Targetfollowing framework for symmetric cone programming
CHEK BENG CHUA
January 3,2011
Abstract.We extend the target map,together with the weighted barriers and
the notions of weighted analytic centers,from linear programming to general
convex conic programming.This extension is obtained from a novel geomet
rical perspective of the weighted barriers,that views a weighted barrier as a
weighted sum of barriers for a strictly decreasing sequence of faces.Using the
Euclidean Jordanalgebraic structure of symmetric cones,we give an algebraic
characterization of a strictly decreasing sequence of its faces,and specialize
this target map to produce a computationallytractable targetfollowing algo
rithm for symmetric cone programming.The analysis is made possible with
the use of triangular automorphisms of the cone,a new tool in the study of
symmetric cone programming.As an application of this algorithm,we demon
strate that starting from any given any pair of primaldual strictly feasible
solutions,the primaldual central path of a symmetric cone program can be
eciently approximated.
2000 Mathematics Subject Classication.90C25;90C51;52A41.
Key words and phrases.Symmetric cone programming;targetfollowing algorithm;target map;
weighted barrier;weighted analytic centers; ags of faces;triangular transformations.
1.Introduction
In this paper,we consider primaldual interior point algorithms for linear op
timization problems over symmetric cones (a.k.a.symmetric cone programming):
(1.1a) sup
8
<
:
m
X
j=1
b
j
y
j
:
m
X
j=1
y
j
a
j
+x = c;x 2 cl(
)
9
=
;
;
where
is a given (open) symmetric cone in a nitedimensional real vector space
with inner product h;i,a
1
;:::;a
m
;c are given vectors,and b
1
;:::;b
m
are given
real numbers.Its dual problem is the symmetric cone program
(1.1b) inffhc;xi:ha
j
;si = b
j
;1 j m;s 2 cl(
)g:
Without any loss of generality,we assume that the vectors a
1
;:::;a
m
are linearly
independent.With this assumption,(y
1
;:::;y
m
) is uniquely determined by each
x satisfying the equality constraints,and we thus view (y
1
;:::;y
m
) as a function
of x.Henceforth,we shall use only the xcomponent when referring to a feasible
solution.For the purpose of studying interior point algorithms,we also assume
that the primaldual symmetric cone programs have strictly feasible solutions;i.e.,
there exists x;s 2
satisfying the equality constraints in their respective problems.
Primaldual interiorpoint algorithmsrst designed for linear programming
(see,e.g.,[27]),and subsequently extended to semidenite programming (see,e.g.,
[26,Part II]),symmetric cone programming (see,e.g.,[19]) and,recently,homoge
neous cone programming [6]are the most widely used interiorpoint algorithms in
1
2 C.B.CHUA
practice.At the same time,they are able to achieve the best iteration complexity
bound known to date.
The development of primaldual algorithms for symmetric cone programming
began from two very dierent perspectives.Yu.Nesterov and M.Todd [19] de
scribed their algorithm in the context of selfconcordant barriers (see the seminal
work of Yu.Nesterov and A.Nemirovski [18]) by specializing general logarithmi
cally homogeneous selfconcordant barriers to selfscaled barriers.L.Faybusovich
[8],on the other hand,obtained his algorithmby extending a primaldual algorithm
for semidenite programming via the theory of Euclidean Jordan algebras.This
Jordanalgebraic approach had been so successful that it is now the most commonly
used tool in designing interiorpoint algorithms for symmetric cone programming
[1,2,4,21].
In the special case of linear programming,various primaldual pathfollowing al
gorithms were simultaneously analyzed under the targetfollowing framework by B.
Jansen,C.Roos,T.Terlaky and J.Ph.Vial [12].The targetfollowing framework
was rst introduced by S.Mizuno [15] for linear complementarity problems.It was
subsequently used by Jansen et al.as a unifying framework for various primaldual
pathfollowing algorithms for linear programming and algorithms that nd analytic
centers of polyhedral sets.The essential ingredient of this framework is the target
map (x;s) 7!(x
1
s
1
;:::;x
n
s
n
),dened for each pair of positive nvectors (x;s).
An important feature of the target map is its bijectivity between the primaldual
strictly feasible region and the cone of positive nvectors R
n
++
[12,14],whence
identifying the primaldual strictly feasible region with the relatively simple cone
R
n
++
known as the target space (or vspace).Interiorpoint algorithms based on
the target map are known as targetfollowing algorithms,which are conceptually
elegant when viewed as following a sequence of targets in the target space.
Various attempts were made to generalize the concept of target maps to semidef
inite programming [16,17,22],symmetric cone programming [11,23] and general
convex conic programming [24].It is noted here that these extensions of the target
map do not result in targetfollowing algorithms as they are generally not injective
on the whole primaldual strictly feasible region;see Section 1.4.
In this paper,we present an extension of the targetfollowing framework to sym
metric cone programming.This extension is obtained from a novel geometrical
perspective of the weighted barriers,that views a weighted barrier as a weighted
sum of barriers for a strictly decreasing sequence of faces.Using the Euclidean
Jordanalgebraic structure of symmetric cones,we give an algebraic characteriza
tion of a strictly decreasing sequence of its faces,and specialize this target map
to produce a computationallytractable targetfollowing algorithm for symmetric
cone programming.The analysis is made possible with the use of triangular auto
morphisms of the cone,a new tool in the study of symmetric cone programming.
As an application of this algorithm,we demonstrate that starting from any given
any pair of primaldual strictly feasible solutions,the primaldual central path of a
symmetric cone program can be eciently approximated.
1.1.Linear programming revisited.Let us begin by revisiting the special case
of linear programming,where
= R
r
++
= (0;1)
r
.In this case,the minimizer of
the weighted barrier problemwhich is the problem of minimizing the function
x 2
7!
r
X
i=1
!
i
log x
i
+
m
X
j=1
b
j
y
j
over the primal strictly feasible regionand the unique set of Lagrange multipliers
satisfying the Lagrange optimality conditions form the pair of primaldual weighted
analytic centers associated with the weights!
1
;:::;!
r
.Under the target map,a
Targetfollowing and symmetric cone programs 3
pair of primaldual strictly feasible solutions (x;s) is mapped to (!
1
;:::;!
r
) if and
only if it is the pair of primaldual weighted analytic centers associated with the
weights!
1
;:::;!
r
.The weighted sum of logarithmic barriers
x 2
7!
r
X
i=1
!
i
log x
i
is called a weighted logarithmic barrier for R
r
++
associated with (!
1
;:::;!
r
) (or
simply a weighted barrier for R
r
++
).
We now describe a generalization of the weighted barriers,and the notion of
weighted analytic centers,to symmetric cone programming,and generally to convex
conic programmingwhere
is an open convex cone.
First we select a sequence of faces of R
r
+
satisfying R
r
+
= F
1
B BF
r
BF
r+1
=
f0g,where FB
e
F means that
e
F is a proper face of F.This choice of faces determines
a permutation on the index set f1;:::;rg such that F
i
= fx 2 R
r
+
:x
(1)
= =
x
(i1)
= 0g for i = 2;:::;r;r +1.In analogy to ag manifolds,we use the term
ag when referring to such sequence of faces.
Denition 1 (Flag of faces).A ag of faces (or simply ag) of a convex cone K
is a strictly decreasing sequence of faces
cl(K) = F
1
B BF
p
BF
p+1
= f0g:
A ag is said to be complete if it is not a subsequence of another ag of the same
cone.
Next,we consider the weighted sum of logarithmic barriers:
(1.2) f
(!;f)
:x 7!
r
X
i=1
(!
i
!
i1
)f
F
i
(x);
where!= (!
0
;!
1
;:::;!
r
) is a nondecreasing sequence 0 =!
0
<!
1
!
r
of real numbers,f = (F
1
;:::;F
r
;F
r+1
) is a complete ag of R
r
+
,and f
F
i
denotes
the (modied) LegendreFenchel conjugate
1
of the logarithmic barrier of the face
F
i
;i.e.,f
F
i
is the barrier x 7!log x
(i)
log x
(r)
for the open dual cone
int(F
i
)
]
= fx 2 R
r
:x
(i)
;:::;x
(r)
> 0g of the interior int(F
i
) of the face F
i
.
When the complete ag f is paired with the nondecreasing sequence!,we call this
pair a weighted complete ag.
Denition 2 (Weighted complete ag).A weighted complete ag of a convex cone
K is a pair (!;f),with!= (!
0
;!
1
;:::;!
r
) a nondecreasing sequence 0 =!
0
<
!
1
!
r
of real numbers,and f = (F
1
;:::;F
r
;F
r+1
) a complete ag of K.The
sequence!is called a weight sequence,and the numbers!
1
;:::;!
r
are called its
weights.
Using partial summation,we can write this weighted sum logarithmic barriers
as
f
(!;f)
(x) =
r
X
i=1
!
i
(f
F
i
(x) f
F
i+1
(x)) =
r
X
i=1
!
i
log x
(i)
=
r
X
i=1
!
1
(i)
log x
i
:
This is precisely a weighted logarithmic barrier for R
r
++
associated with the weights
(!
1
(1)
;:::;!
1
(r)
).Conversely,every weighted logarithmic barrier for R
r
++
can
1
The (modied) LegendreFenchel conjugate of a function f:S!R on a (nonempty) convex
set S in a Euclidean space with inner product h;i is the function f
]
:s 7!supfhs;xi f(x):
x 2 Sg with domain fs:f
]
(s) < +1g.When f is closed (e.g.,continuous f on open domain S),
we have f
]
]
= f.
4 C.B.CHUA
be written as a weighted sum of the form (1.2) once the reordering of the indices
that puts the weights in nondecreasing order is determined.
If we replace each logarithmic barrier f
F
i
in this weighted sum by the image
e
F
i
of the vector of all ones 1 under the duality map rf
F
i
(),we recover the
image of the primaldual weighted analytic centers (x;s) under the target map:the
image e
F
i
is the 01 vector with nonzero entries precisely at positions (i);:::;(r),
whence e
F
i
e
F
i+1
is the (i)'th unit vector,and subsequently
(!
1
(1)
;:::;!
1
(r)
) =
r
X
i=1
!
i
(e
F
i
e
F
i+1
) =
r
X
i=1
(!
i
!
i1
)e
F
i
:
In summary,for a weighted complete ag (!;f) of R
r
+
,
(1) the nonnegative sum of barriers (1.2) is a weighted logarithmic barrier for
R
r
++
,which we call the weighted barrier associated with (!;f);
(2) the pair of primaldual solutions (x;s) to the weighted barrier problem de
termined by this weighted logarithmic barrier is a pair of weighted analytic
centers,which we call the pair of weighted centers associated with (!;f);
and
(3) the weighted sum
P
r
i=1
(!
i
!
i1
)rf
F
i
(1) is the image of the weighted
analytic centers (x;s) under the target map,which we call the target vector
associated with (!;f).
When the weights are not pairwise distinct,a weighted barrier for R
r
++
is asso
ciated with more than one weighted complete ags since the permutation is not
uniquely determined.Thus we group the weighted complete ags into equivalence
classes according on the weighted barriers with which they associate.
Denition 3 (Equivalence of weighted complete ags).Two weighted complete
ags (!;f) and (e!;
e
f) of an open convex cone K are said to be equivalent if they
have the same weights!
1
;:::;!
r
,and
!
i
>!
i1
=) F
i
=
e
F
i
for i = 1;:::;r.
It is straightforward to verify that two weighted complete ags of R
r
+
are equiv
alent if and only if they associate with the same target vector.Hence we have an
alternative denition for the target map (x;s) 7!(x
1
s
1
;:::;x
r
s
r
):
(x;s) 7!
r
X
i=1
(!
i
!
i1
)rf
F
i
(1);
where (!;f) is a weighted complete ag of R
r
+
such that (x;s) is the pair of weighted
centers associated with (!;f).
1.2.Extension to convex conic programming.We now extend this idea to
linear optimization over a general closed convex cone cl(K).In order to associate a
weighted complete ag (!;f) of the open dual cone K
]
with a weighted barrier,we
would need to x,for each and every face F of cl(K
]
),an a priori logarithmically ho
mogeneous selfconcordant barrier f
]
F
.Since the LegendreFenchel conjugate f
cl(K
]
)
is strictly convex,the weighted barrier problemit determines has a unique solution.
With these barriers,we then dene the weighted barrier,pair of weighted centers,
and target vector associated with (!;f),respectively,as
(1) the nonnegative sum of barriers in (1.2),
(2) the pair of primaldual solutions (x;s) to the weighted barrier problem
determined by the weighted barrier (1.2),and
Targetfollowing and symmetric cone programs 5
(3) the weighted sum
P
r
i=1
(!
i
!
i1
)rf
F
i
(e),where e 2 K is the xed point
of the duality map rf
cl(K
]
)
().
Denition 4 (Target map).The target map for a linear optimization problemover
the closure cl(K) of an open convex cone is the map dened over its primaldual
strictly feasible region by
(x;s) 7!
r
X
i=1
(!
i
!
i1
)rf
F
i
(e);
where (!;f) is a weighted complete ag of K
]
such that (x;s) is the pair of weighted
centers associated with (!;f),and e 2 K is the xed point of the duality map
rf
cl(K
]
)
().
One of our main results generalizes the bijectivity of the target map for linear
programming over the set of primaldual strictly feasible solutions.
Theorem 1 (Bijectivity of target map).The target map is a bijection between the
primaldual strictly feasible region and the open dual cone K
]
.
Proof.We rst demonstrate that the association of each pair of primaldual strictly
feasible solutions (x;s) to a weighted barrier f
(!;f)
is a bijection.It suces to show
that each pair of primaldual strictly feasible solutions (x;s) solves the weighted
barrier problem determined by a unique weighted barrier f
(!;f)
.
To this end,we note that the pair of primaldual strictly feasible solutions (x;s)
solves the weighted barrier problem determined by f
(!;f)
if and only if
s =
r
X
i=1
(!
i
!
i1
)rf
F
i
(x):
Since s 2 K
]
and rf
F
1
(x) = rf
cl(K
]
)
(x) 2 K
]
cl(K
]
),we can nd some
positive
1
such that the dierence s
1
(rf
F
1
(x)) is on the boundary of K
]
.
Let F
2
C cl(K
]
) be the minimal face containing the dierence s
1
(rf
F
1
(x)).
If the minimal face F
2
is not the trivial cone f0g,we repeat this process with s
replaced by the dierence s
1
(rf
F
1
(x)) 2 int(F
2
),x replaced by the projection
Proj
F
2
F
2
x 2 int(F
2
)
]
,and K replaced by the cone int(F
2
).
After a nite number (at most the dimension of K) of iterations of this process,
we have a ag and a corresponding strictly increasing sequence of weights f
1
+
+
i
g
p
i=1
satisfying s =
P
p
i=1
i
rf
F
i
(x).The weighted complete ag (!;f),
obtained by extending this ag and sequence of weights to a weighted complete
ag,then denes a weighted barrier f
(!;f)
associated with (x;s).
In the above argument,we have in fact demonstrated a bijection between ele
ments w 2 K
]
and weighted logarithmic barriers via
w =
r
X
i=1
(!
i
!
i1
)f
F
i
(e);
by taking (x;s) = (e;w).Composing these two bijections proves the theorem.
1.3.Specialization to symmetric cone programming.In the special case of
symmetric cone programming,where
is a symmetric cone,we consider the Eu
clidean Jordan algebra J of rank r associated with
,and use the standard log
determinant barriers
x 7!log det(x)
6 C.B.CHUA
for all faces of cl(
);see Section A.1.This results in the following weighted barrier
and target map associated with the weighted complete ag (!;f):
f
(!;f)
:x 7!
r
X
i=1
(!
i
!
i1
) log det Proj
F
i
F
i
(x);
and
(x;s) 7!
r
X
i=1
(!
i
!
i1
)e
F
i
;
where e
F
i
is the identity element in the Euclidean Jordan subalgebra J
F
i
of J
associated with the symmetric cone int(F
i
).
The association between weighted complete ags (!;f) and targeted pairs of
primaldual strictly feasible solutions (x;s) is given by
s =
r
X
i=1
(!
i
!
i1
)(Proj
F
i
F
i
(x))
1
;
where the inverse is taken as an element of the Euclidean Jordan subalgebra J
F
i
.In
the unweighted case where!= 1,we get the familiar expression s = x
1
,which we
readily express in terms of the Jordan product as the perturbed complementarity
condition
x s = e
with e the identity element of the Euclidean Jordan algebra J.In the general
weighted case,however,we would introduce a partition of s into (!
1
!
0
)s
1
+ +
(!
r
!
r1
)s
r
with s
i
= (Proj
F
i
F
i
(x))
1
;i.e.,
Proj
F
i
F
i
(x) s
i
= e
F
i
:
This is done so that we can enjoy the benet of applying well studied numerical
solution methods for the perturbed complementarity condition.In particular,we
choose to use the NesterovTodd method [19] when computing the search direction,
and measure progress via the function
(x;fs
i
g) 7!
1
p
!
1
v
u
u
t
r
X
i=1
(!
i
!
i1
)kP
Proj
F
i
F
i
(x)
1=2s
i
e
F
i
k
2
;
where P
x
denotes the quadratic representation of x;see Section A.A weighted sum
is used here as the computed search directions are orthogonal under the induced
weighted inner product.The multiplicative factor of
1
p
!
1
scales the induced unit
ball centered at the targeted primaldual solutions so that it just sits within the
primaldual feasible region.
The NesterovTodd method applies scalings to the primaldual variables to get
P
p
i
Proj
F
i
F
i
(x) P
p
1
i
s
i
= e
F
i
;
where p
i
2 int(F
i
) is commonly known as a scaling point:it is chosen so that
after scaling,P
p
i
Proj
F
i
F
i
(x) = P
p
1
i
s
i
.This method is chosen for its simplicity
in the analysis of algorithm,as shall be evident in Section 3.1.One drawback of
partitioning s is that we now have a sequence of r dual variables (s
1
;:::;s
r
) to
solve for;i.e.,there is an increased in the size of the Newton system.This can be
circumvented by using triangular transformations A
i
(see Denition 7) instead of
quadratic representations,together with an appropriate choice of complete ag f;
see Section 3.1 for details.
We thus apply Newton's method to solve
A
t
i
Proj
F
i
F
i
(x) A
1
i
s
i
= e
F
i
;
Targetfollowing and symmetric cone programs 7
where A
i
is a triangular automorphism of int(F
i
) satisfying A
t
i
Proj
F
i
F
i
(x) =
A
1
i
s
i
,and measure progress via the proximity measure
d
F
(x;s;!) =
v
u
u
t
1
!
1
r
X
i=1
(
i
(P
x
1=2s) !
i
)
2
!
i
;
which is obtained naturally from the measure for (x;fs
i
g).Here,
i
() denotes the
i'th smallest eigenvalue;see Section A.1.In short,we prove the following quadratic
convergence result.
Proposition 1.If the target
P
r
i=1
(!
i
!
i1
)e
F
i
is selected in such a way that the
current primaldual iterates (x;s) satisfy
s =
r
X
i=1
i
Proj
F
i
F
i
(x)
1
for some
1
;:::;
r
> 0,and d
F
(x;s;!) <
p
51
2
< 1,then taking a full step along
the search directions (
x
;
s
) determined by the NesterovTodd method using a
suitable triangular automorphism keeps the iterates within the primaldual strictly
feasible region,and satises
d
F
(x +
x
;s +
s
;!)
d
F
(x;s;!)
2
1 d
F
(x;s;!)
:
The above inequality allows us to design a globally convergence targetfollowing
algorithm (see Algorithm 2),with which we show that points on the primaldual
central path can be eciently approximated when given any primaldual strictly
feasible solutions within a prescribed wideneighborhood of the central path;i.e.,in
the set f(x;s) 2
2
:
1
(P
x
1=2s)
hx;si
r
g for some 2 (0;1).This is summarized
in the following theorem.
Theorem 2.Suppose 2 (0;1) is xed.Given any pair of primaldual strictly
feasible solutions (bx;bs) for the primaldual symmetric cone programming problems
(1.1),and any positive real number b,there is a sequence of at most
O
p
r
log
hbx;bsi
r
1
(P
bx
1=2bs)
+
log
hbx;bsi
rb
weights such that Algorithm 2 nds a pair of primaldual strictly feasible solutions
(x;s) satisfying kP
x
1=2
s bek b.
1.4.Comparison with existing notions target maps.As earlier mentioned,
there were other attempts at extending the concept of target maps to semidenite
programming [16,17,22],symmetric cone programming [11,23] and general convex
conic programming [24].
In the works [16,17],the authors consider various notions of target map,and
demonstrate that each of these target maps is injective on some neighborhood of the
primaldual central path.However,it is not known if any of these target maps are
injective on the whole strictly feasible region.Thus,unlike our target map and the
targetfollowing algorithms derived from it,any targetfollowing algorithm based
on the target maps in [16,17] requires all targets to stay within some neighborhood
of the central path.
In the work [22],the authors consider the mapping of primaldual strictly feasible
solutions to the diagonal matrix of eigenvalues of their product as the target map.
This target map is only injective along the primaldual central path,hence the
targetfollowing algorithm based on it can only follow the central path.
8 C.B.CHUA
The target maps considered in [11,23] and [24] are all generalizations of the target
map induced by the NesterovTodd method [17] to symmetric cone programming
and general convex conic programming.Hence,we expect that any targetfollowing
algorithmbased on these will again require all targets to stay within some neighbor
hood of the central path.While our algorithm is also based on the NesterovTodd
approach,we do not use selfadjoint automorphisms P
p
for the primaldual scal
ings,but instead employ triangular scalings;this new tool enables our algorithm to
work beyond neighborhoods of the central path.
1.5.Organization of paper.This paper is organized as follows.In Section 2,we
use the Euclidean Jordanalgebraic characterization of symmetric cones to dene
the notion of weighted analytic centers for symmetric cone programming.This
notion allows us to dene the target map,with which we describe and analyze a
targetfollowing algorithm in Section 3.Finally in Section 4,we apply the target
following algorithm to the problem of nding the primaldual central path of a
symmetric cone program.
2.Target map for symmetric cone programming
Throughout this paper,
denotes a symmetric cone,and (J;) denotes a Eu
clidean Jordan algebra of rank r with identity element e such that the associated
symmetric cone
(J) coincides with
;i.e.,the interior of the cone of squares of J
is the symmetric cone
.Here,and throughout,we equip J with the inner product
h;i:(x;y) 7!tr(x y).We refer the reader to the appendix for more details on
Euclidean Jordan algebras,including various notations used in this paper.
We shall denote the automorphism group of
by G(
) and its connected com
ponent containing the identity by G.We note that since
is selfadjoint,so is its
automorphism group G(
);i.e.for each automorphism A 2 G(
),its adjoint A
t
is
also an automorphism of
.
2.1.Weighted barriers and target map.In order to have a Jordanalgebraic
description of the target map,we would rst need a description of ags of faces of
cl(
).
2.1.1.Weighted ags and Jordan frames.We begin with an algebraic characteri
zation of faces of
given by L.Faybusovich [9,Theorem 2]:each face F Ccl(
)
is the cone of squares cl(
(J
c
2
++c
k
)) of the subalgebra J
c
2
++c
k
in the Peirce
decomposition J = J
c
2
++c
k
J
c
2
++c
k
;c
1
J
c
1
with respect to the idempotent
c
2
+ +c
k
,where
1
c
1
+
2
c
2
+ +
k
c
k
is the type I spectral decomposition
of an arbitrary x 2 relint(F) with 0 =
1
<
2
< <
k
.
We now extend this algebraic characterization to one for ag of faces of
.
Theorem3.Given a ag f = (F
1
;:::;F
p
;F
p+1
) of
,there exists a unique complete
system of orthogonal idempotents C = (c
1
;:::;c
p
) such that
F
i
= cl(
(J
c
i
++c
p
)) for i = 1;:::;p.
Proof.We shall prove by induction on the rank r of the Euclidean Jordan algebra
J.When r = 1,the theorem trivially holds with c
1
= e.
Suppose that the theorem holds for every Euclidean Jordan algebra of rank no
more than some r 2.Consider a Euclidean Jordan algebra J of rank r+1.If p = 1,
then the theorem trivially holds with c
1
= e.Otherwise,by the preceding facial
characterization,there is an idempotent ec 6= e such that F
2
is the cone of squares of
J
ec
.This idempotent is the identity element in the subalgebra J
ec
,and is thus unique.
The Euclidean Jordan algebra J
e
c
has rank at most r.By the inductive hypothesis,
there is a unique systemof orthogonal idempotents (c
2
;:::;c
p
) with c
2
+ +c
p
=ec
Targetfollowing and symmetric cone programs 9
such that F
i
is the cone of squares of the subalgebra J
c
i
++c
p
for i = 2;:::;p.With
c
1
= e ec,F
1
= cl(
) is the cone of squares of J
c
1
++c
p
= J
e
= J.
The above description of ags leads to the following denition of weighted Jor
dan frame and its equivalent classes:there is a natural correspondence between
(equivalent) weighted Jordan frames and (equivalent) weighted complete ags of
.
Denition 5 (Weighted Jordan frame).A weighted Jordan frame of a Euclidean
Jordan algebra J is a pair (!;C),with!= (!
0
;!
1
;:::;!
r
) is a nondecreasing
sequence 0 =!
0
<!
1
!
r
of real numbers,and C a Jordan frame of J.The
sequence!is called a weight sequence,and the real numbers!
1
;:::;!
r
are called
its weights.
Denition 6 (Equivalence of weighted Jordan frame).Two weighted Jordan frames
(!;C) and (e!;
e
C) of a Euclidean Jordan algebra J are said to be equivalent if they
have the same weights (!
0
;!
1
;:::;!
r
),and
!
i
>!
i1
=) c
i
+ +c
r
=ec
i
+ +ec
r
for i = 1;:::;r.In other words,two weighted Jordan frames are equivalent if the
weighted complete ags they determined are equivalent.
2.1.2.Triangular automorphisms.In our targetfollowing algorithm,we will employ
special automorphisms of
that respect the structure of ags of faces.These are
called triangular automorphisms.
Denition 7 (Triangular transformation).Given a Jordan frame C = (c
1
;:::;c
r
),
a linear transformation A 2 L[J;J] is said to be Ctriangular if,for each i 2
f1;:::;rg,the subalgebra J
c
i
++c
r
is an invariant subspace of A,and the restric
tion of A to each subspace in the Peirce decomposition of J with respect to C is
some multiple of the identity transformation.
In matrix theory,the Gauss decomposition of a square matrix A is its decompo
sition into the product LU of a lower triangular matrix with an upper triangular
matrix.This decomposition is obtained as a consequence of the Gaussian elimi
nation process.For a symmetric positive denite matrix,we often further require
that the two triangular matrices have positive diagonal entries,and are transposes
of each other.This symmetric Gauss decomposition A= LL
T
is commonly known
as the Cholesky decomposition.The Cholesky decomposition produces the linear
transformation X 7!LXL
T
that is representable by a triangular matrix under a
suitable choice of basis.This triangular transformation is in fact an automorphism
of the cone of positive denite matrices,and we recover the original matrix X by
applying this triangular automorphism to the identity matrix.
We shall brie y see that this Ctriangular automorphisms of the identity compo
nent of G(
) generalizes the triangular automorphisms to the setting of Euclidean
Jordan algebra.Moreover,these triangular automorphisms can be used in place of
quadratic representation for the primaldual scalings in the NesterovTodd method.
Proposition 2 (Symmetric Gauss decomposition,cf.Theorem VI.3.6 of [7]).For
each Jordan frame C of J,
(1) each x 2
can be uniquely expressed as x = Ae =
e
A
t
e,where A;
e
A 2 G
are Ctriangular;
(2) each A 2 G can be uniquely decomposed into A = BQ =
e
Q
e
B,where B;
e
B 2 G
are Ctriangular and Q;
e
Q 2 G are orthogonal;
Proof.All statements,except the last expression in each item,are proved in The
orem VI.3.6 of [7].To prove these last expressions,we reverse the ordering of
primitive idempotents in C before applying Theorem VI.3.6 of [7].
10 C.B.CHUA
Example 1.For the Jordan algebra of r r real symmetric matrices,a Jordan
frame C is the rtuple (q
1
q
T
1
;:::;q
r
q
T
r
) with the vectors q
1
;:::;q
r
taken from
the columns of an orthogonal matrix Q,and a Ctriangular automorphism A 2 G
takes the form X 7!QLQ
T
XQL
T
Q
T
for some lower triangular matrix L with
positive diagonal entries,and an orthogonal automorphism Q 2 G takes the form
X 7!PXP
T
for some orthogonal matrix P.Thus the rst item in the above
corollary gives the Cholesky and inverse Cholesky decompositions,and the second
item is the QRdecomposition.
Theorem 4 (Triangular NesterovTodd scaling).For each pair (x;s) 2
2
and
each Jordan frame C of J,there exists a unique Ctriangular automorphism A 2 G
satisfying A
t
x = A
1
s.
Proof.By the preceding proposition,there exists a unique Ctriangular automor
phism
e
A 2 G satisfying
e
A
t
e = x,and a unique Ctriangular automorphism
e
B 2 G
satisfying
e
Be = (
e
As)
1=2
.The theorem follows from
A
t
x = A
1
s () (
e
AA)
t
e = (
e
AA)
1
(
e
As) = (
e
AA)
1
P
(
e
As)
1=2
e
() (
e
AA)(
e
AA)
t
e = P
e
Be
e
(Lemma A.2)
=
e
BP
e
e
B
t
e =
e
B
e
B
t
e
(Lemma A.4)
() (
e
AAe)
2
= (
e
Be)
2
()
e
AAe =
e
Be
(Proposition 2)
()
e
AA =
e
B:
Finally,we show that we can always nd a Jordan frame C = (c
1
;:::;c
r
) for
which the unique Ctriangular automorphismin the above proposition scales a given
primaldual pair in
2
to a\diagonal"element;i.e.,an element d with d
c
i
;c
j
= 0
for all i < j in its Peirce decomposition d =
P
r
i=1
d
c
i
+
P
i<j
d
c
i
;c
j
.This will be
subsequently used to give an algebraic proof of bijectivity of the target map.
Theorem5.For each pair (x;s) 2
2
,there exists a Jordan frame C = (c
1
;:::;c
r
)
of J and a unique Ctriangular automorphism A 2 G satisfying
A
t
x = A
1
s =
r
X
i=1
p
i
(P
x
1=2s)c
i
:
Proof.Take any spectral decomposition P
x
1=2s =
P
r
i=1
i
(P
x
1=2s)
e
c
i
.By Propo
sition 2,there is a
e
Ctriangular automorphism B 2 G and an orthogonal auto
morphism Q 2 G such that P
x
1=2
= BQ.We take A to be the automorphism
P
1
x
1=2
P
(P
x
1=2
s)
1=4
Q = Q
t
B
1
P
(P
x
1=2
s)
1=4
Q 2 G,and C = (c
1
;:::;c
r
) to be the r
tuple (Q
t
e
c
1
;:::;Q
t
e
c
r
);so that A
1
s = Q
t
P
1
(P
x
1=2
s)
1=4
P
x
1=2s = Q
t
(P
x
1=2s)
1=2
and
A
t
x = Q
t
P
(P
x
1=2
s)
1=4
P
1
x
1=2
x = Q
t
P
(P
x
1=2
s)
1=4
e = Q
t
(P
x
1=2
s)
1=2
,with
Q
t
(P
x
1=2s)
1=2
= Q
t
r
X
i=1
p
i
(P
x
1=2s)ec
i
=
r
X
i=1
p
i
(P
x
1=2s)c
i
:
It remains to check that C is a Jordan frame and A is a Ctriangular transforma
tion.The former holds since orthogonal automorphisms in G are automorphisms
of J that stabilizes the identity element e;see Theorem A.7.The latter follows
from A = Q
t
B
1
P
(P
x
1=2
s)
1=4
Q and the fact that both B
1
and P
(P
x
1=2
s)
1=4
are
e
Ctriangular.
2.1.3.Weighted analytic centers.In dening weighted barriers for
,we use the
logdeterminant barriers for faces F of cl(
):s 2 int(F) 7!log det
F
(s),where
det
F
(s) denotes the determinant of s as an element of the Jordan subalgebra J
F
that is the linear span of the face F (so that int(F) =
(J
F
)).Its LegendreFenchel
Targetfollowing and symmetric cone programs 11
conjugate is then the composition of the orthogonal projection onto this subalgebra,
and the logdeterminant barrier of the associated symmetric cone.
Remark 1.While the choice of barriers is irrelevant to the bijectivity of the target
map,this choice is taken here for the convenience of designing and analyzing the
targetfollowing algorithmbased on it.In fact,it gives an algebraic means of nding
a weighted Jordan frame that associates with a given pair of weighted centers;see
proof of Theorem 6.
The onetoone correspondence between weighted complete ags and weighted
Jordan frames results in the following denition of weighted barriers.
Denition 8 (Weighted logdeterminant barrier for symmetric cone).The weighted
logdeterminant barrier for
associated with the weighted Jordan frame (!;C) of
the Euclidean Jordan algebra J (or simply weighted barrier) is the function
f
(!;C)
:x 2
7!
r
X
i=1
(!
i
!
i1
) log det
c
i:r
(x
c
i:r
);
where c
i:r
denotes the idempotent c
i
+ +c
r
,and det
c
i:r
(x) denotes the deter
minant of x as an element of the Jordan subalgebra J
c
i:r
.
Up to equivalence of weighted Jordan frame,there is a onetoone correspondence
between the weighted barriers for
and the weighted Jordan frames of J.
The logdeterminant barrier x 7!log det(x) for the symmetric cone
is strictly
convex,and all barriers x 7!log det
c
i:r
(x
c
i:r
) are convex.Therefore the weighted
barrier f
(!;C)
is strictly convex,and hence the weighted barrier problem
inf
8
<
:
f
(!;C)
(x) +
p
X
j=1
b
j
y
j
:
m
X
j=1
y
j
a
j
+x = c;x 2
9
=
;
has a unique solution.We call this the primal weighted analytic center associated
with the weighted Jordan frame (!;C) for the symmetric cone program.
We now consider the Lagrange optimality conditions for this weighted barrier
problem.The gradient of the logdeterminant barrier at the element x is the nega
tion of its inverse x
1
.Therefore the gradient of the weighted barrier at x is
P
r
i=1
(!
i
!
i1
)x
1
c
i:r
,and the Lagrange optimality conditions are
(WCE
(!;C)
)
m
X
j=1
y
j
a
j
+x = c;x 2
;
ha
j
;si = b
j
;1 j m;
s =
r
X
i=1
(!
i
!
i1
)x
1
c
i:r
:
We shall call these conditions the weighted center equations given by the weighted
Jordan frame (!;C),and the unique s satisfying the above conditions the dual
weighted analytic center associated with the weighted Jordan frame (!;C).
By following the proof of Thereom 1,we can show that every pair of primaldual
strictly feasible solutions to the symmetric cone programs (1.1) is a pair of weighted
analytic centers.
Theorem 6 (Completeness of weighted logdeterminant barriers).Given any pair
of primaldual strictly feasible solutions (x;s) to the symmetric cone programs (1.1),
there exist a weighted Jordan frame (!;C) such that (x;s) is the unique solution
to the weighed central equations (WCE
(!;C)
).Moreover,up to equivalence,the
weighted Jordan frame (!;C) is uniquely determined by the pair (x;s).
12 C.B.CHUA
Proof.While this follows fromthe constructive proof of Theorem1 as a special case,
we can instead use the proof Theorem 5 to nd the weighted Jordan frame (!;C).
Indeed in the proof of Theorem5,we construct a Jordan frame C = (c
1
;:::;c
r
) such
that there is unique a Ctriangular automorphism A 2 G satisfying A
t
x = A
1
s =
P
r
i=1
p
i
(P
x
1=2
s)c
i
.
2
From Lemma 1,we deduce (x
c
i:r
)
1
= (A
t
i
(A
t
x)
c
i:r
)
1
,
where A
i
denotes the restriction of A to the subalgebra J
c
i:r
.From Lemma A.2,
we see that this expression is equivalent to A
i
((A
t
x)
c
i:r
)
1
.Therefore,for any
weight sequence!,we have
r
X
i=1
(!
i
!
i1
)(x
c
i:r
)
1
=
r
X
i=1
(!
i
!
i1
)A
i
((A
t
x)
c
i:r
)
1
(Lemma 1) = A
r
X
i=1
(!
i
!
i1
)((A
t
x)
c
i:r
)
1
= A
r
X
i=1
(!
i
!
i1
)
r
X
j=i
1
p
j
(P
x
1=2
s)
c
j
:
In particular,for the weight sequence!= (0;
1
(P
x
1=2s);:::;
r
(P
x
1=2s)),the above
expression simplies to A
P
r
i=1
p
i
(P
x
1=2s)c
i
= s.
2.1.4.Target map.For each pair of primaldual strictly feasible solutions (x;s) of
the symmetric programs (1.1),Theorem 6 states that,up to equivalence,there
exists a unique weighted Jordan frame (!;C) satisfying s =
P
r
i=1
(!
i
!
i1
)x
1
c
i:r
,
where c
i:r
denotes the idempotent c
i
+ +c
r
.With this weighted Jordan frame
(!;C),we dene the target map as
T:(x;s) 7!
r
X
i=1
(!
i
!
i1
)c
i:r
=
r
X
i=1
!
i
c
i
:
We note here that the idempotent c
i:r
= e
c
i:r
= r(log det
c
i:r
)(e
c
i:r
),where the
identity element e is the xed point of the duality map y 7!r(log det(y)) =
y
1
.
The following theoremis a special case of Theorem1,and has a constructive alge
braic proof,obtained by replacing the geometric argument in the proof of Theorem
1 with the algebraic version in the proof of Theorem 6.
Theorem 7 (Bijectivity of target map for symmetric cone programming).The
target map for the symmetric cone programs (1.1) is a bijection between the primal
dual strictly feasible region and the symmetric cone
.
3.Targetfollowing algorithms for symmetric cone programming
Using the target map T dened in the previous section,we propose the following
targetfollowing framework.
Algorithm 1.(Targetfollowing framework for symmetric cone programming)
Given a pair of primaldual strictly feasible solutions (x
in
;s
in
) and a target w
out
2
.
(1) Set (x
+
;s
+
) = (x
in
;s
in
),and w
+
= T (x
in
;s
in
).
(2) Repeat the following steps until w
+
is close to w
out
,
(a) Select the next target w
++
leading towards w
out
.
(b) Compute an approximation (x
++
;s
++
) of the preimage T
1
(w
++
).
(c) Update (x
+
;s
+
) (x
++
;s
++
) and w
+
w
++
.
(3) Output (x
out
;s
out
) = (x
+
;s
+
).
2
This involves a QRdecomposition in the case of semidenite programming.
Targetfollowing and symmetric cone programs 13
The two main steps in this framework are the selection of the next target w
++
and the computation of the next pair of iterates (x
++
;s
++
).In the next section,
we consider the problem of computing the next pair of iterates.
3.1.Approximating weighted analytic centers.We consider the problem of
approximating the weighted analytic centers determined by the weighted center
equations (WCE
(!;C)
),given a weighted Jordan frame (!;C) that denes the next
target w
++
and a pair of current iterates (x
+
;s
+
).For simplicity of notations,we
shall denote by c
i:r
the idempotent c
i
+ + c
r
,by J
i
and
i
,respectively,the
Jordan subalgebra J
c
i:r
and its associated symmetric cone
(J
c
i:r
),and by G
i
the
identity component of G(J
i
).
3.1.1.NesterovTodd scaling.We begin by writing the last equation in the weighted
center equations (WCE
(!;C)
) as
s
r
X
i=1
(!
i
!
i1
)s
i
= 0(3.1a)
x
c
i:r
x
i
= 0;i = 1;:::;r(3.1b)
x
i
s
i
= c
i:r
;i = 1;:::;r:(3.1c)
For the moment,we cast aside the rst two equations and consider the application
of a primaldual NesterovToddtype scaling in (3.1c):
(fx
i
g;fs
i
g) 7!(fA
t
i
x
i
g;fA
1
i
s
i
g)
where A
i
is some automorphism in G
i
such that A
t
i
x
i
= A
1
i
s
i
for each i.The
bilinear equations (3.1c) are invariant under this transformation since x
i
s
i
= c
i:r
if and only if A
t
i
x
i
A
1
i
s
i
= c
i:r
for any A
i
2 G
i
;see Lemma A.5.The advantage
of using the NesterovToddtype scaling is that we can simplify the linearization of
A
t
i
x
i
A
1
i
s
i
= c
i:r
by scaling with L
1
A
t
i
x
i
= L
1
A
1
i
s
i
to get
A
t
i
x
i
+A
1
i
s
i
= (A
t
i
x
i
)
1
A
t
i
x
i
:
As we turn out attention to the rst two equations in (3.1),we quickly realize
that the automorphisms A
i
used in the primaldual scalings of the bilinear equations
(3.1c) should be chosen so that for some automorphism A 2 G,
(3.2) (A
t
x)
c
i:r
= A
t
i
x
c
i:r
8x 2 J 8i 2 f1;:::;rg;
and
(3.3) A
1
r
X
i=1
(!
i
!
i1
)s
i
=
r
X
i=1
(!
i
!
i1
)A
1
i
s
i
8s
i
2 J
i
:
The next lemma shows that these conditions do hold if we take A 2 G to be
Ctriangular,and take A
i
to be its restriction to the subalgebra J
i
.
Lemma 1.If A
i
is the restriction of a nonsingular Ctriangular transformation
A,then both (3.2) and (3.3) hold.
Proof.By denition of A
i
,we have As
i
= A
i
s
i
for all s
i
2 J.Therefore for all
s
i
2 J
i
with i 2 f1;:::;rg,
A
r
X
i=1
(!
i
!
i1
)A
1
i
s
i
=
r
X
i=1
(!
i
!
i1
)AA
1
i
s
i
=
r
X
i=1
(!
i
!
i1
)s
i
:
Since J
i
is invariant under A and the subspaces in a Peirce decomposition are
pairwise orthogonal,we have for all s
i
2 J
i
A
t
i
x
c
i
;s
i
= hx
c
i
;A
i
s
i
i = hx;As
i
i =
A
t
x;s
i
=
(A
t
x)
c
i
;s
i
:
14 C.B.CHUA
In summary,we will transform the primaldual variables by (x;s) 2 J
2
7!
(A
t
x;A
1
s) and (x
i
;s
i
) 2 J
2
i
7!(A
t
i
x
i
;A
1
s
i
),where A 2 G is Ctriangular and
A
i
2 G
i
is the restriction of A to J
i
.We further require that A
t
i
x
i
= A
1
i
s
i
for
each i 2 f1;:::;rg at the current primaldual iterates (x
+
;s
+
),where x
i
= (x
+
)
c
i
and s
i
2 J
i
is such that s
+
=
P
r
i=1
(!
i
!
i1
)s
i
.
From x
i
= (x
+
)
c
i
,A
t
i
x
i
= A
1
i
s
i
and s
+
=
P
r
i=1
(!
i
!
i1
)s
i
,we arrive at
s
+
=
r
X
i=1
(!
i
!
i1
)A
i
A
t
i
(x
+
)
c
i
=
r
X
i=1
(!
i
!
i1
)A(A
t
x
+
)
c
i
= AMA
t
x
+
;
where M:x 7!
P
r
i=1
(!
i
!
i1
)x
c
i
.Since Mis not an automorphism of
,we
cannot expect AMA
t
x
+
2
in general,thence the above equation may not be
satised with any choice of A 2 G.On the other hand,if (A
t
x
+
)
c
i
;c
j
= 0 for all
i < j,then we can replace Mwith the automorphism D = P
P
r
i=1
p
!
i
c
i
2 G.This
happens whenever the next target w
++
is selected in such a way that the current
primaldual iterates (x
+
;s
+
) satisfy
(3.4) s
+
=
r
X
i=1
(e!
i
e!
i1
)(x
+
)
1
c
i:r
for some weight sequence e!;i.e.,we select f to be some complete ag
e
f such that
(x
+
;s
+
) is the pair of weighted analytic centers associated with the weighted com
plete ag (e!;
e
f).We note that this complete ag
e
f can be obtained from the con
struction in the proof of Theorem 6.
Remark 2.This assumption means that we only need to (and,in fact,only allowed
to) choose the weight sequence!when selecting the next target.Thus the analysis
in this paper only allow the algorithm to target at the collection of targets with
specic weight sequence,but nonspecic complete ag;i.e.,targets that share the
same set of eigenvalues.This is not an issue if the nal target is a multiple of the
identity element e;i.e.,if the algorithm aims to locate points on the central path.
For all other purposes,we would need to resort to another approach described in
[5],which unfortunately has a more involved analysis.
Under the above assumption,the following lemma shows that with the choice
D = P
P
r
i=1
p
!
i
c
i
2 G,we are able to nd Ctriangular automorphism A 2 G
satisfying s
+
= ADA
t
x
+
.
Lemma 2.There exists a unique Ctriangular automorphism A 2 G satisfying
A
t
x
+
= D
1
A
1
s
+
,where D 2 G is the automorphism P
P
r
i=1
p
!
i
c
i
.Moreover,if
the next target w
++
is selected in such a way that the current primaldual iterates
(x
+
;s
+
) satisfy (3.4) for some weight sequence e!,then A
t
x
+
= D
1
A
1
s
+
=
P
r
i=1
p
e!
i
=!
i
c
i
Proof.By Theorem 4,there is a unique Ctriangular automorphism
e
A 2 G satis
fying
e
A
t
x
+
=
e
A
1
s
+
.We can then take A to be the Ctriangular automorphism
e
AD
1=2
2 G,and check that A
t
x
+
= D
1=2
e
A
t
x
+
= D
1=2
e
A
1
s
+
= D
1
A
1
s
+
.
Uniqueness of A follows from that of
e
A.Moreover,if (3.4) holds,we see from
the proof of Theorem 6 that
e
A can be chosen such that
e
A
t
x
+
=
e
A
1
s
+
=
P
r
i=1
p
e!c
i
;subsequently,D
1
A
1
s
+
= D
1=2
e
A
1
s
+
= D
1=2
P
r
i=1
p
e!
i
c
i
=
P
r
i=1
p
e!
i
=!
i
c
i
.
Let A 2 G be the Ctriangular automorphism in the lemma,and denote its
restriction to J
i
by A
i
.We then choose s
i
= A
i
A
t
i
x
i
2 J
i
,and check that s
+
=
Targetfollowing and symmetric cone programs 15
P
r
i=1
(!
i
!
i1
)s
i
:using lemma 1,we deduce x
i
= (A
t
d)
c
i:r
= A
t
i
d
c
i:r
,where
d denotes the element
P
r
i=1
p
e!
i
=!
i
c
i
,whence s
i
= A
i
d
c
i:r
;we then have
r
X
i=1
(!
i
!
i1
)s
i
=
r
X
i=1
(!
i
!
i1
)A
i
d
c
i:r
=
r
X
i=1
(!
i
!
i1
)Ad
c
i:r
= ADd = s
+
:
With the primaldual scaling (x;s;fx
i
g;fs
i
g) 7!(A
t
x;A
1
s;fA
t
i
x
i
g;fA
1
i
s
i
g),the
linearization of the rewritten last equation (3.1) of the weighted center equations at
the current iterate (x
+
;s
+
;fx
i
= (x
+
)
c
i:r
g;fs
i
= A
i
A
t
i
x
i
g),after scaling by L
1
d
c
i:r
,
becomes
s
r
X
i=1
(!
i
!
i1
)
s
i
= 0;(3.5a)
(
x
)
c
i
x
i
= 0;i = 1;:::;r;(3.5b)
A
t
i
x
i
+A
1
s
i
= d
1
c
i:r
d
c
i:r
=
r
X
j=i
!
j
e!
j
p
e!
j
!
j
c
j
;i = 1;:::;r:(3.5c)
The weighted sum of these equations,with weights (!
i
!
i1
),is
r
X
i=1
(!
i
!
i1
)(A
t
x
1
)
c
i:r
+A
1
r
X
i=1
(!
i
!
i1
)
s
i
=
r
X
j=i
r
!
j
e!
j
(!
j
e!
j
)c
j
:
Thus the search directions (
x
;
s
) are obtained by solving
(3.6)
m
X
j=1
y
j
a
j
+
x
= 0;
r
X
i=1
ha
j
;
s
i = 0;1 j m;
r
X
i=1
(!
i
!
i1
)(A
t
x
)
c
i:r
+A
1
s
=
r
X
j=i
r
!
j
e!
j
(!
j
e!
j
)c
j
:
Since the linear operator x 2 J 7!
P
r
i=1
(!
i
!
i1
)x
c
i:r
is positive denite,the
search directions are uniquely determined.
3.1.2.Proximity measure.Proximity of the iterates to the weighted analytic centers
is measured in terms of (fx
i
g;fs
i
g) via the backward error
(3.7) d
F
(fy
i
g;fw
i
g;(!;C))
def
=
1
p
!
1
v
u
u
t
r
X
i=1
(!
i
!
i1
)kP
y
1=2
i
w
i
c
i:r
k
2
dened on
(J
1
J
r
) (J
1
J
r
).This error is induced by the inner
product h;i
!
:(fu
i
g;fv
i
g) 7!
1
!
1
P
r
i=1
(!
i
!
i1
) hu
i
;v
i
i on the Cartesian product
J
1
J
r
of Euclidean Jordan algebras,which is chosen because the search
directions (f
x
i
g;f
s
i
g) satisfy hf
x
i
g;f
s
i
gi
!
= 0.The factor 1=
p
!
1
is the
greatest factor so that
d
F
(fy
c
i:r
g;fw
i
g;(!;C)) < 1 =)
r
X
i=1
(!
i
!
i1
)w
i
2
;
which is a consequence of the following lemma.
16 C.B.CHUA
Lemma 3.For w =
P
r
i=1
(!
i
!
i1
)w
i
,
d
F
(fy
c
i:r
g;fw
i
g;(!;C))
v
u
u
t
1
!
1
r
X
i=1
(
i
(P
y
1=2
w) !
i
)
2
!
i
:
Proof.By Lemma A.3,P
y
1=2
i
w
i
and A
y
i
w
i
shares the same spectrum for any au
tomorphism A
y
i
2 G(
i
) satisfying A
t
y
i
c
i:r
= y
i
.In particular,we may use a C
triangular transformation A
y
satisfying A
t
y
e = y (see Proposition 2) and take A
y
i
to be the restriction of A
y
to J
i
;this results in kP
y
1=2
c
i:r
w
i
c
i:r
k = kA
y
i
w
i
c
i:r
k =
kA
y
w
i
c
i:r
k,whence!
1
d
F
(fy
i
g;fw
i
g;(!;C))
2
=
P
r
i=1
(!
i
!
i1
)kA
y
w
i
c
i:r
k
2
.
In terms of the Pierce decomposition with respect to C,
!
1
d
F
(fy
i
g;fw
i
g;(!;C))
2
=
r
X
i=1
(!
i
!
i1
)
r
X
j=i
((A
y
w
i
)
c
j
1)
2
+
r
X
i=1
2(!
i
!
i1
)
X
ij<k
k(A
y
w
i
)
c
j
;c
k
k
2
=
r
X
j=1
j
X
i=1
(!
i
!
i1
)((A
y
w
i
)
c
j
1)
2
+2
X
j<k
j
X
i=1
(!
i
!
i1
)k(A
y
w
i
)
c
j
;c
k
k
2
:
Using Cauchy's inequality and the triangle inequality,we get
!
1
d
F
(fy
i
g;fw
i
g;(!;C))
2
r
X
j=1
1
!
j
j
X
i=1
(!
i
!
i1
)(A
y
w
i
)
c
j
!
j
!
2
+2
X
j<k
1
!
j
j
X
i=1
(!
i
!
i1
)(A
y
w
i
)
c
j
;c
k
2
=
r
X
j=1
1
!
j
(A
y
w)
c
j
!
j
2
+2
X
j<k
1
!
j
(A
y
w)
c
j
;c
k
2
;
where w =
P
r
i=1
(!
i
!
i1
)w
i
.The lemma then follows fromthe next theorem.
Theorem 8 (cf.Lemma 3 of [3];cf.HomanWielandt Inequality).For any
0 <!
1
!
n
,any x 2 J and any Jordan frame C,
r
X
i=1
1
!
i
(x
c
i
!
i
)
2
+2
X
i<j
1
!
i
x
c
i
;c
j
2
r
X
i=1
1
!
i
k
i
(x) !
i
k
2
:
Proof.By expanding both sides of the desired inequality,it is clear that it suces
to bound the sum
P
r
i=1
1
!
i
x
2
c
i
+2
P
i<j
1
!
i
x
c
i
;c
j
2
from below by
P
r
i=1
1
!i
i
(x)
2
.
Let x =
P
i
(x)ec
i
be a spectral decomposition.From x
2
=
P
i
(x)
2
ec
i
,we get
2
6
4
(x
2
)
c
1
.
.
.
(x
2
)
c
r
3
7
5
=
2
6
4
(ec
1
)
c
1
(ec
r
)
c
1
.
.
.
.
.
.
.
.
.
(ec
1
)
c
r
(ec
r
)
c
r
3
7
5
2
6
4
1
(x)
2
.
.
.
n
(x)
2
3
7
5
;
where the matrix on the right side of the equation is doublystochastic.By the
HardyLittlewoodPolya Theorem [10],we have
P
r
i=k
(x
2
)
c
i
P
r
i=k
i
(x)
2
for any
Targetfollowing and symmetric cone programs 17
k 2 f1;:::;rg.Consequently
r
X
i=1
1
!
i
x
2
c
i
+2
X
i<j
1
!
i
x
c
i
;c
j
2
=
1
!
1
r
X
i;j=1
x
c
i
;c
j
2
r
X
k=2
1
!
k1
1
!
k
r
X
i;j=k
x
c
i
;c
j
2
1
!
1
r
X
i=1
(x
2
)
c
i
r
X
k=2
1
!
k1
1
!
k
r
X
i=k
(x
2
)
c
i
1
!
1
r
X
i=1
i
(x)
2
r
X
k=2
1
!
k1
1
!
k
r
X
i=k
i
(x)
2
=
r
X
i=1
1
!
i
i
(x)
2
proves the lemma.
We note that for x
i
= (x
+
)
c
i:r
and s
i
= A
i
A
t
i
x
i
,
d
F
(fx
i
g;fs
i
g;(!;C)) =
1
p
!
1
v
u
u
t
r
X
i=1
(!
i
!
i1
)
r
X
j=i
e!
j
!
j
!
j
2
=
v
u
u
t
1
!
1
r
X
i=1
(e!
i
!
i
)
2
!
i
:
This suggest the following proximity measure for (x;s):
(3.8) d
F
(x;s;!)
def
=
v
u
u
t
1
!
1
r
X
i=1
(
i
(P
x
1=2
s) !
i
)
2
!
i
:
From this denition,it is straightforward to deduce that
i
(P
x
1=2
s)
!
i
2 [1 d
F
(x;s;!);1 +d
F
(x;s;!)]:
Lemma 4.For all 2 [0;1],
d
F
(fx
i
+
x
i
g;fs
i
+
s
i
g;(!;C)) (1 ) +
2
!
1
r
X
i=1
(!
i
e!
i
)
2
e!
i
(1 ) +
2
2
1
;
where denotes d
F
(fx
i
g;fs
i
g;(!;C)).
Proof.Denote by ex
i
,es
i
,
ex
i
and
es
i
the scaled elements A
t
i
x
i
,A
1
i
s
i
,A
t
i
x
i
and
A
1
i
s
i
respectively.
We rst show that d
F
is invariant under the primaldual scaling (fy
i
g;fw
i
g) 7!
(fA
t
i
y
i
g;fA
t
i
w
i
g).From Lemma A.2,we have P
A
t
i
y
i
= A
t
i
P
y
i
A
i
.We can then
follow the method of F.Alizadeh and S.H.Schmieta [1,Proposition 21] to show that
P
y
1=2
i
w
i
and P
(A
t
i
y
i
)
1=2A
1
i
w
i
share the same set of eigenvalues by demonstrating
that their quadratic representations are similar to each other.This shows that each
summand in d
F
,and hence d
F
,is invariant under the primaldual scaling.
18 C.B.CHUA
From the rst two sets of equations (3.5a) and (3.5b) of the linearization,we see
that the search directions (f
x
i
g;f
s
i
g) are orthogonal under the inner product
h;i
!
.Thus
kf
x
i
gk
2
!
+kf
s
i
gk
2
!
= kf
x
i
+
s
i
gk
2
!
=
r
X
i=1
(!
i
!
i1
)
r
X
j=i
!
j
e!
j
p
e!
j
!
j
!
2
=
r
X
i=1
!
i
!
i
e!
i
p
e!
i
!
i
2
=
r
X
i=1
(!
i
e!
i
)
2
e!
i
where we have used the scaled third set of equations (3.5c) in the second equal
ity.Therefore kf
x
i
gk
!
;kf
s
i
gk
!
q
P
r
i=1
(!
i
e!
i
)
2
e!
i
.For each i 2 f1;:::;rg,
we have k
x
i
k = k(
x
)
c
i:r
k k
x
k = k
x
1
k,and hence k
x
i
k k
x
1
k
1
p
!
1
kf
x
i
gk
!
1
p
!
1
q
P
r
i=1
(!
i
e!
i
)
2
e!
i
.
From the third set of equations (3.5c) of the linearization,we have
(ex
i
+
e
x
i
) (es
i
+
e
s
i
) c
i:r
=
e
x
i
e
s
i
+(
ex
i
e
s
i
+
e
x
i
es
i
) +
2
ex
i
es
i
c
i:r
= (1 )(ex
i
es
i
c
i:r
) +(ex
i
es
i
+
ex
i
es
i
+ex
i
es
i
c
i:r
) +
2
ex
i
es
i
= (1 )(ex
i
es
i
c
i:r
) +
2
ex
i
es
i
:
Therefore,by Lemma 30 of [1],
kP
(ex
i
+
e
x
i
)
1=2(es
i
+
es
i
) c
i:r
k k(ex
i
+
ex
i
) (es
i
+
es
i
) c
i:r
k
(1 )kex
i
es
i
c
i:r
k +
2
k
ex
i
kk
es
i
k
= (1 )kP
ex
1=2
i
es
i
c
i:r
k +
2
k
ex
i
kk
es
i
k
= (1 )kP
x
1=2
i
s
i
c
i:r
k +
2
k
ex
i
kk
es
i
k;
where the rst equality follows from the fact that
e
x
i
=
e
s
i
.Consequently,
d
F
(fx
i
+
x
i
g;fs
i
+
s
i
g;(!;C))
= d
F
(fex
i
+
e
x
i
g;fes
i
+
e
s
i
g;(!;C))
=
1
p
!
1
v
u
u
t
r
X
i=1
(!
i
!
i1
)kP
(ex
i
+
ex
i
)
1=2(es
i
+
es
i
) c
i:r
k
2
1
p
!
1
v
u
u
t
r
X
i=1
(!
i
!
i1
)kP
x
1=2
i
s
i
c
i:r
k
2
+
2
p
!
1
v
u
u
t
r
X
i=1
(!
i
!
i1
)k
ex
i
k
2
k
es
i
k
2
(1 )d
F
(fx
i
g;fs
i
g;(!;C)) +
2
!
1
r
X
i=1
(!
i
e!
i
)
2
e!
i
proves the lemma.
With this lemma,we prove the following quadratic convergence result.
Proposition 3.If next target w
++
=
P
r
i=1
!
i
c
i
is selected in such a way that the
current primaldual iterates (x
+
;s
+
) satisfy
s
+
=
r
X
i=1
b
i
(x
+
)
1
c
i:r
Targetfollowing and symmetric cone programs 19
for some
b
1
;:::;
b
r
> 0,and d
F
(x
+
;s
+
;!) <
p
51
2
< 1,then taking a full step along
the directions (
x
;
s
),determined by the linear system (3.6) with A 2 G a C
triangular automorphism satisfying P
P
r
i=1
p
!
i
c
i
A
t
x
+
= A
1
s
+
,keeps the iterates
within the primaldual strictly feasible region,and
d
F
(x
+
+
x
;s
+
+
s
;!) d
F
(fx
i
+
x
i
g;fs
i
+
s
i
g;(!;C))
d
F
(x
+
;s
+
;!)
2
1 d
F
(x
+
;s
+
;!)
< 1;
where x
i
= (x
+
)
c
i:r
,s
i
= A
i
A
t
i
x
i
,
x
i
= (
x
)
c
i:r
,
s
i
= x
1
i
s
i
A
i
A
t
i
x
i
,
and A
i
is the restriction of A to J
c
i:r
.
Proof.Recall that
x
i
= (
x
)
c
i:r
and
s
=
P
r
i=1
(!
i
!
i1
)
s
i
,whence d
F
(x+
x
;s +
s
;!) d
F
(fx
i
+
x
i
g;fs
i
+
s
i
g;(!;C)) by Lemma 3.Thus it suces
to show that x
i
+
x
i
;s
i
+
s
i
2
i
for 2 [0;1],and that
d
F
(fx
i
+
x
i
g;fs
i
+
s
i
g;(!;C))
d
F
(x;s;!)
2
1 d
F
(x;s;!)
:
By Lemma 3,if x
i
+
x
i
2 cl(
i
)n
i
for any 2 [0;1],then d
F
(fx
i
+
x
i
g;fs
i
+
s
i
g;(!;C)) 1.Assuming := d
F
(x;s;!) < 1,the previous lemma states that
d
F
(fx
i
+
x
i
g;fs
i
+
s
i
g;(!;C)) (1 ) +
2
2
1
< 1;
for all 2 [0;1],whence x
i
+
x
i
2
i
for all 2 [0;1] by the continuity of
7!x
i
+
x
i
.We then apply Lemma 3 once again,together with the above
inequality,to conclude that s
i
+
s
i
2
i
for all 2 [0;1].
3.2.Choice of targets.The analysis in the preceding section requires the as
sumption that the next target w
++
is selected in such a way that the current
primaldual iterates (x
+
;s
+
) satisfy
s
+
=
r
X
i=1
b
i
(x
+
)
1
c
i:r
for some
b
1
;:::;
b
r
> 0.This,in general,decides the choice of the complete ag
f.Thus,we now only need to decide the values of the weights!;see Remark 2.
In light of the proximity measure d
F
(;;!),we shall use the following proximity
measure on the set of weights:
(3.9) d
F
(e!;!)
def
=
v
u
u
t
1
!
1
r
X
i=1
(e!
i
!
i
)
2
!
i
;
With this choice of targets and proximity measure,the targetfollowing framework
is now specialized to the following:
Algorithm 2.(Targetfollowing algorithm for symmetric cone programming)
Given a pair of primaldual strictly feasible solutions (x
in
;s
in
) and target weights
!
out
.
(1) Pick some 2 (0;1) and a sequence of weights f!
k
g
N
k=0
such that
s
+
=
r
X
i=1
(!
(0)
i
!
(0)
i1
)(x
+
)
1
c
i
++c
r
for some Jordan frame (c
1
;:::;c
r
),d
F
(!
k
;!
k1
) for k = 1;:::;N,
and!
N
=!
out
.
(2) Set (x
+
;s
+
) = (x
in
;s
in
).
(3) For k = 1;:::;N,
20 C.B.CHUA
(a) Solve the linear system (3.6) with (c
1
;:::;c
r
) a Jordan frame from
Theorem 6 for (x
+
;s
+
),with (!
1
;:::;!
r
) the weights in!
k
,and with
A 2 G the automorphism from Lemma 2.
(b) Update (x
+
;s
+
) (x
+
+
x
;s
+
+
s
).
(4) Output (x
out
;s
out
) = (x
+
;s
+
).
3.3.Analysis of algorithm.Consider one iteration of Algorithm 2.Recall from
Proposition 3 that we can take a full step when d
F
(x
+
;s
+
) < (
p
51)=2.This can
be enforced during the update of weights via the following lemma.
Lemma 5.If d
F
(x;s;!) and d
F
(e!;!) for some ; 2 (0;1),then
d
F
(x;s;e!)
+
1
:
Proof.We have
d
F
(x;s;e!)
v
u
u
t
1
e!
1
r
X
i=1
(
i
(P
x
1=2s) !
i
)
2
e!
i
+
v
u
u
t
1
e!
1
r
X
i=1
(!
i
e!
i
)
2
e!
i
(d
F
(x;s;!) +d
F
(e!;!)) max
i
!
i
e!
i
:
If d
F
(e!;!) ,then
2
1
!
1
r
X
i=1
(e!
i
!
i
)
2
!
i
r
X
i=1
e!
i
!
i
1
2
;
whence min
i
e!
i
!
i
1 .
We now give the main theoremof this section,which states that for all > 0 suf
ciently small,say
1
6
,then Algorithm 2 terminates with a good approximation
of T
1
(
P
r
i=1
!
out
i
c
i
) for some Jordan frame C = (c
1
;:::;c
r
).
Theorem 9.In Algorithm 2,if 2 (0;1) is such that there exists some 2
(0;
p
51
2
) satisfying
(3.10)
( +)
2
(1 )(1 )
;
then (x
+
;s
+
) is welldened and strictly feasible in each iteration,and the algorithm
terminates with d
F
(x
out
;s
out
;!
out
)
min
,where
min
is the least satisfying the
inequality.
Proof.We shall prove the theorem by induction that the iterates (x
+
;s
+
) are
strictly feasible and d
F
(x
+
;s
+
;!
k
)
min
at the beginning of each iteration.This
is certainly true for the rst iteration.By Lemma 5,we have d
F
(x
+
;s
+
;!
k+1
)
(
min
+)=(1 ).If the hypothesis (3.10) holds,then we may apply Proposition
3 to deduce that the iterates (x
+
+
x
;s
+
+
s
) are strictly feasible with
d
F
(x
+
+
x
;s
+
+
s
)
((
min
+)=(1 ))
2
1 (
min
+)=(1 )
=
(
min
+)
2
(1
min
)(1 )
min
:
This completes the induction.
4.Finding analytic centers
In this section,we consider an algorithm that nds the analytic center T
1
(be)
for any given b > 0.This algorithm can be used to nd analytic centers of compact
sets described by linear matrix inequalities and convex quadratic constraints.It
can also be combined with a pathfollowing algorithm to solve the symmetric cone
program (1.1).
Targetfollowing and symmetric cone programs 21
Given a pair of primaldual strictly feasible solutions (bx;bs),we shall construct a
nite sequence of targets f!
k
g
N
k=0
such that
s
+
=
r
X
i=1
(!
(0)
i
!
(0)
i1
)(x
+
)
1
c
i
++c
r
for some Jordan frame (c
1
;:::;c
r
),
(4.1) d
F
(!
k
;!
k1
) =
v
u
u
t
1
!
k1
1
r
X
i=1
(!
k
i
!
k1
i
)
2
!
k1
i
for 1 k N;
and!
N
= b1,with satisfying the hypothesis of Theorem 9,thus allowing us to
apply Algorithm 2 to approximate T
1
(be).
Any sequence f!
k
g
N
k=0
satisfying (4.1) is called a sequence,and N is called its
length;see [15].In [3],the author gave an upper bound on the length of a shortest
sequence from any weight sequence!
0
to the ray f1: > 0g.For the sake of
completeness,we repeat the argument here.
Consider the local metric dened by the inner product
h;i
!
:(u;v) 2 R
r
R
r
7!
1
!
n
r
X
i=1
u
i
v
i
!
i
at each weight sequence!.We denote by kk
!
the norminduced by the above inner
product.In terms of this local metric,an sequence f!
k
g
N
k=0
is one that satises
!
k
!
k1
!
k1
for 1 k N:
The length of a piecewise smooth curve :[0;1]!W,where W R
r
++
denotes
the set of weight sequences,is dened to be
Z
1
0
d(t)
dt
(t)
dt =
Z
1
0
v
u
u
t
1
n
n
X
i=1
_
2
i
i
dt;
and denoted by l().The next lemma gives an upper bound on a shortest 
sequence between any two weight sequences in terms of the length of a piecewise
smooth curve joining them.Its proof can be obtained by adapting the proof of a
similar result in [20],and is thus omitted here.
Lemma 6 (c.f.Lemma 3.3 of [20]).For every piecewise smooth curve :[0;1]!W
and every 2 (0;1),there exists an sequence f!
k
g
N
k=0
with!
0
= (0),!
1
= (1)
and length
N
l()
1
2
2
:
Next we show the existence of a piecewise smooth curve from a given weight
sequence!to the ray f1: > 0g with length O(log(
1
r!
1
P
r
i=1
!
i
)).
Lemma 7.For each weight sequence!,there exists a piecewise smooth curve
:[0;1] 7!W with (0) =!,(1) = 1,and length
l()
p
r log
4
!
1
;
where denotes the average weight
1
r
P
r
i=1
!
i
.
Proof.The lemma is trivially true when!= 1.Otherwise,!
1
= =!
p
<
!
p+1
!
r
for some p 2 f1;:::;r 1g.
Consider the straight line segment
e
:[0;
t] 7!Wstarting from!,along which the
weights of least value increases at the same rate,with the other weights decreasing
22 C.B.CHUA
at rates proportional to their values,while maintain the average weight throughout,
and ending when the weights of least value coincide with the next higher value;i.e.,
e
is dened by
(4.2)
e
(t)
1
= =
e
(t)
p
=!
1
+t
r p!
1
p
and
e
(t)
i
=!
i
t!
i
for i = p +1;:::;r,
where
t 2 (0;1) is such that
e
(
t)
1
= =
e
(
t)
p
=
e
(
t)
p+1
;as required
e
(t)
1
+ +
e
(t)
r
= r is independent of t.Its length is
Z
t
0
0
@
1
!
1
+t
rp!
1
p
0
@
p
X
i=1
(
rp!
1
p
)
2
!
1
+t
rp!
1
p
+
r
X
i=p+1
!
2
i
!
i
t!
i
1
A
1
A
1=2
dt
=
Z
t
0
0
@
p
rp!
1
p!
1
rp!
1
+t
0
@
r p!
1
p!
1
rp!
1
+t
+
r
X
i=p+1
!
i
1 t
1
A
1
A
1=2
dt
=
p
p
Z
t
0
1
p!
1
rp!
1
+t
1
p!
1
rp!
1
+t
+
1
1 t
!!
1=2
dt
=
p
p
Z
t
0
q
r
rp!
1
(
p!
1
rp!
1
+t)
p
1 t
dt
=
p
plog
q
r
rp!
1
+1
q
r
rp!
1
1
p
plog
q
r
rp!
1
+
p
1
t
q
r
rp!
1
p
1
t
:
From the denition of
e
(t)
1
,we can simplify
q
r
rp!
1
+
p
1
t
q
r
rp!
1
p
1
t
=
1 +
q
(1
t)
rp!
1
r
1
q
(1
t)
rp!
1
r
=
1 +
q
rp(
t)
1
r
1
q
rp(
t)
1
r
= R(
p
e
(
t)
1
r
);
and
q
r
rp!
1
+1
q
r
rp!
1
1
=
1 +
q
rp!
1
r
1
q
rp!
1
r
= R(
p!
1
r
) = R(
p
e
(0)
1
r
);
where R:(0;1]![1;1) is the decreasing function u 7!(1+
p
1 u)=(1
p
1 u)
satisfying R(u) (1 +1)=(1 (1
1
2
u)) = 4=u.This gives
l(
e
) =
p
plog R(
p
e
(0)
1
r
)
p
plog R(
p
e
(
t)
1
r
):
As long as
e
(
t) 6= 1,we repeat this process to construct another straight line
segment starting from
e
(
t).Eventually,we get a piecewise linear curve joining!
to 1 with q r straight line segments,and total length
l() =
q
X
i=1
p
p
i
log R(
p
i
!
i
1
r
)
p
p
i
log R(
p
i
!
i+1
1
r
)
;
where!
i
is the weight sequence at the start of the i'th straight segment,!
q+1
denotes the weights 1,and p
i
is the number of weights of least value in!
i
.We
claim that for any a > 1,the function u 2 (0;1=a] 7!log R(u) log R(au) is
Targetfollowing and symmetric cone programs 23
increasing
3
.Thus,since both f!
i
1
g
q
i=1
increasing by construction,we have the
upper bound
l() <
q
X
i=1
p
p
i
log R(
r!
i
1
r
) log R(
r!
i+1
1
r
)
p
r
q
X
i=1
log R(
!
i
1
) log R(
!
i+1
1
)
=
p
r(log R(
!
1
) log R(1))
p
r log
4
!
1
:
From the above two lemmas,we deduce the following upper bound on the length
of a shortest sequence from a given weight sequence!to the ray f1: > 0g.
Theorem 10.For every weight sequence!
0
and every 2 (0;1),there exists an
sequence f!
k
g
N
k=0
with!
N
= 1,where =
P
r
i=1
!
0
i
=r,and length
N
p
r
1
2
2
log
4
!
0
1
:
Corollary 1.Suppose 2 (0;
p
51
2
) is xed.Let 2 (0;1) be a number satisfying
the inequality (3.10) in Theorem 9.Given any pair of primaldual strictly feasible
solutions (bx;bs) for the primaldual symmetric cone programming problems (1.1),
there is a sequence of at most
N
p
r
1
2
2
log
4hbx;bsi
r
1
(P
bx
1=2bs)
:
weights such that Algorithm 2 nds a pair of primaldual strictly feasible solutions
(x;s) satisfying kP
x
1=2s ek = d
F
(x;s;1) ,where =
1
r
hbx;bsi.
Combining the corollary with an sequence on the central path,we have the
following theorem.
Theorem 11.Suppose 2 (0;1) is xed.Given any pair of primaldual strictly
feasible solutions (bx;bs) for the primaldual symmetric cone programming problems
(1.1),and any positive real number b,there is a sequence of at most
O
p
r
log
hbx;bsi
r
1
(P
bx
1=2
b
s)
+
log
hbx;bsi
rb
weights such that Algorithm 2 nds a pair of primaldual strictly feasible solutions
(x;s) satisfying kP
x
1=2
s bek = bd
F
(x;s;b1) b.
As an immediate corollary,we have the following worstcase iteration bound on
solving symmetric cone problems using Algorithm 2.
Corollary 2.Given any pair of primaldual strictly feasible solutions (bx;bs) and
any"> 0,there is a sequence of at most
O
p
r
log
hbx;bsi
r
r
(P
bx
1=2bs)
+
log"
1
targets such that Algorithm 2 nd a pair of primaldual strictly feasible solutions
(x;s) satisfying hx;si "hbx;bsi.
Proof.If (x;s) 2
2
satises kP
x
1=2s ek for some 2 (0;1) and some
> 0,then hx;si r = he;P
x
1=2s ei
p
r.Apply the preceding theorem
with b ="hbx;bsi =(
p
r +r).
3
We have
R(u)
R(au)
= a
1+
p
1u
1+
p
1au
2
,and it is straightforward to check that u 7!
1+
p
1u
1+
p
1au
is
increasing when a > 1.
24 C.B.CHUA
5.Conclusion
We extend the target map (x;s) 7!(x
1
s
1
;:::;x
n
s
n
),together with the weighted
barriers x 7!
P
n
i=1
!
i
log x
i
and the notions of weighted analytic centers,fromlin
ear programming to general convex conic programming.This extension is obtained
from a geometrical perspective of the weighted barriers,via the facial structure of
the nonnegative orthant,that views a weighted barrier as a weighted sumof barriers
for a strictly decreasing sequence of faces.When we replace decreasing sequences of
faces of the nonnegative orthant with decreasing sequences of faces of an arbitrary
closed convex cone,we arrive at weighted barriers for the convex cone;provided
that we have made a priori choices of barriers for all faces of the convex cone.This
potentially opens the door to ecient targetfollowing algorithms for general convex
conic programming,once we know how to design and analyze ecient primaldual
algorithms for general convex conic programming.
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E.B.Vinberg.The structure of the group of automorphisms of a homogeneous convex cone.
Tr.Mosk.Mat.O.va,13:56{83,1965.
[26] H.Wolkowicz,R.Saigal,and L.Vandenberghe,editors.Handbook of Semidenite Program
ming:Theory,Algorithms,and Applications.SpringerVerlag,BerlinHeidelbergNew York,
2000.
[27] S.J.Wright.PrimalDual InteriorPoint Methods.SIAM Publication,Philadelphia,PA,
USA,1997.
[section]
Appendix A.Euclidean Jordan algebras
In this section,we give a very brief introduction to Euclidean Jordan algebras,
stating various known results and proving some new ones that are necessary for
the development of this paper.For a comprehensive discussion on symmetric cones
and Jordan algebras,we refer the reader to the excellent exposition by J.Faraut
and A.Koranyi [7].
A Jordan algebra (J;;+) is a commutative algebra whose multiplication oper
ator satises the Jordan identity x (x
2
y) = x
2
(x y) for all x;y 2 J,where
x
2
denotes x x.The multiplication operator is often called the Jordan product
of the Jordan algebra.We use L
x
to denote the Lyapunov operator y 2 J 7!x y.
A Jordan algebra (J;) is said to be formally real if
(x
2
+y
2
= 0 =) x = y = 0) 8x;y 2 J:
A formally real Jordan algebra has a identity element e:an element that satises
e x = x;see [13].It is also noted in [13] that a formally real Jordan algebra
is power associative
4
:if we recursively dene,for each element x 2 J,the k'th
power by x
0
= e and x
k
= x
k1
x for k = 1;2;:::,then x
k+l
= x
k
x
l
for
all nonnegative integers k and l (i.e.,the collection of all powers of an element x
forms a semigroup).This important fact results in the existence of the minimal
polynomial for each element x 2 J:the monic polynomial in R[X] that generates
the principle ideal fp 2 R[X]:p(x) = 0g.The maximum degree of all minimal
polynomials is called the rank of the Jordan algebra.An element x 2 J is said to
be regular if the degree of its minimal polynomial coincides with the rank of J.
Henceforth,r shall denote the rank of J.
The minimal polynomials also give us two important functions.For a regular
element x,its trace is the coecient of the second highest power in its minimal
polynomial,and its determinant is the constant term in its minimal polynomial.
As the set of regular elements is dense in J,and both functions are continuously
extendable to J as polynomials,we can dene the trace and determinant of all
elements of J;see [7,Proposition II.2.1].
4
This is actually true in general for all Jordan algebras;see [7,Proposition II.1.2].
26 C.B.CHUA
Example A.1.The space of r r real symmetric matrices S
r
equipped with the
symmetrized product
1
2
(AB+BA) is a formally real Jordan algebra with identity
I.The notions of minimal polynomial,trace and determinant are as we commonly
dened.
It is known (see,e.g.,[7,Proposition VIII.4.2]) that a Jordan algebra with iden
tity is formally real if and only if it is a Euclidean Jordan algebra;i.e.,there is a
symmetric positive denite bilinear functional B:J
2
7!R that is associative;i.e.,
B(xy;z) = B(y;xz) for all x;y;z 2 J.Equivalently,a Euclidean Jordan algebra
is a Jordan algebra with identity such that the bilinear function (x;y) 7!tr(xy) is
positive denite;see [7,Proposition III.1.5].Thus a Euclidean Jordan algebra can
be given a Euclidean structure with the inner product h;i:(x;y) 2 J
2
7!tr(xy)
in such a way that L
x
is selfadjoint.
It is known that if (J;) is a Euclidean Jordan algebra,then the interior of its
cone of squares fxx:x 2 Jg is a symmetric cone in the Euclidean space (J;h;i).
We denote this interior by
(J).Moreover,this symmetric cone coincides with the
set of elements with positive denite Lyapunov operators;see [7,Theorem III.2.1].
A key ingredient in showing the homogeneity of the cone of squares is the quadratic
representation of J:P
x
:x 2 J 7!2L
2
x
L
x
.The collection of quadratic repre
sentations at all x 2
(J) gives a transitive subset of automorphisms of
(J).In
particular,to each x 2
(J) is associated a unique y 2
(J) such that P
y
e = x.
We denote such y by x
1=2
,and called it the square root of x.
Conversely,given any symmetric cone
,there is an Euclidean Jordanalgebraic
structure such that the symmetric cone coincides with the interior of the cone of
squares.Moreover the closure cl(
) of the symmetric cone coincides with the cone
of squares;see [7,Theorem III.3.1].
Alternatively,the symmetric cone
(J) can be dened as the connected com
ponent of the set of invertible elements containing the identity element;see [7,
Theorem III.2.1].An element x is said to be invertible if there exists a linear com
bination of powers of x whose Jordan product with x is the identity element.This
linear combination of powers is called the inverse of x,and denoted by x
1
.It is
unique since the subalgebra generated by x and the identity element e is associative.
An element x is invertible if and only if its quadratic representation is nonsingular,
and in this case,P
1
x
= P
x
1;see [7,Theorem II.3.1].
Example A.2.For the Jordan algebra of r r real symmetric matrices,the cone
of squares is the cone of all positive semidenite matrices.The quadratic represen
tation of a matrix X is Y 7!XYX.The notions of square root and inverse are as
we commonly dened.
A.1.Spectral decompositions.A key ingredient in the study of formally real
Jordan algebra by P.Jordan et al.[13] is the set of idempotents.An idempotent of
J is a nonzero element c 2 J satisfying cc = c.Amongst the idempotents are the
primitive ones:idempotents that cannot be written as the sum of two idempotents.
Two idempotents c and d are said to be orthogonal if c d = 0.Orthogonal
idempotents are indeed orthogonal with respect to the inner product h;i since
hc;di = hc e;di = he;c di:
From its denition,it is straightforward to check that the sum of two orthogonal
idempotents is an idempotent,and that an element c is an idempotent if and only if
the element ec is an idempotent.A complete system of orthogonal idempotents is
a set of idempotents that are pairwise orthogonal and sum to the identity element
e.A Jordan frame is a complete system of orthogonal primitive idempotents.The
number of elements in any Jordan frame always coincide with the rank of J;see
paragraph immediately after Theorem III.1.2 of [7].
Targetfollowing and symmetric cone programs 27
Example A.3.For the Jordan algebra of r r real symmetric matrices,an idem
potent is a product PP
T
where the matrix P has orthogonal columns of unit length.
It is primitive if and only if it is of rank one.A complete system of orthogonal
idempotents is a ptuple (P
1
P
T
1
;:::;P
p
P
T
p
) with the columns of P
1
;:::;P
p
taken
from the columns of an orthogonal matrix.A Jordan frame C is then a complete
system of r orthogonal idempotents with each P
i
a columnmatrix.
For each x 2 J,there exists numbers
1
r
and a Jordan frame
fc
1
;:::;c
r
g such that x =
1
c
1
+ +
r
c
r
.This is known as a spectral de
composition of type II of x;see Theorem III.1.2 of [7].Moreover,the set of values
of
i
's (with their multiplicities) remain unchanged over all such Jordan frames.
When the primitive idempotents corresponding to the same eigenvalues are com
bined,we have the spectral decomposition of type I:x =
1
ec
1
+ +
k
ec
k
,where
1
< <
k
are the distinct eigenvalues of x,and ec
i
is the sum of the primitive
idempotents corresponding to the eigenvalue
i
;see also Theorems III.1.1 [7].This
spectral decomposition is unique.
The values
i
in a type II spectral decomposition are called the eigenvalues
of x,and are denoted by
i
(x),with
1
(x)
r
(x).In terms of the
spectral decompositions,the inverse of an invertible element x is the element
x
1
=
1
(x)
1
c
1
+ +
r
(x)
1
c
r
,and the square root of an element x in the
symmetric cone
(J) is the element x
1=2
=
1
(x)
1=2
c
1
+ +
r
(x)
1=2
c
r
.
For an element x with the type I spectral decomposition x =
1
c
1
+ +
k
c
k
,
the orthogonality of the idempotents implies that a polynomial p 2 R[X] is in
the principle ideal generated by the minimal polynomial if and only if p(
1
) =
= p(
k
) = 0.Thus the minimal polynomial of an element x is t 7!(t
1
) (t
k
).Consequently,an element is regular if and only if it has distinct
eigenvalues.Moreover,the trace of x is the sum
1
(x) + +
r
(x),and its
determinant is the product
1
(x)
r
(x).The norm of an element x is then
p
1
(x)
2
+ +
r
(x)
2
.In particular,the square of the norm of an idempotent is
the number of pairwise orthogonal primitive idempotents summing up to it.
The logarithmof the determinant plays an important role in interiorpoint meth
ods for symmetric cone programming:its negation serves as a barrier (called the
logdeterminant barrier) for the symmetric cone
.We note that the gradient of
this logdeterminant barrier at x 2
is x
1
,and its Hessian is P
1
x
= P
x
1
;see
[7,Proposition II.3.3 and Proposition III.4.2]
5
.
A.2.Peirce decomposition.For any idempotent c,its Lyapunov operator L
c
can only have eigenvalues 0,1=2 or 1;see [7,p.62].We denote by J(c;0),J(c;1=2)
and J(c;1) the (possibly empty) eigenspaces associated with the eigenvalues 0,1=2
and 1,respectively.Since L
e
is the identity map,the eigenspaces of the orthogonal
idempotents c and e c satisfy J(c;0) = J(e c;1),J(c;1=2) = J(e c;1=2) and
J(c;1) = J(e c;0).Recall that L
c
is selfadjoint.Thus we have the orthogonal
decomposition
J = J(c;1) +J(c;1=2) +J(c;0)
= J(c;1) +J(c;1=2)\J(e c;1=2) +J(e c;1):
This is known as the Peirce decomposition of J with respect to c.
5
Part (ii) of [7,Proposition III.4.2],while stated only for simple Euclidean Jordan algebras,
is in fact true for general Euclidean Jordan algebras.This follows from the fact that when a
Euclidean Jordan algebra is written as the direct sum J
1
J
of simple Euclidean Jordan al
gebras,the determinant det(x
1
;:::;x
()
) decomposes into the product det
1
(x
1
) det
()
(x
()
) of
determinants on each component,and the quadratic representation P
(x
1
;:::;x
()
)
is block diagonal
with the quadratic representations P
x
1
;:::;P
x
()
as the diagonal blocks.
28 C.B.CHUA
For simplicity of notation,we shall use J
c
to denote the (nonempty) eigenspace
J(c;1),and for each pair of orthogonal idempotents (c;c
0
),we shall use J
c;c
0 to
denote the (nonempty) common eigenspace J(c;1=2)\J(c
0
;1=2).The above Peirce
decomposition J = J
c
+J
c;ec
+J
ec
with respect to c can be generalized to one
with respect to a collection of pairwise orthogonal idempotents that sums up to the
identity element.
Theorem 12 (Peirce decomposition,cf.Theorem IV.2.1 of [7]).For each complete
system of orthogonal idempotents C = fc
1
;:::;c
p
g,the space J decomposes into the
orthogonal direct sum
J =
p
M
i=1
J
c
i
M
i<j
J
c
i
;c
j
in such a way that
(1) J
c
i
is a Jordan subalgebra of J with identity element c
i
;
(2) the orthogonal projector onto J
c
i
is P
c
i
,and that onto J
c
i
;c
j
is 4L
c
i
L
c
j
;
and
(3) for each 1 i;j;k;l p with fi;jg\fk;lg =;,
J
c
i
;c
j
J
c
i
;c
j
J
c
i
+J
c
j
;J
c
i
J
c
i
;c
k
J
c
i
;c
k
;
J
c
i
;c
j
J
c
j
;c
k
J
c
i
;c
k
;J
c
i
;c
j
J
c
k
;c
l
= f0g:
Proof.This theorem follows from Theorems 8 and 9 of [13],and their proofs.
For each x 2 J,its decomposition into x =
P
p
i=1
x
c
i
+
P
i<j
x
c
i
;c
j
with x
c
i
=
2P
c
i
(x) and x
c
i
;c
j
= 4L
c
i
(L
c
j
(x)) is called its Peirce decomposition with respect
to the complete system of idempotents fc
1
;:::;c
p
g.
Example A.4.For the Jordan algebra of rr real symmetric matrices,the Peirce
decomposition of a matrix X with respect to a complete system of orthogonal idem
potents (P
1
P
T
1
;:::;P
p
P
T
p
) is X =
P
p
i=1
P
i
P
T
i
XP
i
P
T
i
+
P
i<j
P
i
P
T
i
XP
j
P
T
j
+
P
j
P
T
j
XP
i
P
T
i
.
The Peirce decomposition allows us to express the eigenvalues and eigenspaces
of the Lyapunov operator L
x
in terms of the spectral decomposition of x:if x =
1
c
1
+ +
k
c
k
is the type I spectral decomposition of x,then the subspace
J
c
i
;c
j
,if nonempty,is an eigenspace of L
x
associated with the eigenvalue
1
2
(
i
+
j
).
Subsequently,the eigenvalues and eigenspaces of the quadratic representation P
x
can be similarly obtained:the subspace J
c
i
;c
j
,if nonempty,is an eigenspace of
P
x
associated with the eigenvalue
i
j
.These observations leads to the following
lemma.
Lemma A.1 (cf.Lemma 12 of [1]).If L
x
(resp.,P
x
) and L
y
(resp.,P
y
) are
similar to each other,then x and y have the same set of eigenvalues.
Proof.In the Peirce decomposition with respect to a complete system of orthogo
nal idempotents (c
1
;:::;c
p
),the subspace J
c
i
is generated by any set of orthogonal
idempotents summing up to c
i
,and has dimension kc
i
k
2
.Thus if two Lyapunov
operators (or quadratic representations) are similar to each other,then the corre
sponding elements share the same eigenvalues,and each eigenvalue occurs the same
number of times in each type II spectral decomposition.
A.3.Some new results.
Lemma A.2.For each automorphism A in the identity component G of the auto
morphism group G(
) of
and all x 2
,
log det(Ax) = log det(x) +c
A
;(A
t
x)
1
= A
1
x
1
and P
A
t
x
= A
t
P
x
A:
Targetfollowing and symmetric cone programs 29
Proof.When A is a quadratic representation,the rst equation follows from [7,
Proposition III.4.2].In general,we decompose A into the product P
p
Q of the
quadratic representation of some p 2
and some orthogonal automorphism Q
in the identity component of G(
) (see [7,Theorem III.5.1]),and note that the
determinant is invariant under automorphisms of J (see [7,TheoremII.4.2]),whence
invariant under Q (see Theorem A.7).Dierentiating the rst equation twice gives
A
t
(Ax)
1
= x
1
and A
t
P
1
Ax
A = P
1
x
.Since
is selfdual implies that G(
)
t
=
G(
),the other two equations follows.
Lemma A.3.For each x 2
and each A 2 G satisfying Ax = e,the elements
z 2 J and AP
x
1=2z always have the same set of eigenvalues.
Proof.We shall show that the quadratic representations of z and AP
x
1=2z are
similar to each other,whence by Lemma A.1,both elements have the same set of
eigenvalues.By the choice of A,AP
x
1=2e = Ax = e.Therefore Q:= AP
x
1=2 2 G
is orthogonal by Theorem A.7.By Lemma A.2,P
AP
x
1=2
z
= QP
z
Q
T
is similar to
P
z
.
Lemma A.4.For each A 2 G,AA
t
e = (Ae)
2
.
Proof.By Lemma A.2,AA
t
e = AP
e
A
t
e = P
Ae
e = (Ae)
2
.
Lemma A.5.For any A 2 G,and any x;s 2
2
,x s = e if and only if A
t
x
A
1
s = e.
Proof.Since every automorphism A decomposes into A = P
p
Q
t
for some orthog
onal automorphism Q in the identity component of G(
) and some p 2
(see [7,
Theorem III.5.1]),we can write A
t
x A
1
s = QP
p
x QP
p
1s.
From the fact that the orthogonal subgroup of G(
) coincides with both the
automorphism group G(J) and the stabilizer subgroup G(
)
c
i:r
of the unit c
i:r
in
G(
) (see TheoremA.7),we have A
t
xA
1
s = QP
p
xQP
p
1s = Q(P
p
xP
p
1s),
and subsequently,A
t
x A
1
s = c
i:r
if and only if P
p
x P
p
1
s = e.The lemma
then follows from Lemma 28 of [1] (cf.Theorem 3.1 (i) of [19]).
Example A.5.For the Jordan algebra of r r real symmetric matrices,an au
tomorphism of the cone of positive denite matrices takes the form X 7!PXP
T
for some invertible matrix P.It is in the identity component G if and only if
det(P) > 0.The rst lemma specializes to the wellknown facts log det(PXP
T
) =
log det(X)log det(P)
2
,(P
T
XP)
1
= P
1
X
1
P
T
,and (P
T
XP)Y(P
T
XP) =
P
T
(X(PYP
T
)X)P.The second lemma follows easily from XS + SX = 2I ()
XS = I.
A.4.Automorphisms of Euclidean Jordan algebras.In Section II.1 of [25],
it was stated without proof that if (J;) is a Euclidean Jordan algebra and
is
its associated symmetric cone,then the stabilizer subgroup G(
)
e
of the unit e in
G(
) coincide with the group of automorphisms G(J) of J.Here we give a proof of
this fact.
Theorem A.6.Given a Euclidean Jordan algebra (J;) with unit e and associated
symmetric cone
,the stabilizer subgroup G(
)
e
of the unit e in G(
) coincide with
the group of automorphisms G(J) of J.
Proof.Consider the inner product h;i:(x;y) 7!trace L
xy
,where L
x
denotes
the linear map y 7!x y.Let O(J) denote the orthogonal group of the Euclidean
space (J;h;i);i.e.,O(J) = fA 2 GL(J):hAx;Ayi = hx;yi 8x;y 2 Jg.
Let'be the characteristic function of
;i.e.,
':x 2
7!
Z
]
e
hx;yi
dy;
30 C.B.CHUA
where dy denotes the Euclidean measure on (J;h;i).Let x
]
denote the negative
gradient of the logarithmic derivative of'at x.We deduce from Propositions
II.3.4 and III.2.2 of [7] that expL
x
2 G(
).Thus by Proposition I.3.1 of [7],we
have log'(expL
x
e) = log'(e) log det expL
x
= log'(e) trace L
x
.Dier
entiating this at 0 gives trace L
h
= Dlog'(e)[h].Since trace L
h
= hh;ei and
Dlog'(e)[h] =
e
]
;h
,it follows that e is a xed point of the map x 2
7!x
]
.
Proposition I.4.3 of [7] then states that G(
)\O(J) = G(
)
e
.
We now show that G(J) coincides with G(
)\O(J) = G(
)
e
.It is straightfor
ward to check that every automorphism of J is an automorphism of
(which is
the interior of the cone of squares) that stabilizes the unit e.For the other direc
tion,it suces to show that every linear map A 2 G(
)\O(J) = G(
)
e
preserves
orthogonality of idempotents and maps every primitive idempotent to some prim
itive idempotent,for if x =
P
i
c
i
is the spectral decomposition,then we have
A(x
2
) = A
P
2
i
c
i
=
P
2
i
A(c
i
) = (
P
i
A(c
i
))
2
= (Ax)
2
,whence A 2 G(J) by
polarization.Suppose A 2 G(
)\O(J) = G(
)
e
.Two idempotents are c and d are
orthogonal if and only if hc;di = 0.One direction of this statement follows fromthe
denition of h;i.For the other direction,suppose that c and d are two idempotents
satisfying hc;di = 0.Since the inner product h;i is associative (see Proposition
II.4.3 of [7]),L
c
is selfadjoint.Proposition III.1.3 of [7] then implies that L
c
is
positive semidenite.Thus it has a selfadjoint,positive semidenite square root
L
1=2
c
.Hence 0 = hc;di =
c;d
2
= hc d;di = hL
c
d;di =
D
L
1=2
c
d;L
1=2
c
d
E
shows
that L
1=2
c
d = 0,whence c d = L
1=2
c
L
1=2
c
d = 0;i.e.,c and d are orthogonal.
Since A is orthogonal,it follows that orthogonal idempotents remain orthogonal
under A.Proposition IV.3.2 of [7] states c is a primitive idempotent if and only if
fc: 0g is an extreme ray of
.Since A 2 G(
),it maps each extreme ray
to some extreme ray of
.Thus it maps each primitive idempotent c to a positive
multiple d of some primitive idempotent d.In fact, must be unit since
0 < hd;di =
d
2
;e
= hd;ei = hd;Aei =
1
hAc;Aei
=
1
hc;ei =
1
c
2
;e
=
1
hc;ci =
1
hAc;Aci = hd;di
Hence A maps each primitive idempotent to some primitive idempotent.
The proof of the theorem shows that both G(
)
e
and G(J) coincide with certain
orthogonal subgroup of G(
).The next theorem gives a similar result.
Theorem A.7.Given a Euclidean Jordan algebra (J;) with unit e and associated
symmetric cone
,the groups G(
)
e
and G(J) are both equivalent to the orthogonal
subgroup of G(
) under the inner product h;i:(x;y) 7!tr(x y).
Proof.Let O(
) denote the orthogonal subgroup of G(
) under h;i.By Propo
sition II.4.2 of [7],if A 2 G(J),then tr(Ax Ay) = tr A(x y) = tr(x y) for all
x;y 2 J,whence A is orthogonal.Therefore G(J) = G(
)
e
O(
).According to
Proposition I.1.8 of [7] and the paragraph following it,G(
)
e
is a maximal compact
subgroup of G(
).Hence O(
) G(
)
e
.
Example A.6.For the Jordan algebra of r r real symmetric matrices,an auto
morphism of the Jordan algebra takes the form X 7!QXQ
T
for some orthogonal
matrix Q,which clearly stabilizes the identity I.It is also orthogonal under the
trace inner product:tr(QXQ
T
)(QYQ
T
) = tr XY.
Division of Mathematical Sciences,School of Physical & Mathematical Sciences,
Nanyang Technological University,Singapore 637371,Singapore
Email address:cbchua@ntu.edu.sg
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