A set,
, in a real inner product space is convex if for all
and
, where
, it holds that
.
The set of solutions to an LMI is convex.
That is, the set
is a convex set, where
is an LMI.
An LMI,
, in the variable
is an expression of the form
where
and
,
.
Consider
and
, and suppose that
and
satisfy Lemma 1.2.
The LMI
is convex, since
From Lemma 1.1, it is known that an optimization problem with a convex objective function and LMI constraints is convex.
The following is a non-exhaustive list of scalar convex objective functions involving matrix variables that can be minimized in conjunction with LMI constraints to yield a semi-definite programming (SDP) problem.
, where
,
,
, and
.
- Special case when
and
, where
,
, and
.
- Special case when
2
,
, and
, where
.
, where
,
,
,
,
, and
.
- Special case when
and
, where
,
, and
.
- Special case when
,
and
, where
.
- Special case when
,
and
, where
,
.
- Special case when
,
,
, and
, where
.
, where
and
.
The definiteness of a matrix can be found relative to another matrix.
For example,
Consider the matrices
and
. The matrix inequality
is equivalent to
or
.
Knowing the relative definiteness of matrices can be useful.
For example,
If in the previous example we have
and also know that
, when we know that
.
This follows from
.