Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method
We establish in this paper a group of closed-form formulas for calculating the global maximum and minimum ranks and inertias of the quadratic Hermitian matrix function ϕ ( X ) = Q − X P X ∗ with respect to the variable matrix X by using a linearization method and some known formulas for extremum ran...
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| Vydáno v: | Nonlinear analysis Ročník 75; číslo 2; s. 717 - 734 |
|---|---|
| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Amsterdam
Elsevier Ltd
2012
Elsevier |
| Témata: | |
| ISSN: | 0362-546X, 1873-5215 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | We establish in this paper a group of closed-form formulas for calculating the global maximum and minimum ranks and inertias of the quadratic Hermitian matrix function
ϕ
(
X
)
=
Q
−
X
P
X
∗
with respect to the variable matrix
X
by using a linearization method and some known formulas for extremum ranks and inertias of linear Hermitian matrix functions, where both
P
and
Q
are complex Hermitian matrices and
X
∗
is the conjugate transpose of
X
. We then derive the global maximum and minimum ranks and inertias of the two quadratic Hermitian matrix functions
ϕ
1
(
X
)
=
Q
1
−
X
P
1
X
∗
and
ϕ
2
(
X
)
=
Q
2
−
X
∗
P
2
X
subject to a consistent matrix equation
A
X
=
B
, respectively, by using some pure algebraic operations of matrices and their generalized inverses. As consequences, we establish necessary and sufficient conditions for the solutions of the matrix equation
A
X
=
B
to satisfy the quadratic Hermitian matrix equalities
X
P
1
X
∗
=
Q
1
and
X
∗
P
2
X
=
Q
2
, respectively, and for the quadratic matrix inequalities
X
P
1
X
∗
>
(
⩾
,
<
,
⩽
)
Q
1
and
X
∗
P
2
X
>
(
⩾
,
<
,
⩽
)
Q
2
in the Löwner partial ordering to hold, respectively. In addition, we give complete solutions to four Löwner partial ordering optimization problems on the matrix functions
ϕ
1
(
X
)
and
ϕ
2
(
X
)
subject to
A
X
=
B
. Examples are also presented to illustrative applications of the equality-constrained quadratic optimizations in some matrix completion problems. |
|---|---|
| Bibliografie: | ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23 |
| ISSN: | 0362-546X 1873-5215 |
| DOI: | 10.1016/j.na.2011.09.003 |