1
SEMIDEFINITE PROGRAMMING
RELAXATIONS OF NON-CONVEX
PROBLEMS IN CONTROL AND
COMBINATORIAL OPTIMIZATION
Stephen Boyd and Lieven Vandenb erghe
Information Systems Laboratory, Stanford University
In celebration of Tom Kailath's contributions, on his 60th birthday.
DedicatedtoTom Kailath: mentor, model, col league, teacher, friend.
ABSTRACT
We p oint out some connections b etween applications of semide nite programming in
control and in combinatorial optimization. In b oth elds semide nite programs arise
as convex relaxations of NP-hard quadratic optimization problems. We also show that
these relaxations are readily extended to optimization problems over bilinear matrix
inequaliti es.
In a semide nite program SDP we minimize a linear function of a variable
m
x 2 R sub ject to a matrix inequality:
T
minimize c x
1.1
sub ject to F x 0
where
m
X
x F : F x = F +
i i 0
i=1 1
2 Chapter 1
m
The problem data are the vector c 2 R and m + 1 symmetric matrices
nn
F ;:::;F 2 R . The inequality sign in F x 0 means that F x is
0 m
n
T
p ositive semide nite, i.e., z F x z 0 for all z 2 R .We call the inequality
F x 0a linear matrix inequality LMI .
Semide nite programs can b e regarded as an extension of linear programming
where the comp onentwise inequalities b etween vectors are replaced by matrix
inequalities, or, equivalently, the rst orthant is replaced by the cone of p osi-
tive semide nite matrices. Semide nite programming uni es several standard
problems e.g., linear and quadratic programming , and nds many applica-
tions in engineering and combinatorial optimization see [Ali95], [BEFB94 ],
[VB96] . Although semide nite programs are much more general than linear
programs, they are not much harder to solve. Most interior-p oint metho ds for
linear programming have b een generalized to semide nite programs. As in lin-
ear programming, these metho ds have p olynomial worst-case complexity, and
p erform very well in practice.
2 SEMIDEFINITE PROGRAMMING AND
COMBINATORIAL OPTIMIZATION
Semide nite programs playavery useful role in non-convex or combinatorial
optimization. Consider, for example, the quadratic optimization problem
minimize f x
0
1.2
sub ject to f x 0;i=1;:::;L
i
T T
where f x = x Ax+2b x+c , i =0;1;:::;L. The matrices A can b e
i i i i
i
inde nite, and therefore problem 1.2 is a very hard, non-convex optimization
problem. For example, it includes all optimization problems with p olynomial
ob jective function and p olynomial constraints see [NN94, x6.4.4], [Sho87] .
For practical purp oses, e.g., in branch-and-b ound algorithms, it is imp ortantto
have go o d and cheaply computable lower b ounds on the optimal value of 1.2 .
Shor and others have prop osed to compute suchlower b ounds by solving the
Semide nite programming relaxations 3
semide nite program with variables t and
i
maximize t
A b A b A b
0 0 1 1 L L
sub ject to + + + 0
1 L
T T T
b c t b c b c
0 1 L
0 1
L
0;i=1;:::;L:
i
1.3
One can easily verify that this semide nite program yields lower b ounds for 1.2 .
Supp ose x satis es the constraints in the nonconvex problem 1.2 , i.e.,
T
x A b x
i i
f x = 0
i
T
1 b c 1
i
i
for i =1;:::;L, and t, ,..., satisfy the constraints in the semide nite
1 L
program 1.3 . Then
T
x A b A b A b x
0 0 1 1 L L
0 + + +
1 L
T T T
1 b c t b c b c 1
0 1 L
0 1
L
= f x t + f x + + f x
0 1 1 L L
f x t:
0
Therefore t f x for every feasible x in 1.2 , as desired. Problem 1.3 can
0
also b e derived via Lagrangian duality; for a deep er discussion, see Shor [Sho87],
or Poljak, Rendl, and Wolkowicz [PRW94].
Most semide nite relaxations of NP-hard combinatorial problems seem to b e
related to the semide nite program 1.3 , or the related one,
T
minimize TrXA +2b x+ c
0 0
0
T
sub ject to TrXA +2b x+c 0;i=1;:::;L
i i
i
1.4
X x
0;
T
x 1
k k k
T
where the variables are X = X 2 R and x 2 R . It can b e shown
that 1.4 is the semide nite programming dual of Shor's relaxation 1.3 ; the
two problems 1.3 and 1.4 yield the same b ound.
Note that the constraint
X x
0 1.5
T
x 1
4 Chapter 1
T
is equivalentto X xx . The semide nite program 1.4 can therefore b e
directly interpreted as a relaxation of the original problem 1.2 , which can b e
written as
T
minimize TrXA +2b x+ c
0 0
0
T
sub ject to TrXA +2b x+c 0;i=1;:::;L
1.6
i i
i
T
X = xx :
The only di erence b etween 1.6 and 1.4 is the replacement of the non-
T T
convex constraint X = xx with the convex relaxation X xx . It is also
interesting to note that the relaxation 1.4 b ecomes the original problem 1.6
if we add the nonconvex constraint that the matrix on the left hand side
of 1.5 is rank one.
As an example, consider the 1; 1 -quadratic program
T T
minimize x Ax +2b x
1.7
2
sub ject to x =1;i=1;:::;k;
i
which is NP-hard. The constraint x 2f 1;1gcan b e written as the quadratic
i
2 2
equality constraint x = 1, or, equivalently,astwo quadratic inequalities x 1
i i
2 T
and x 1. Applying 1.4 we nd that the semide nite program in X = X
i
and x
T
minimize TrXA +2b x
sub ject to X =1;i=1;:::;k
ii
1.8
X x
0
T
x 1
yields a lower b ound for 1.7 . In a recent pap er on the MAX-CUT problem,
which is a sp eci c case of 1.7 where b = 0 and the diagonal of A is zero,
Go emans and Williamson have proved that the lower b ound from 1.8 is at
most 14 sub optimal see [GW94] and [GW95] . This is much b etter than any
previously known b ound. Similar strong results on semide nite programming
relaxations of NP-hard problems have b een obtained by Karger, Motwani, and
Sudan [KMS94].
The usefulness of semide nite programming in combinatorial optimization was
recognized more than twentyyears ago see, e.g., Donath and Ho man [DH73] .
Many p eople seem to have develop ed similar ideas indep endently.We should
however stress the imp ortance of the work by Grotschel, Lov asz, and Schrij-
ver [GLS88, Chapter 9], [LS91] who have demonstrated the p ower of semidef-
inite relaxations on some very hard combinatorial problems. The recent de-
velopment of ecientinterior-p oint metho ds has turned these techniques into
Semide nite programming relaxations 5
powerful practical to ols; see Alizadeh [Ali92b, Ali91, Ali92a], Kamath and Kar-
markar [KK92, KK93], Helmb erg, Rendl, Vanderb ei and Wolkowicz [HRVW94].
For a more detailed survey of semide nite programming in combinatorial opti-
mization, we refer the reader to the recent pap er by Alizadeh [Ali95].
3 SEMIDEFINITE PROGRAMMING AND
CONTROL THEORY
Semide nite programming problems arise frequently in control and system
theory; Boyd, El Ghaoui, Feron and Balakrishnan catalog many examples
in [BEFB94 ]. We will describ e one simple example here.
Consider the di erential inclusion
dx
= Ax t +Bu t ; y t =Cx t ; ju t j jy t j;i=1;:::;p 1.9
i i
dt
l p p
where x t 2 R , u t 2 R , and y t 2 R . In the terminology of control
theory, this is describ ed as a linear system with uncertain, time-varying, unity-
b ounded, diagonal feedback.
We seek an invariant ellipsoid, i.e., an ellipsoid E such that for any x and u
that satisfy 1.9 , x T 2E implies x t 2E for all t T . The existence
of such an ellipsoid implies, for example, that all solutions of the di erential
inclusion 1.9 are b ounded.
T T
The ellipsoid E = fx j x Px 1g, where P = P > 0, is invariant if and
T
only if the function V t =x t Px t is nonincreasing for any x and u that
satisfy 1.9 . In this case wesay that V is a quadratic Lyapunov function that
proves stability of the di erential inclusion 1.9 .
We can express the derivativeof V as a quadratic form in x t and u t :
T
T
d
x t A P + PA PB x t
V x t = : 1.10
T
u t B P 0 u t
dt
We can express the conditions ju t j jy t j as the quadratic inequalities
i i
T
T
x t c c 0 x t
i
2 2
i
t y t = 0; i =1;:::;p; u
i i
u t 0 E u t
ii 1.11
6 Chapter 1
where c is the ith rowofC, and E is the matrix with all entries zero except
i ii
the ii entry, whichis1.
Putting it all together we nd that E is invariant if and only if 1.10 holds
whenever 1.11 holds. Thus the condition is that one quadratic inequality
should hold whenever some other quadratic inequalities hold, i.e.:
l+p
T T
for all z 2 R ; z T z 0;i=1;:::;p = z T z 0 1.12
i 0
where
T T
A P + PA PB c c 0
i
i
T = ; T = ;i=1;:::;p:
0 i
T
B P 0 0 E
ii
In the general case, simply verifying that 1.12 holds for a given P is very
dicult. But an obvious sucient condition is
there exist 0;:::; 0 such that T T + + T : 1.13
1 p 0 i 1 p p
Replacing the condition 1.12 with the stronger condition 1.13 is called the
S -pro cedure in the Soviet literature on control theory, and dates back at least
to 1944 see [BEFB94, p.33], [FY79], [LP44] . Note the similaritybetween
Shor's b ound see 1.2 and 1.3 and the S -pro cedure 1.12 and 1.13 .
Indeed Shor's b ound is readily derived from the S -pro cedure, and vice versa.
Returning to our example, we apply the S -pro cedure to obtain a sucient
condition for invariance of the ellipsoid E : for some D = diag ;:::; ,
1 p
T T
A P + PA + C DC PB
0: 1.14
T
P D B
T
This is a linear matrix inequality in the matrix variables P = P and diag-
onal D . Hence, by solving a semide nite feasibility problem we can nd an
invariant ellipsoid if the problem is feasible . One can also optimize various
quantities over the feasible set; see [BEFB94]. Note that 1.14 is really a convex
relaxation of the condition that E be invariant, obtained via the S -pro cedure.
The close connections b etween the S -pro cedure, used in control theory to form
semide nite programming relaxations of hard control problems, and the various
semide nite relaxations used in combinatorial optimization, do not app ear to
be well known.
Semide nite programming relaxations 7
4 EXTENSION TO BILINEAR MATRIX
INEQUALITIES
Wenow consider an extension of the SDP 1.1 , obtained by replacing the
linear matrix inequality constraints by bilinear or bi-ane matrix inequalities
BMIs ,
T
minimize c x
m m
X X
1.15
sub ject to F + x F + x x G 0:
0 i i j k jk
i=1
j;k =1
m
The problem data are the vector c 2 R and the symmetric matrices F ,
i
nn
G 2 R .
jk
Bilinear matrix inequality problems are NP-hard, and include a wide varietyof
+
control problems see, e.g., [GLTS94], [GSP94], [SGL94], [GTS 94], [GSL95] .
They also include all quadratic problems when the matrices in 1.15 are di-
agonal , all p olynomial problems, all f0; 1g and integer programs, etc.
Several heuristic metho ds for BMI problems have b een presented in the litera-
ture cited ab ove and rep orted to b e useful in practice. Our purp ose here is to
p oint out that the semide nite relaxations for quadratic problems can b e easily
extended to bilinear matrix inequalities.
We rst express the BMI problem as
T
minimize c x
m m
X X
sub ject to F + x F + w G 0
0 i i jk jk
i=1
j;k =1
w = x x ;j;k=1;:::;m;
jk j k
and then relax the second constraintasanLMI
T
minimize c x
m m
X X
sub ject to F + x F + w G 0
0 i i jk jk
i=1
j;k =1
W x
0:
T
x 1
This is an SDP in the variables W , x. Its optimal value is a lower b ound for
the optimal value of problem 1.15 . As in Section 2, this SDP can also b e
interpreted as the dual of the Lagrangian relaxation of problem 1.15 .
8 Chapter 1
We should p oint out that the SDP relaxation may require some manipulation
e.g., when some of the matrices G are zero , just as in the general inde nite
ii
quadratic programming case.
We do not yet haveanynumerical exp erience with this SDP relaxation of the
BMI problem.
5 CONCLUSION
The simultaneous discovery of semide nite programming applications in con-
trol and combinatorial optimization is remarkable and raises several interesting
questions. For example,
can we obtain Go emans and Williamson-typ e results in control theory, i.e.,
solve a p olynomial-time SDP and get a guaranteed b ound on sub opti-
mality?
what is the practical p erformance of semide nite relaxations in nonconvex
quadratic or BMI problems?
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