ROBUST SOLUTIONS TO UNCERTAIN
SEMIDEFINITE PROGRAMS
*
LAURENT EL GHAOUI
†
, FRANCOIS OUSTRY
†
,
AND
HERV
´
E LEBRET
†
SIAM J. O
PTIM
.
c
±
1998 Society for Industrial and Applied Mathematics
Vol. 9, No. 1, pp. 33–52
Abstract.
In this paper we consider semideﬁnite programs (SDPs) whose data depend on some
unknown but bounded perturbation parameters. We seek “robust” solutions to such programs, that
is, solutions which minimize the (worst-case) objective while satisfying the constraints for every
possible value of parameters within the given bounds. Assuming the data matrices are rational
functions of the perturbation parameters, we show how to formulate suﬃcient conditions for a robust
solution to exist as SDPs. When the perturbation is “full,” our conditions are necessary and suﬃcient.
In this case, we provide suﬃcient conditions which guarantee that the robust solution is unique and
continuous (H¨older-stable) with respect to the unperturbed problem’s data. The approach can thus
be used to regularize ill-conditioned SDPs. We illustrate our results with examples taken from linear
programming, maximum norm minimization, polynomial interpolation, and integer programming.
Key words.
convex optimization, semideﬁnite programming, uncertainty, robustness, regular-
ization
AMS subject classiﬁcations.
93B35, 49M45, 90C31, 93B60
PII.
S1052623496305717
Notation.
For a matrix
X
,
k
X
k
denotes the largest singular value. If
X
is
square,
X
±
0 (resp.,
X
²
0) means
X
is symmetric and positive semideﬁnite (resp.,
deﬁnite). For
X
±
0,
X
1
/
2
denotes the symmetric square root of
X
. The notation
I
p
denotes the
p
×
p
identity matrix; the subscript is omitted when it can be inferred
from context.
1. Introduction.
A semideﬁnite program (SDP) consists of minimizing a linear
objective under a linear matrix inequality (LMI) constraint; precisely,
P
0
:
minimize
c
T
x
subject to
F
(
x
)=
F
0
+
m
X
i
=1
x
i
F
i
±
0
,
(1)
where
c
∈
R
m
-{
0
}
and the symmetric matrices
F
i
=
F
T
i
∈
R
n
×
n
,i
=0
,...,m
,
are given. SDPs are convex optimization problems and can be solved in polynomial
time with, e.g., primal-dual interior-point methods [24, 35, 26, 19, 2]. SDPs include
linear programs and convex quadratically constrained quadratic programs, and arise
in a wide range of engineering applications; see, e.g., [12, 1, 35, 22].
In the SDP (1), the “data” consist of the objective vector
c
and the matrices
F
0
,...,F
m
. In practice, these data are subject to uncertainty. An extensive body of
work has concentrated on the sensitivity issue, in which the perturbations are assumed
to be inﬁnitesimal, and regularity of optimal values and solution(s), as functions of
the data matrices, is studied. Recent references on sensitivity analysis include [30,
31, 10] for general nonlinear programs, [33] for semi-inﬁnite programs, and [32] for
semideﬁnite programs.