,−hF*

*(x), vi ≤*

^{0}*−ν. Hence, the quadratic formt*^{2}+ 2t+*ν, with−ν* *≤t≤*0 achieves its maximum at the
end point of the interval. Thus, we may conclude

(θ*x*(v*−x))*^{2} *≤ hF** ^{0}*(x), vi

^{2}+ 2hF

*(x), vi+*

^{0}*ν≤ν*

^{2}

*.*(3.10) Now, using (3.10) and Cauchy-Schwarz inequality, we have for any

*v∈S*and

*g∈R*

^{n}*hg, vi*=

*hg, xi*+

*hg, v−xi ≤ hg, xi*+

*νθ*

_{x}*(g). (3.11) If*

^{∗}*y*+

*∈S, let us takev*=

*y*+ and

*g*=

*s*in (3.11). Then,

*hs,yi*¯ +*θ*_{y}^{∗}_{¯}(s)*≤ hs, xi*+*νθ*^{∗}* _{x}*(s). (3.12)
Now, if

*y*

_{−}*∈S, let us takev*=

*y*

*and*

_{−}*g*=

*−s*in (3.11). Then,

*−hs,yi*¯ +*θ*_{y}^{∗}_{¯}(s)*≤ −hs, xi*+*νθ*^{∗}* _{x}*(s). (3.13)

Together, the inequalities (3.12) & (3.13) are equivalent to
*θ*^{∗}_{y}_{¯}(s)*≤ |hs, x−yi|*¯ +*νθ*^{∗}* _{x}*(s)

*≤*2νθ

^{∗}*(s).*

_{x}It remains to note that in view of homogeneity of the barrier*F*(x) we have
(θ_{y}^{∗}_{¯}(s))^{2}=*h[F** ^{00}*(¯

*y)]*

^{−}^{1}

*s, si*=

*ν*

^{2}

*hF** ^{0}*(x), yi

^{2}

*h[F*

*(y)]*

^{00}

^{−}^{1}

*s, si*=

*ν*

^{2}

*hF** ^{0}*(x), yi

^{2}(θ

_{y}*(s))*

^{∗}^{2}

*.*

*2*

Now we can give a complexity estimate for (3.7).

**Theorem 4** *The scheme (3.7) terminates no more than after* *N* *iterations, where* *N*
*satisfies the following inequality:*

(1*−*∆)^{N}*ν*^{2}

"

*ρ*^{2}*γ**u*(ˆ*x) + 8γ**u*(ˆ*x)ν*^{2}

*γ**l*(ˆ*x)* + 16
*ρ*^{4}*γ*_{l}^{2}(ˆ*x)*

Ã

4ν+ *ρ*^{2}
16*γ**l*(ˆ*x)*

!#

*≤β.*

**Proof:**

Note that*ψ*^{0}_{t}_{0}(x0) = 0, and thus

*θ*_{x}^{∗}_{0}(ψ_{t}^{0}_{0}(x0)) =*σ*^{∗}_{x}_{0}(ψ^{0}_{t}_{0}(x0)) = 0*≤β.*

Hence, we can apply the path-following scheme (3.7), starting from*x*0. In view of Theorem
2 and relation (2.2), for all *k≥*0, we have

*β* *≥θ*_{x}^{∗}* _{k}*(ψ

^{0}

_{t}*(x*

_{k}*k*))

*≥λ*

^{∗}

_{x}*(ψ*

_{k}

_{t}

^{0}*(x*

_{k}*k*)) =

*λ*

^{∗}

_{x}*(−t*

_{k}*k*Φ

*(x0) + Φ*

^{0}*(x*

^{0}*k*)). (3.14) Thus,

*λ*^{∗}_{x}* _{k}*(Φ

*(x*

^{0}*k*))

*≤β*+

*t*

*k*

*λ*

^{∗}

_{x}*(Φ*

_{k}*(x0)). (3.15) We shall try to bound the right-hand side of (3.15). To this end, we introduce the quantity*

^{0}*ω*

*k*=

*−hF*

*(x0), x*

^{0}*k*

*i. In view of Lemma 3,*

[F* ^{00}*(x

*k*)]

^{−}^{1}

*¹*4ω

^{2}

*[F*

_{k}*(x0)]*

^{00}

^{−}^{1}

*.*Equivalently,

*F*

*(x*

^{00}*k*)

*º*

_{4ω}

^{1}2

*k*

*F** ^{00}*(x0). Therefore,

(λ^{∗}_{x}* _{k}*(Φ

*(x0)))*

^{0}^{2}=

*h[Q*+

*F*

*(x*

^{00}*k*)]

^{−}^{1}Φ

*(x0),Φ*

^{0}*(x0)i,*

^{0}*≤ h*[Q+ _{4ω}^{1}2
*k*

*F** ^{00}*(x0)]

^{−}^{1}Φ

*(x0),Φ*

^{0}*(x0)*

^{0}*i,*

=*h[Q*+ _{4ω}^{1}2
*k*

*F** ^{00}*(x0)]

^{−}^{1}(Q

*−F*

*(x0))x0*

^{00}*,*(Q

*−F*

*(x0))x0*

^{00}*i.*

Using the same argument as in Lemma 1, we check that
(Q*−F** ^{00}*(x0))[Q+ 1

4ω_{k}^{2}*F** ^{00}*(x0)]

^{−}^{1}(Q

*−F*

*(x0))*

^{00}*¹*(Q+ 4ω

_{k}^{2}

*F*

*(x0)).*

^{00}Thus,

(λ^{∗}_{x}* _{k}*(Φ

*(x0)))*

^{0}^{2}

*≤ h(Q*+ 4ω

^{2}

_{k}*F*

*(x0))x0*

^{00}*, x*0

*i ≤ hQx*0

*, x*0

*i*+ 4νω

_{k}^{2}

*.*(3.16) In view of (3.15) this implies that the algorithm terminates as soon as

*t**k*[hQx0*, x*0*i*+ 4νω^{2}* _{k}*]

*≤β.*(3.17) Note that

*t*

*k*= (1

*−*∆)

*and*

^{k}*hQx*0*, x*0*i*=*ρ*^{2}*hQˆx,xi ≤*ˆ *ρ*^{2}*γ**u*(ˆ*x)hF** ^{0}*(ˆ

*x),xi*ˆ

^{2}=

*ρ*

^{2}

*γ*

*u*(ˆ

*x)ν*

^{2}

*.*(3.18) Therefore, in order to justify our complexity estimate, we need only to find a uniform upper bound for the special quantity

*ω*

*k*. From (3.17) it is clear that the bound on the number of iterations will involve the quantity

*ω*

*k*only under a logarithm. Consequently, it is not necessary to look for a sharp bound on

*ω*

*k*.

Since*x**k* is close to the central path, we shall derive the bound on*ω**k*from a closer look
at that path. Let*x(t) denote the central path trajectory for problem (3.1). By definition,*
*ψ*^{0}* _{t}*(x(t)) = 0; thus,

Φ* ^{0}*(x(t)) =

*Qx(t) +F*

*(x(t)) =*

^{0}*tΦ*

*(x0), 0*

^{0}*≤t≤*1.

Since Φ(x) is a convex function, we have

Φ(x0) *≥* Φ(x(t)) +*hΦ** ^{0}*(x(t)), x0

*−x(t)i,*

= Φ(x(t)) +*thΦ** ^{0}*(x0), x0

*−x(t)i,*

*≥* Φ(x(t)) +*t(Φ(x*0)*−*Φ(x(t))).

Thus, Φ(x(t))*≤*Φ(x0), for all 0*≤t≤*1. Recall that by (3.14)
*λ*^{∗}_{x}* _{k}*(ψ

^{0}

_{t}*(x*

_{k}*k*))

*≤β.*

Since the function*ψ**t*(x) is self-concordant, we have by inequality 4.1.15 in [2]

*ψ**t** _{k}*(x

*k*)

*≤ψ*

*t*

*(x(t*

_{k}*k*)) +

*ω*

*(β), where*

_{∗}*ω*

*(t) =*

_{∗}*−t−*ln(1

*−t). Note that*

Φ(x*k*) =*ψ**t** _{k}*(x

*k*) +

*t*

*k*

*hΦ*

*(x0), x*

^{0}*k*

*i,*

*≤ψ**t** _{k}*(x(t

*k*)) +

*t*

*k*

*hΦ*

*(x0), x*

^{0}*k*

*i*+

*ω*

*(β),*

_{∗}= Φ(x(t*k*)) +*t**k**h*Φ* ^{0}*(x0), x

*k*

*−x(t*

*k*)

*i*+

*ω*

*(β),*

_{∗}= Φ(x(t*k*)) +*hΦ** ^{0}*(x(t

*k*)), x

*k*

*−x(t*

*k*)i+

*ω*

*(β), (from the central path)*

_{∗}*≤*Φ(x(t*k*)) +*hΦ** ^{0}*(x

*k*), x

*k*

*−x(t*

*k*)i+

*ω*

*(β), (by monotony of Φ*

_{∗}*)*

^{0}*≤*Φ(x0) +*λ*^{∗}_{x}* _{k}*(Φ

*(x*

^{0}*k*))

*·λ*

*x*

*(x*

_{k}*k*

*−x(t*

*k*)) +

*ω*

*(β). (by Cauchy-Schwarz)*

_{∗}Moreover, in view of inequality (4.1.16) in [2], we have
*λ**x** _{k}*(x

*k*

*−x(t*

*k*))

*≤*

*β*

1*−β.*
Using the last two inequalities and inequality (2.4) we get

1

2*hQx**k**, x**k**i*+*F*(x*k*)*≤* ^{1}_{2}*hQx*0*, x*0*i*+*F*(x0) + *β*

1*−β*[hQx*k**, x**k**i*+*ν]*^{1/2}+*ω** _{∗}*(β). (3.19)
Consider the function

*f*(u) =

^{1}

_{2}

*u−β/(1−β)√*

*u*+*ν, withu*=*hQx**k**, x**k**i. With our choice*
of*β(= 1/9), the function* *f(u) is increasing. We can therefore replacehQx**k**, x**k**i*in (3.19)
by any appropriate lower bound. Precisely, by definition of *γ**l*(ˆ*x) and the basic property*
of self concordant barriers, one may write

*hQx**k**, x**k**i ≥γ**l*(ˆ*x)hF** ^{00}*(ˆ

*x)x*

*k*

*, x*

*k*

*i ≥*

^{γ}

^{l}^{(ˆ}

_{ν}

^{x)}*hF*

*(ˆ*

^{0}*x), x*

*k*

*i*

^{2}=

^{1}

_{ν}*ρ*

^{2}

*γ*

*l*(ˆ

*x)ω*

^{2}

_{k}*.*Also, by definition of

*ω*

*k*,

*F*(x*k*)*≥F*(x0) +*hF** ^{0}*(x0), x

*k*

*−x*0

*i*=

*F(x*0) +

*ν−ω*

*k*

*.*Using these bounds in (3.19), we obtain:

1

2ν*ρ*^{2}*γ**l*(ˆ*x)ω*^{2}_{k}*−ω**k* *≤* ^{1}_{2}*hQx*0*, x*0*i*+_{1}_{−}^{β}* _{β}*
h1

*ν**ρ*^{2}*γ**l*(ˆ*x)ω*_{k}^{2}+*ν*
i_{1/2}

+*ω** _{∗}*(β)

*−ν,*

*≤* ^{1}_{2}*hQx*0*, x*0*i*+_{1}_{−}^{β}* _{β}*
q

*γ*

*(ˆ*

_{l}*x)*

*ν* *ρω**k*+ _{1}_{−}^{β}_{β}*√*

*ν*+*ω** _{∗}*(β)

*−ν.*

Since *β* = 1/9, one easily checks that _{1}_{−}^{β}_{β}*√*

*ν*+*ω** _{∗}*(β)

*−ν <*0. Thus, using the bound (3.18) on

*hQx*0

*, x*0

*i, and replacing*

_{1}

_{−}

^{β}*=*

_{β}^{1}

_{8}by its value, we get the following quadratic inequality:

*aω*^{2}_{k}*≤b*+*cω**k**,*
with*a*= _{2ν}^{1} *ρ*^{2}*γ**l*(ˆ*x),b*= ^{1}_{2}*ρ*^{2}*γ**u*(ˆ*x)ν*^{2} and *c*=

µ
1 +^{ρ}_{8}

q*γ** _{l}*(ˆ

*x)*

*ν*

¶ . From the above inequality, we derive the trivial bound

*ω**k**≤*
s

*b*
*a*+ *c*

*a,*
i.e.,

*ω**k**≤*
s

*γ**u*(ˆ*x)ν*^{3}

*γ**l*(ˆ*x)* + 1
*ρ*^{2}*γ**l*(ˆ*x)*

µ
2ν+ *ρ*

4 q

*γ**l*(ˆ*x)ν*

¶
*.*

Finally, squaring the above inequality and using the obvious inequality (u+v)^{2} *≤*2(u^{2}+v^{2})
twice, we get

*ω*^{2}_{k}*≤*2*γ**u*(ˆ*x)ν*^{3}

*γ**l*(ˆ*x)* + 4ν
*ρ*^{4}*γ*_{l}^{2}(ˆ*x)*

Ã

4ν+*ρ*^{2}
16*γ**l*(ˆ*x)*

!
*.*

Inserting the bound on*ω*_{k}^{2} into (3.17), we conclude that the algorithm terminates as soon
as

(1*−*∆)^{k}*ν*^{2}

"

*ρ*^{2}*γ**u*(ˆ*x) + 8γ**u*(ˆ*x)ν*^{2}

*γ**l*(ˆ*x)* + 4
*ρ*^{4}*γ*_{l}^{2}(ˆ*x)*

Ã

16ν+*ρ*^{2}
4 *γ**l*(ˆ*x)*

!#

*≤β.*

*2*
Theorem 4 implies the following complexity estimates on the path-following scheme.

If we can choose *ρ* = 1/^{p}*γ**u*(ˆ*x), then the number of iterations of that scheme is of the*
order of

*O(√*

*ν*ln*ν) +O(ln√*

*νκ(ˆx)),*

where *κ(ˆx) =γ**u*(ˆ*x)/γ**l*(ˆ*x) is a kind of condition number for problem (3.1) with starting*
point ˆ*x.*

If*γ**u*(ˆ*x) cannot be estimated beforehand, we can still chooseρ*as an absolute constant.

The number of iterations of the path-following scheme remains comparable and still quite moderate:

*O(√*

*ν*ln*ν) +O(√*

*ν*ln*γ**u*(ˆ*x)) +O(√*

*ν*ln 1
*γ**l*(ˆ*x)*).

This result basically improves Theorem 3 by a factor *√*
*ν.*

**4** **Constrained analytic center**

Let us look now at the problem of finding a constrained analytic center of the augmented barrier:

*φ** ^{∗}* = min

*x* [ ^{1}_{2}*hQx, xi*+*F*(x) : *Ax*= 0 ], (4.1)
where *Q* is a symmetric positive definite matrix and the matrix *A* *∈* *R*^{m}^{×}* ^{n}* is non-
degenerate (m < n).

Of course, this problem can be solved in two stages. The first stage aims to find a
point ¯*x*in the interior of the cone

*F* =*{x∈K* : *Ax*= 0}.

The second stage computes the analytic center of the augmented barrier restricted to
the subspace *{x* : *Ax* = 0}, by an interior point scheme starting form ¯*x. However, it*
turns out that problem (4.1) has a dual that can be solved in a single stage and yields
a solution to the primal (4.1). (Problem (4.1) is studied in [1] in the framework of the
positive orthant.)

Denote by *Q*^{∗}* _{A}* the following matrix:

*Q*^{∗}* _{A}*=

*Q*

^{−}^{1}

*−Q*

^{−}^{1}

*A*

^{T}^{h}

*AQ*

^{−}^{1}

*A*

^{T}^{i}

^{−}^{1}

*AQ*

^{−}^{1}

*.*Since

*m < n, thenQ*

^{∗}*is a degenerate matrix.*

_{A}**Theorem 5** *Assume that problem (4.1) is strictly feasible. Consider the associated prob-*
*lem*

*φ** _{∗}* = min

*s* [ ^{1}_{2}*hs, Q*^{∗}*A**si*+*F** _{∗}*(s) ]. (4.2)

*Then,*

*1) Problems (4.1) and (4.2) are both solvable. They are dual to one another with a zero*
*duality gap:* *φ** ^{∗}*+

*φ*

*= 0.*

_{∗}*2) The solutions* *x*^{∗}*and* *s*^{∗}*are unique and satisfy the following relations:*

*x** ^{∗}*=

*−F*

_{∗}*(s*

^{0}*),*

^{∗}*s*

*=*

^{∗}*−F*

*(x*

^{0}*), (4.3)*

^{∗}*Q*^{∗}_{A}*s** ^{∗}* =

*x*

^{∗}*,*(4.4)

*hQx*^{∗}*, x*^{∗}*i*=*hs*^{∗}*, Q*^{∗}_{A}*s*^{∗}*i*=*ν.* (4.5)
**Proof:**

Let us show first that both problems achieve their minimum value and that*φ** ^{∗}*+

*φ*

_{∗}*≥*0.

Since int*F 6=* *∅* and *Q* *Â* 0, the solution *x** ^{∗}* of the primal problem exists. Using the
inequality

*F*(x) +

*F*

*(s)*

_{∗}*≥ −hs, xi, we have*

*φ** ^{∗}* = min[

^{1}

_{2}

*hQx, xi*+

*F(x) :*

*Ax*= 0],

*≥ −F**∗*(s) + min

*x:Ax=0*[^{1}_{2}*hQx, xi − hs, xi],*

= [−^{1}_{2}*hs, Q*^{∗}*A**si −F** _{∗}*(s)].

This shows that the function

Φ* _{∗}*(s) =

^{1}

_{2}

*hs, Q*

^{∗}*A*

*si*+

*F*

*(s)*

_{∗}is bounded on *K* from below. Since Φ* _{∗}*(s) is a self-concordant function, its minimum

*s*

*exists. Finally, since*

^{∗}*K*

*is a proper cone, the Hessian*

^{∗}*F*

*(s) is non-degenerate at any*

_{∗}*s∈*int

*K*

*. Thus,*

^{∗}*s*

*is a unique solution to (4.2). By taking the maximum over*

^{∗}*s∈*int

*K*

*, we obtain*

^{∗}*φ*

*+*

^{∗}*φ*

_{∗}*≥*0.

Let us show now that the duality gap is zero. In view of the first order optimality conditions for problem (4.2), we have

*−F*_{∗}* ^{0}*(s

*) =*

^{∗}*Q*

^{∗}

_{A}*s*

^{∗}*.*(4.6)

Since *F** _{∗}*(s) is logarithmically homogeneous,

*hs*^{∗}*, Q*^{∗}_{A}*s*^{∗}*i*=*−hs*^{∗}*, F*_{∗}* ^{0}*(s

*)i=*

^{∗}*ν.*

Denote ¯*x* = *−F*_{∗}* ^{0}*(s

*). Then*

^{∗}*F*

*(¯*

^{0}*x) =*

*−s*

*. Since*

^{∗}*AQ*

^{∗}*= 0, then (4.6) implies*

_{A}*A¯x*= 0.

Besides, we have

Φ* ^{0}*(¯

*x) =*

*Q¯x*+

*F*

*(¯*

^{0}*x) =−QF*

_{∗}*(s*

^{0}*)*

^{∗}*−s*

^{∗}*,*

= *QQ*^{∗}_{A}*s*^{∗}*−s** ^{∗}* =

*−A*

^{T}^{h}

*AQ*

^{−}^{1}

*A*

^{T}^{i}

^{−}^{1}

*AQ*

^{−}^{1}

*s*

^{∗}*.*(4.7) In other words, ¯

*x*solves the first order optimality conditions of problem (4.1), and thus

*x*

^{∗}*≡x.*¯

Using (4.7), we also have

*hQx*^{∗}*, x*^{∗}*i*+*hF** ^{0}*(x

*), x*

^{∗}

^{∗}*i*= 0.

Since *F* is logarithmically homogeneous, we conclude
*hQx*^{∗}*, x*^{∗}*i*=*−hF** ^{0}*(x

*), x*

^{∗}

^{∗}*i*=

*ν.*

Therefore

*φ** ^{∗}* =

^{1}

_{2}

*hQx*

^{∗}*, x*

^{∗}*i*+

*F*(x

*) =*

^{∗}^{1}

_{2}

*ν*+

*F*(

*−F*

_{∗}*(s*

^{0}*)),*

^{∗}=*−*^{1}_{2}*ν−F** _{∗}*(s

*) =*

^{∗}*−*

^{1}

_{2}

*hQ*

^{∗}

_{A}*s*

^{∗}*, s*

^{∗}*i −F*

*(s*

_{∗}*) =*

^{∗}*−φ*

_{∗}*.*

*2*
Theorem 5 shows that the solution of the linearly constrained minimization problem
(4.1) can be obtained by solving problem (4.2), which is of the type studied in the previous
section. Clearly, one can apply a damped Newton Method, or a path-following scheme,
starting from any arbitrary point in int*K** ^{∗}*.

Note that the problem (4.2) has *no linear constraints. All necessary information on*
the matrix *A* is incorporated in the matrix *Q*^{∗}* _{A}*. Thus, this approach can be seen as

*an*

*infeasible start dual technique*for solving the constrained minimization problem (4.1).

**5** **Applications**

In this section we consider some applications of the results of Sections 3 and 4. Theorem
4 tells us that the complexity of the path-following scheme depends on the choice of the ˆ*x*
and on the characteristics *γ**u*(ˆ*x) and* *γ**l*(ˆ*x). In particular, if one can estimate* *γ**u*(ˆ*x), it is*
possible to set the scaling factor at the value*ρ*= 1/^{p}*γ**u*(ˆ*x)) in the path-following scheme*
(3.7) and get the best complexity estimate. In the applications we review, we shall discuss
these issues and propose ways to compute those critical entities. In particular, we shall
see that *γ**u*(ˆ*x) is easily computable in all cases.*

**5.1** **Conic feasibility problem**

Assume we need to solve the following*conic feasibility problem:*

Find*x*^{∗}*∈ F* =*{x∈K*: *Ax*= 0}. (5.1)

To apply the techniques previously devised, we choose a point ˆ*x∈*int*K, and we transform*
the feasibility problem (5.1) into the auxiliary optimization problem

Find *x** ^{∗}* = arg min

*x* [F(x) : *Ax*= 0, *hF** ^{00}*(ˆ

*x)x, xi ≤ν*]. (5.2) Using the first order optimality conditions and the fact that

*F*is logarithmically ho- mogeneous, we can readily verify that (5.2) is equivalent to

min*x* [ ^{1}_{2}*hF** ^{00}*(ˆ

*x)x, xi*+

*F*(x) :

*Ax*= 0 ]. (5.3)