A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$

This paper is concerned with some rank and inertia optimization problems of the Hermitian matrix-valued functions $A + BXB^{*}$ subject to restrictions. We first establish several groups of explicit formula for calculating the maximum and minimum ranks and inertias of matrix sum $A + X$ subject to a Hermitian matrix $X$ that satisfies a fixed-rank and semi-definiteness restrictions by using some discrete and matrix decomposition methods. We then derive formulas for calculating the maximum and minimum ranks and inertias of the matrix-valued function $A + BXB^*$ subject to a Hermitian matrix $X$ that satisfies a fixed-rank and semi-definiteness restrictions, and show various properties 
 $A + BXB^{*}$ from these ranks and inertias formulas. In particular, we give necessary and sufficient conditions for 
 the equality $A + BXB^* = 0$ and the inequality $A + BXB^* \succ 0\, (\succeq 0, \prec 0, \, \preceq 0)$ to hold respectively for these specified Hermitian matrices $X$.

Problem 1.2. For the given A + BXB * in (1.1), solve the following constrained rank/inertia optimization problems maximize r( A + BXB * ) s.t. X ∈ C n H and p r(X) q, (1.10) minimize r( A + BXB * ) s.t. X ∈ C n H and p r(X) q, (1.11) maximize i ± ( A + BXB * ) s.t. X ∈ C n H and p r(X) q, (1.12) minimize i ± ( A + BXB * ) s.t. X ∈ C n H and p r(X) q, (1.13) maximize r( A + BXB * ) s.t. X ∈ C n H , ±X 0 and p r(X) q, (1.14) minimize r( A + BXB * ) s.t. X ∈ C n H , ±X 0 and p r(X) q, (1.15) maximize i ± ( A + BXB * ) s.t. X ∈ C n H , ±X 0 and p r(X) q, (1.16) minimize i ± ( A + BXB * ) s.t. X ∈ C n H , ±X 0 and p r(X) q. (1.17) The LMF φ(X) in (1.1) and its variations were extensively studied in the literature from theoretical and applied points of view due to its simple and symmetric pattern, and many results on behaviors of φ(X) were obtained. Some recent work done by the present author and his collaborators on φ(X) is summarized below: (i) expansion formulas for calculating the rank/inertia of φ(X), as well as algebraic methods for finding maximum and minimum rank/inertias of φ(X) when X running over C n H [8,18,27]; (ii) characterizations of algebraic properties of φ(X), such as, the nonsingularity, positive definiteness, rank invariance, inertia invariance, range invariance, etc., of φ(X) [18,27]; (iii) canonical forms of φ(X) under generalized singular value decompositions and their algebraic properties [8]; (iv) solutions and least-squares solutions of the matrix equation φ(X) = 0 and their algebraic properties [7,8,11,21,23,25]; (v) solutions of the matrix inequalities φ(X) 0 ( 0, ≺ 0, 0) and their properties [18,22]; (vi) formulas for calculating the maximum and minimum ranks/inertias of φ(X) under X ∈ C n H , r(X) q and/or ±X 0 [18,27]; (vii) formulas for calculating the maximum and minimum ranks/inertias of φ(X) subject to Hermitian solutions of one or two consistent matrix equations [9,26]; (viii) formulas for calculating the maximum and minimum ranks/inertias of the Schur complement A + BC − B * , where C − is a Hermitian generalized inverse of a Hermitian matrix C, [9,24]. The maximization and minimization of ranks/inertias of LMFs could be regarded as a special kind of continuous-integer optimization problems. However, this kind of optimization problems cannot be handled by various known methods for solving continuous or discrete optimization problems because matrix multiplications occurred in matrix-valued functions are not necessarily commutative. In fact, there does not exist a rigorous mathematical theory for solving optimization problems on ranks/inertias of LMFs, but we are really able to solve some special cases, as mentioned above, by using matrix decompositions, generalized inverses of matrices, and some tricky matrix operations.
Note that for any two integers p and q satisfying 0 p q m, the following decompositions of the generalized Stiefel manifolds {X ∈ C m H | p r(X) q} = {X ∈ C m H | r(X) = p} ∪ { X ∈ C m H | r(X) = p + 1} ∪ · · · ∪ {X ∈ C m H | r(X) = q} , (1.18) and {0 X ∈ C m H | p r(X) q} = {0 X ∈ C m H | r(X) = p} ∪ {0 X ∈ C m H | r(X) = p + 1} ∪ · · · ∪ {0 X ∈ C m H | r(X) = q} (1. 19) hold. Once Problem 1.1 is solved, solutions of Problem 1.2 can be obtained consequently from the above matrix set decompositions. The constrained optimization problems formulated in (1.2)-(1.17) consist of determining the global maximum and minimum numbers of ranks/inertias of φ(X), and finding the constrained variable matrix X such that the corresponding φ(X) attains the maximum and minimum numbers, respectively. The Hermitian matrix X with r(X) = q is not unique, and can be characterized by the following canonical decomposition under the *congruence transformation: where U is any nonsingular matrix of order n, and 0 s q with r(X) = q. This canonical decomposition shows that a Hermitian matrix X with rank q is characterized by both a variable integer s with 0 s q and an arbitrary nonsingular matrix U . So that both BXB * and A + BXB * depend on the choices of both s and U . In this setting, (1.2)-(1.17) can be classified as some special types of constrained integer optimization problem, and thus we can only derive the maximum and minimum ranks/inertias of φ(X) via pure algebraic operations of matrices. The tasks described in (1.2)-(1.17) are challenging, but fortunately, we now are able to derive closed-form solutions by using many known and new formulas on ranks/inertias of matrices and some tricky matrix operations.
2. Preliminary results. The following are some known assertions on ranks /inertias of matrices, which will be used in the latter part of this paper for solving the previous problems.
This lemma indicates that rank/inertia of matrix can be used as quantitative tools to characterize some fundamental algebraic properties of a given matrix set. In particular, whether a given matrix-valued function is semi-definite everywhere is ubiquitous in matrix theory and applications. Lemma 2.1(e)-(h) assert that if certain explicit formulas for calculating the maximum and minimum inertias of Hermitian matrix-valued functions are established, we can use them, as demonstrated in Sections 2-5 below, to derive explicit necessary and sufficient conditions for the matrix-valued functions to be definite or semi-definite. In addition, we are able to use these inertia formulas to establish various matrix inequalities in the Löwner partial ordering, and to solve many matrix optimization problems in the Löwner partial ordering. 3) and Q ∈ C m×n , and assume that P ∈ C m×m is nonsingular. Then, 14]). Let A ∈ C m×n , B ∈ C m×k , and C ∈ C l×n . Then, the following rank expansion formulas hold Lemma 2.5 ( [18]). Let A ∈ C m H , B ∈ C m×n , and D ∈ C n H , and let

YONGGE TIAN
Then, the following inertia/rank expansion formulas hold In particular, the following results hold.
(a) If A 0, then The following three lemmas will play essential roles in solving Problems 1.1 and 1.2.
Lemma 2.6. Let X ∈ C m×n and Y ∈ C n H be two variable matrices, and let Then, the following results hold.
For any integer q with 0 q min{ m + n, 2n }, there exist X ∈ C m×n and Y ∈ C n H such that r[φ(X, Y )] = q.
(2.28) (c) For any integer q with 0 q n, there exist X 1 , X 2 ∈ C m×n and Y 1 , hold, respectively.
For any integer q with 0 q min{ m + n, 2n }, there exist X ∈ C m×n and 0 Y ∈ C n H such that r[φ(X, Y )] = q.
(2.36) (f) For any integer q with 0 q n, there exist X ∈ C m×n and 0 Y ∈ C n H such that (2.37) (g) For any integer q with 0 q min{ m, n }, there exist X ∈ C m×n and 0 Y ∈ C n H such that Proof. Let r(X) = min{m, n} and Y = I n in (2.23). Then, we find from (2.20) that For any integer 0 q n, setting X = 0 and r(Y ) = q in (2.23) leads to for any integer n < q min{ m + n, 2n }, setting r(X) = q − n and Y = I n in (2.23) and applying (2.20) lead to For any integer 0 q n, setting X = 0 and Y 1 = diag( I q , 0 ) and Y 2 = diag( −I q , 0 ) in (2.23), and applying (2.20) leads to Let r(X) = min{m, n} and Y = I n 0 in (2.23). Then, we find from (2.20) that r[φ(X, Y )] = r[ X, Y ] + r(X) = n + r(X) = min{ m + n, 2n }, establishing (2.30). Setting X = 0 and Y = 0 in (2.23) leads to (2.31). For any X and Y = I n 0 in (2.23), we find from (2.20) that establishing (2.32). Setting X = 0 and Y = 0 in (2.23) leads to (2.33). Setting r(X) = min{ m, n } and Y 0, we find from (2.20) that establishing (2.34). Setting X = 0 and Y = 0 in (2.23) leads to (2.35). For any integer 0 q n, setting X = 0 and Y = diag(I q , 0 ) 0 in (2.23) leads to r[φ(X, Y )] = r(Y ) = q; for any integer n < q min{m + n, 2n} , setting r(X) = q − n and Y = I n 0 in (2.23), and applying (2.20) lead to For any integer 0 q n, setting X = 0 and Y = diag( I q , 0 ) 0 in (2.23) For any integer 0 q min{m, n} , setting r(X) = q and Y = 0 in (2.23) and applying (2.20) H be given, X ∈ C m×n and Y ∈ C n H be two variable matrices, and let Then, the following results hold.
Proof. Without lost generality, we assume that A is given by Correspondingly, φ(X, Y ) in (2.39) can be written as ( (2.68) Applying Lemma 2.6(a) to (2.66)-(2.68) leads to   ). Let A ∈ C m H and B ∈ C m×n be given. Then, there exist a unitary matrix U ∈ C n×n and a nonsingular matrix P ∈ C m×m such that The structures of the blocks in (2.81)-(2.85) and the scalars in (2.86)-(2.93) are formulated more explicitly, which improve the results in [29] and [6]. 3. Formulas for the rank/inertia of A + X subject to rank and semidefinite restrictions. One of the special cases in (1.1) is the ordinary sum A + X. Many results on equalities and inequalities of rank/inertia of sum of two Hermitian matrices were established in the literature; see, e.g., [2,5,18]. Note that the rank/inertia of A + X may vary with respect to the choice of the variable Hermitian matrix X. In this section, we derive explicit formulas for calculating the extremum ranks/inertias A + X subject to X ∈ C m H and X 0, respectively. The formulas obtained will be used in Sections 4 and 5 for deriving general rank/inertia formulas of A + BXB * .
H be given, X ∈ C m H be a variable matrix, and assume that p and q are two integers satisfying 0 p q m. (3.1) Then, The following results hold. (i) For any integer t 1 between the two quantities on the right-hand sides of (3.2) and (3.3), there exists an X ∈ C m×m H with r(X) = q such that r( A + X ) = t 1 .
(ii) For any integer t 2 between the two quantities on the right-hand sides of (3.4) and (3.5), there exists an X ∈ C m×m H with r(X) = q such that i + ( A + X ) = t 2 .
(iii) For any integer t 3 between the two quantities on the right-hand sides of (3.6) and (3.7), there exists an X ∈ C m×m H with r(X) = q such that i − ( A + X ) = t 3 . (iv) There exists an X ∈ C m H with r(X) = q such that A + X is nonsingular if and only if r(A) m − q.
The following statements hold.
(i) For any integer t 1 between the two quantities on the right-hand sides of (3.8) and (3.9), there exists an X ∈ C m×m H with p r(X) q such that r( A + X ) = t 1 .
(ii) For any integer t 2 between the two quantities on the right-hand sides of (3.10) and (3.11), there exists an X ∈ C m×m H with p r(X) q such that i + ( A + X ) = t 2 .
(iii) For any integer t 3 between the two quantities on the right-hand sides of (3.12) and (3.13), there exists an X ∈ C m×m H with p r(X) q such that i − ( A + X ) = t 3 . (iv) There exists an X ∈ C m H with p r(X) q such that A + X is nonsingular if and only if r(A) m − q. The Hermitian matrices Xs satisfying these equalities can be formulated from the canonical form of A under the Hermitian congruence.
Proof. It is easy to see from (2.1)-(2.4) that the right-hand sides of (3.2), (3.4), and (3.6) are upper bounds, while the right-hand sides of (3.3), (3.5), and (3.7) are lower bounds. Without loss of generality, we assume that A is of the form The establishments of (3.2)-(3.7) are based on the following assertions:   When the variable matrix X in A±X runs over the cone of positive semi-definite matrices, we have the following results.
H be given, X ∈ C m H be a variable matrix, and assume that p and q are two integers satisfying 0 p q m. Then, the following results hold.
(a) The rank/inertia of A ± X satisfy max 0 X, r(X)=q The following statements hold.
(i) For any integer t 1 between the two quantities on the right-hand sides of (3.39) and (3.40), there exists a 0 X ∈ C m×m H with r(X) = q such that r( A + X ) = t 1 .
(ii) For any integer t 2 between the two quantities on the right-hand sides of (3.41) and (3.42), there exists a 0 X ∈ C m×m H with r(X) = q such that i + ( A + X ) = t 2 .
(iii) For any integer t 3 between the two quantities on the right-hand sides of (3.43) and (3.44), there exists a 0 X ∈ C m×m H with r(X) = q such that i − ( A + X ) = t 3 .
(iv) For any integer t 4 between the two quantities on the right-hand sides of (3.45) and (3.46), there exists a 0 X ∈ C m×m H with r(X) = q such that r( A − X ) = t 4 .
(v) For any integer t 5 between the two quantities on the right-hand sides of (3.47) and (3.48), there exists a 0 X ∈ C m×m H with r(X) = q such that i + ( A − X ) = t 5 .
(vi) For any integer t 6 between the two quantities on the right-hand sides of      4. Rank and inertia formulas of A + BXB * when X is Hermitian with a fixed rank. Based on the results in the previous two sections, we are able to derive explicit solutions to Problems 1.1 and 1.2. To do so, we need the following known result on the canonical form of A + BXB * ; see [10]. for Y = U XU * and X ∈ C n H .
(iii) Partition the Hermitian matrix Y as , (4.5) where Y ii = Y * ii , i = 1, 2, 3, 4. Then, 6) and the following expansion formulas hold (4.10) It can be seen from (4.7)-(4.9) that max X∈C n H r( A + BXB * ) = 2f + k + max In the past two decades, many numerical methods for solving rank minimization problems were developed. It seems, however, that there is no solid matrix theory to support the conclusions obtained by various approximation methods on the rank minimization problems. Because all formulas in the previous sections are given in analytical forms, people can utilize the results in this paper as test examples to check the validity of various numerical methods for solving rank minimization problems.