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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| Published in | Nonlinear analysis Vol. 75; no. 2; pp. 717 - 734 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Amsterdam
Elsevier Ltd
2012
Elsevier |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0362-546X 1873-5215 |
| DOI | 10.1016/j.na.2011.09.003 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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 phi (X)=Q-XPX* 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 phi sub(1)(X)=Q sub(1)-XP sub(1)X* ; and phi sub(2)(X)=Q sub(2)-X*P sub(2) X subject to a consistent matrix equation AX=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 AX=B to satisfy the quadratic Hermitian matrix equalities XP sub(1)X*=Q sub(1) and X*P sub(2)X=Q sub(2), respectively, and for the quadratic matrix inequalities [inline image] and [inline image] in the Lowner partial ordering to hold, respectively. In addition, we give complete solutions to four Lowner partial ordering optimization problems on the matrix functions phi sub(1)(X) and phi sub(2)(X) subject to AX=B. Examples are also presented to illustrative applications of the equality-constrained quadratic optimizations in some matrix completion problems. |
| Author | Tian, Yongge |
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| Keywords | 65K15 Löwner partial ordering Rank Matrix inequality 15B57 15A09 Generalized inverse Linearization method Optimization 15A63 15A24 15B10 65K10 Inertia Quadratic matrix function Linear matrix function Matrix equation Optimization method Partial ordering Lowner partial ordering Mathematical model Linearization Nonlinear analysis Matrix function Algebraic method Problem solving |
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| Snippet | 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... |
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| SubjectTerms | Algebra Calculus of variations and optimal control Exact sciences and technology Finite differences and functional equations Generalized inverse Inertia Linear and multilinear algebra, matrix theory Linear matrix function Linearization method Löwner partial ordering Mathematical analysis Mathematics Matrices Matrix equation Matrix inequality Nonlinear algebraic and transcendental equations Nonlinearity Numerical analysis Numerical analysis. Scientific computation Optimization Order disorder Quadratic matrix function Rank Sciences and techniques of general use |
| Title | Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method |
| URI | https://dx.doi.org/10.1016/j.na.2011.09.003 https://www.proquest.com/docview/963907356 |
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