Construction of a Class of Sharp Löwner Majorants for a Set of Symmetric Matrices

The Löwner partial order is taken into consideration in order to define Löwner majorants for a given finite set of symmetric matrices. A special class of Löwner majorants is analyzed based on two specific matrix parametrizations: a two-parametric form and a four-parametric form, which arise in the context of so-called zeroth-order bounds of the effective linear behavior in the field of solid mechanics in engineering. The condensed explicit conditions defining the convex parameter sets of Löwner majorants are derived. Examples are provided, and potential application to semidefinite programming problems is discussed. Open-source MATLAB software is provided to support the theoretical findings and for reproduction of the presented results. The results of the present work offer in combination with the theory of zeroth-order bounds of mechanics a highly efficient approach for the automated material selection for future engineering applications.

In the field of materials science, semidefinite programming problems arise, e.g., in the context of zeroth-order bounds of linear elastic properties of solids. e zeroth-order bounds were introduced by [15] in the context of statistical bounds of linear elastic properties, further analyzed by [16], and corrected by [17,18]. For instance, an N-phase solid is constituted of N materials with corresponding symmetric positive-definite stiffness matrices, say, A (1) � , . . . , A (N) � ∈ R n×n with n � 6 for three-dimensional linear elasticity. A zerothorder bound of the effective linear material behavior of the N-phase solid is a symmetric matrix B � ∈ R n×n which satisfies 0 ≤ x T (B � − A (i) � )x , ∀x ∈ R n , ∀i ∈ 1, . . . , N { } for all possible realizations of the solid. In solid mechanics, from realization to realization of a composite material, the orientation of the material constituents may change, i.e., the direction of the eigensystem of each of the stiffnesses A (i) � can vary from realization to realization. For practical reasons, see, e.g., [17] or [19] for details, the zeroth-order bound is chosen as an isotropic stiffness such that the orientation of the eigensystems of the stiffnesses A (i) problem can be tackled with a high number of general techniques of semidefinite programming. But, since the zeroth-order bounds are chosen as isotropic, the number of free components of B � (β) reduces significantly, in linear elasticity to two, and the structure of B � (β) is highly specific. An analogous problem can be formulated for linear thermoelasticity, cf. [19], where the number of parameters is four. e low number of parameters in these problems immediately suggests to condense the optimization constraints to a couple of explicit conditions in terms of the parameters β, which are then easily and more efficiently treatable with standard methods.
is would significantly reduce the computational time of the optimization problem defining the zeroth-order bounds.
is efficiency perspective is particularly of practical importance since, as described in [19], large material databases for solids with multiple constituents could then be scanned, greatly benefiting the material selection in material design problems of engineering applications.
is is not only relevant for linear elasticity but also for many associated physical problems, e.g., linear heat or electric conductivity, see, e.g., [20,21] or [16], such that an investigation of the optimization problem defining the zeroth-order bounds in elasticity would greatly benefit all associated physical problems and corresponding material selection approaches. e focus of the present work is the proper algebraic analysis and condensation of the optimization constraints to more easily treatable explicit conditions for the two-and four-parametric forms B � (β). is parameter constraint condensation has not been conducted in either [15][16][17][18] or [19]. For the present investigation, the Löwner partial order is considered, and the concept of Löwner majorants of a single matrix-representing, e.g., a single stiffness matrixand of a finite matrix set-corresponding to the set of material stiffnesses of a composite material-is introduced. e derived explicit conditions for the parameters β offer compact results which allow to solve nonlinear semidefinite programming problems over the set of resulting Löwner majorants. Most importantly, the evaluation of the constraints is independent of the matrix dimension (modulo a once in a lifetime setup cost that can be performed before the optimization initiation), and it does not require repeated evaluation of subdeterminants of matrices. Additionally, a deterministic construction of the solution is possible for linear cost functions that are of complexity O(N) for the two-parametric case. Furthermore, geometric interpretations of the stationary condition are provided. e parametrizations are kept as general as possible such that the results of the present work may be of use for other semidefinite programming problems as an efficient generator of simplified solutions or initial guesses. e manuscript is structured as follows: In Section 2, Löwner majorants of a single symmetric matrix and finite set of symmetric matrices are defined. e two-parametric form is investigated in Section 3, and the explicit conditions defining all corresponding majorants are derived. Examples for the parameter sets of the two-parametric Löwner majorants are given at the end of the section. en, in Section 4, the four-parametric form is examined building upon the results of Section 3. Examples for the parameter sets of the Löwner majorants are demonstrated at the end of the section. Furthermore, Section 5 shows an example application of the results in nonlinear semidefinite programming problems and shortly discusses the importance of the examination of the optimization domain and functions to be optimized, since the existence of a minimum is not always assured, even for convex optimization domains and convex functions. Conclusions and potential applications are discussed in Section 6. For full transparency, the authors offer through the GitHub repository [22] https://github.com/EMMA-Group/LoewnerMajorant open-source MATLAB software containing all programs and data required for the reproduction of all shown examples of the present work.
Notation. roughout this manuscript, the set of real numbers is denoted by R. Column vectors over R n , n ∈ 2, 3, . . . { }, are denoted by single-underlined characters, e.g., x, y. Rectangular matrices over R m×n are denoted by doubleunderlined characters, e.g., A � , B � . Square matrices over R n×n are referred to as matrices of order n. e set of symmetric matrices of order n with finite eigenvalues is denoted by S n . e transposition is denoted by the superscript (·) T , e.g., a T b equals the scalar product of the vectors, and a b T yields the outer product of the vectors. e eigenvalues of a symmetric matrix A � are denoted by λ i (A) ∈ R with i � 1, . . . , n. e multiplicity of an eigenvalue λ is denoted as m λ . e l 2 norm of a vector a and the Frobenius norm of a matrix A are simply noted as ‖a‖ and ‖ A ‖. e identity matrix is noted as I. For a compact notation, the orthogonality of two vectors p and q or two vector spaces P and Q is denoted simply as p ⊥ q and P ⊥ Q, respectively. e Moore-Penrose inverse of a matrix is denoted by A + .

Löwner Order and Majorants for Symmetric Matrices
In this work, we consider the Löwner partial order of symmetric matrices A, B ∈ S n (see, e.g., [6]): where S + n ⊂ S n denotes the positive semidefinite cone. Any B ∈ S n is referred to, in this work, as a (Löwner) majorant of a given A ∈ S n if A ⪯ B holds. Furthermore, any B ∈ S n is referred to as a sharp (Löwner) majorant of a given A ∈ S n if A ⪯ B and det(B − A) � 0 hold, i.e., if B − A ∈ zS + n holds. For given A ∈ S n , the corresponding majorant set B ⊂ R n×n and sharp majorant set zB are denoted as Of course, B is nonempty, unbounded, and convex. e trivial majorant T ∈ zB of a given A is defined as 2 Journal of Applied Mathematics (4) For a finite set A � A (1) , . . . , A (N) of symmetric matrices, we define the analogous majorant set (5) whose boundary zB A represents the set of all sharp majorants of A: e trivial element on zB A is

Construction of Two-Parametric Majorants for a Single
Matrix. For given A ∈ S n , from the corresponding majorant set B, we investigate a two-parametric form with given (i.e., fixed) normalized vector p 0 ∈ R n , p 0 � 1, defined as a rank-one perturbation of a scaled identity via In the context of zeroth-order bounds of the effective linear elastic behavior of solid materials, , 0, 0, 0) is given for isotropic zeroth-order bounds if a specific parametrization is chosen, cf., e.g., [16] or [17] for details. e goal of this section is the derivation of condensed conditions for the parameters β such that B � 0 2 ∈ B holds. For the sake of a clearer perspective in this section, we consider the spectral factorization of the given A � U Λ U T with ordered eigenvalue diagonal matrix Λ with diagonal entries and corresponding orthogonal eigenvector matrix U, i.e., U U T � U T U � I. We denote the set of eigenvectors of Λ as E and the space of eigenvectors corresponding to a particular eigenvalue as E i , i.e., A change of basis does not alter the Löwner partial order, i.e., and note ‖ p ‖ � ‖p 0 ‖ � 1. For the remainder of this section, we seek the conditions on the parameters β � (β 1 , β 2 ) describing the parameter set Due to linear dependency of B 2 on β and the general properties of B ⊂ R n×n , the parameter set B 2 ⊂ R 2 is nonempty, unbounded, and convex. Note that β ∈ zB 2 ⟺ B � 0 2 (β) ∈ zB holds. Due to the length and technical nature of several passages and proofs, the current section is: (i) Preparations: introduction of several expressions and relations needed for all following lemmas and remarks (ii) Lemma 1: closed form description of B 2 for p ∈ E (iii) Remark 1: remarks for cases on zB 2 for Lemma 1 (iv) Lemma 2: closed form description of B 2 for p ∉ E and β 1 bounded from below by Λ 1 or Λ 2 (v) Remark 2: remarks for cases on zB 2 for Lemma 2 (vi) Lemma 3: closed form description of B 2 for p ∈ E and β 1 bounded from below by a constant between Λ 1 and Λ 2 (vii) Corollary 1: recapitulation of some topological properties of B 2 relevant for semidefinite programming problems (viii) Remark 3: remarks for cases on zB 2 for Lemma 3 (ix) Remark 4: summarizing remarks and perspective for corresponding semidefinite programming problems Preparations. Before we derive the conditions describing B 2 in (12), we introduce some quantities and relations needed throughout this section. e condition Λ ≺ B 2 is equivalent to the positive semidefinitness of the difference matrix C: i.e., We denote the dimension of the kernel of the difference matrix C as We define the mutually orthogonal vector spaces P � span p , and introduce a matrix Q ∈ R n×(n− 1) such that the concatenated matrix (p, Q) ∈ R n×n is orthogonal, i.e., the columns of Q form an orthonormal basis of Q. We define where μ Q corresponds to the largest eigenvalue of the reduced matrix Q T Λ Q ∈ S n− 1 . Inserting vectors from P and Q into (14), respectively, yields the two elementary necessary conditions: Journal of Applied Mathematics Note that μ Q is bounded by Λ 1 due to Furthermore, the relation is obtained based on the following arguments: erefore, generally holds. Denote the algebraic multiplicity of the maximum eigenvalue Λ 1 by m Λ 1 . It is noted that and based on (22), holds. Based on these preparations, we now proceed to the description of the parameter set B 2 through three lemmas.

Lemma 1.
If p ∈ E holds, then the parameter set B 2 is described by Proof. If p ∈ E, corresponding to Λ i for some i ∈ 1, . . . , n { }, then (18) and (19) simplify to For this case, (27) is equivalent to Λ ⪯ B 2 such that B 2 defined in (12) simplifies to (26).
□ Remark 1. For Lemma 1, all β ∈ zB 2 (i.e., at least one of the inequalities in (26) turns into an equality) are sharp majorants and induce a singular C � B 2 − Λ with nonempty kernel ker(C) of dimension κ, cf. (15). e special cases for β ∈ zB 2 deliver a one-dimensional kernel of C. It is noted that only (29) ensures κ � 1, independent of the multiplicity m μ Q of μ Q , and, more importantly, that ker(C) ⊥ P holds. is property is of interest for Section 4.

Lemma 2.
If p ∉ E and μ Q ∈ Λ 1 , Λ 2 hold, then the parameter set B 2 is described: (1) For μ Q � Λ 1 and eigenspace E 1 corresponding to Λ 1 as , Λ 2 based on the following cases: Due to the restriction Λ 1 < β 1 of this case, the diagonal matrix D � β 1 I − Λ is positive definite such that, based on the determinant lemma, cf., e.g., [23] or [24], we consider which is a single-valued constraint in terms of meaning that (34) describes for every β 1 ∈ (Λ 1 , ∞) the corresponding β 2 yielding a sharp majorant, i.e., (34) specifies a portion of zB 2 . Due to the convexity of positive semidefinite. We consider the matrices which contain in the respective columns a basis of E 1 for Λ 1 and of the corresponding orthogonal space E ⊥ 1 . We examine the following cases: (a) If P ⊥ E 1 holds, then C is already singular, independently of β 2 , since holds, meaning that ker(C) is at least one-dimensional. Based on (35) and (14) can be further reduced to e matrix D � is positive definite, and since p is not an eigenvector of Λ, p ≠ o holds. For C to be singular, its determinant has to vanish. Analogous application of the matrix determinant lemma as in (33) yields that C is singular iff where D + denotes the Moore-Penrose inverse of D. e choice (38) induces a second vanishing eigenvalue in C such that its kernel is then at least two-dimensional. (b) If P ⊥ E 1 holds, then using (35) and (14) yields the necessary condition is condition is also sufficient for Λ ⪯ B 2 . For β 2 � 0, one retrieves for the current scenario the trivial majorant β � (Λ 1 , 0) ∈ zB 2 , cf. (4).
If μ Q � Λ 1 holds, cf. (24), then cases 1 and 2 of this proof describe β min 2 , at what c Q ≠ 0, cf. (17), is identified in (38), yielding (30)-cf. Lemma 2.1 as depicted in Figure 1 in the ramification for p ∉ E and μ Q � Λ 1 . If Λ 2 < Λ 1 and μ Q � Λ 2 hold, cf. (25), then case 1 of this proof still applies, but the following cases relevant for Lemma 2.2 and Lemma 2.3, depicted in Figure 1, also need to be considered: indefinite and regular such that the determinant lemma can be applied analogously as in (33), but (34) is not immediately clear since the term p T D − 1 p may vanish for such indefinite D. More precisely, the term p T D − 1 p may vanish, iff D − 1 p ∈ Q holds. We search now for a vector q � Q ξ ∈ Q such that p � D q holds, with the necessary condition is means that ξ would be required to correspond to the eigenvector of the reduced matrix Q T Λ Q for eigenvalue β 1 . But, we consider β 1 larger than the maximum eigenvalue μ Q � Λ 2 of Q T Λ Q such that no such ξ exists, and therefore, the term p T D − 1 p cannot vanish. Solving (33) for β 2 yields again (34). (4′) β 1 � Λ 2 : Due to P ⊥ E 2 , analogous reasoning as in 2(a) is applied to this case with corresponding E 2 and E � ⊥ 2 up to the corresponding equation e difference to 2(a) here is that D is not positive definite but regular and indefinite. For C to be singular, its determinant must vanish. Application of the matrix determinant lemma as in (33) yields Here, the term p T D − 1 p � p T D + p equals the quantity c Q for μ Q � Λ 2 , see (17). e quantity c Q may or may not vanish, depending solely on Λ and p. is means that if c Q ≠ 0, then (42) can be solved for β 2 as in (38), yielding β 1 ∈ [μ Q , ∞) and (31)-cf. Lemma 2.2 as depicted in Figure 1. But, if c Q � 0 holds, then there exists no β 2 for β 1 � Λ 2 such that (β 1 , β 2 ) ∈ B 2 holds since det(C) cannot vanish. erefore, c Q � 0 yields (32) and, more importantly, excludes μ Q � Λ 2 from the range of β 1 for the current scenario, i.e., β 1 ∈ (μ Q , ∞), cf. Lemma 2.3 illustrated in Figure 1.

Journal of Applied Mathematics
Remark 2. It is noted that, for Lemma 2, κ � 1, cf. (15), holds in a number of scenarios. In order to visualize the following argument, the position of the upcoming cases in the set B 2 is depicted in Figure 2.
In cases 1 and 3′ of the proof of Lemma 2, the rank-one perturbation β 2 p p T can only induce a one-dimensional kernel on the difference matrix C � D +β 2 p p T with regular D for β 2 fulfilling (34). is corresponds to the majority of points described through β min 2 in Lemma 2, only excluding the special case β 1 � Λ 1 . e corresponding kernel vector n with ker(C) � span n is obtained as holds since 0 ≠ p T n � p T D − 1 p holds. e corresponding points β ∈ zB 2 based on β min 2 inducing (44) are indicated in Figure 2 at the lower border of B 2 with κ � 1, and more importantly, due to the consequences of Section 4, ker(C) ⊥ P.
For μ Q � Λ 1 , i.e., Lemma 2.1 is considered, the point at β 1 � μ Q � Λ 1 is to be examined. Hereby, cases 2(a) and 2(b) of the proof of Lemma 2 need some attention. In case 2(a), i.e., P ⊥ E 1 , C is already singular (C e 1 � o, i.e., e 1 ∈ ker(C)) and for β 2 fulfilling (38), the rank-one perturbation β 2 p p T induces a further vanishing eigenvalue of C such that ker(C) is at least two-dimensional. is special case corresponds to the black point in Figure 2. For β 2 above the critical value given in (38), ker(C) is then at least one-dimensional such that κ � 1 is only possible at β 1 � μ Q � Λ 1 under the current assumptions only for m Λ 1 � 1, i.e., In case 2(b), i.e., P ⊥ E 1 , p is not perpendicular to the complete eigenspace of Λ 1 , but for m Λ 1 ≥ 2, there always exists at least one eigenvector to Λ 1 which is perpendicular to p.
is property is of interest for Section 4.

Journal of Applied Mathematics
Proof. We investigate the interval [μ Q , ∞) for β 1 in B 2 based on the following cases: (1) β 1 ∈ (Λ 1 , ∞): the results for this case are identical to case 1 of the proof of Lemma 2. (2) β 1 � Λ 1 : due to μ Q ≠ Λ 1 ⟺ m Λ 1 � 1 ∧ P ⊥ E 1 , this case follows the reasoning of case 2(b) of the proof of Lemma 2. (3) β 1 ∈ (μ Q , Λ 1 ): the results for this case are identical to case 3′ of the proof of Lemma 2. (4) β 1 � μ Q : due to Λ 2 < μ Q < Λ 1 , D � μ Q I − Λ is regular, and the determinant lemma can be applied as in (33). e vector q � D − 1 p can be defined based on the regularity of D such that only one vector q exists fulfilling p � D q. e vector p then fulfills which can only be achieved by one specific eigenvector of the reduced matrix Q T Λ Q corresponding to its maximum eigenvalue μ Q , i.e., q � Q ξ with and, consequently, 0 � p T q � p T D − 1 p. Since the term p T D − 1 p vanishes, then, regarding (33), β 2 cannot render the determinant of C to zero, i.e., for β 1 � μ Q , there exists no β 2 delivering a point on zB 2 . More explicitly, μ Q is excluded from the range of β 1 . □ Corollary 1. e set B 2 is always an unbounded closed convex set. In particular, it is never compact, which has implications for optimization problems over B 2 .
is is concluded following Remark 2, see reasoning for (44) and (47). is property is of interest for Section 4.

Remark 4.
Wrapping this section up, the reader solely needs to differentiate the cases: (i) Case 1: p is an eigenvector of Λ ⟶ Lemma 1 (ii) Case 2: p is not an eigenvector of Λ and μ Q ∈ Λ 1 , Λ 2 } ⟶ Lemma 2 (iii) Case 3: p is not an eigenvector of Λ and Λ 2 < μ Q < Λ 1 ⟶ Lemma 3 It should be noted that if a function is to be minimized over B 2 , then the spectral factorization of given A and the transformation of given p 0 to p � U T p 0 can be carried out before the minimization in order to check Lemma 1, Lemma 2, and Lemma 3, at what (26), (30)-(32) or (49) then correspond to B 2 . e minimization can then be carried out at its peak efficiency over B 2 , if a minimum exists. A short discussion of the existence of minima is given in Section 5.

Majorization of a Set of Matrices.
Consider the following finite set of N given symmetric matrices and a given vector e corresponding convex sets are the majorant sets for each of the matrices of A. Denote the respective spectral factorizations as and define the corresponding vectors Since holds, the results of the previous section describe the corresponding sets. e intersection of all sets B (i) 2 delivers the majorant set for the set of matrices A, denoted as Note that, due to β 1 , β 2 ⟶ ∞ being admissible for any bounded matrix, the set B A 2 is always nonempty and, due to the intersection of convex sets, also convex.

Examples: Majorization of a Single Matrix.
In the following, majorants for the matrix Λ are sought-after for three different vectors p 1 , p 2 , and p 3 Λ � 5/2 0 0 leading to the aforementioned cases, see demo1.m in the provided software [22]. Consideration of p 1 � e 1 induces an instance of Lemma 1, in which p is contained in the eigenspace of Λ, cf. (27). Figure 3 shows the parameter domain B 2 . e boundary of B 2 is indicated by the black line in Figure 3(a) defined by β 2 � μ P − β 1 , μ P � Λ 1 and by the vertical line corresponding to β 1 � μ Q � Λ 2 . e value of the quadratic forms x T Λ x and x T B 2 x for normalized vectors || x || � 1 within the (1,2)-plane are shown as contours in Figure 3(b). It is readily seen that all parameters β 1 , β 2 on the boundary of B 2 imply the existence of tangential contact points of the contours of the majorant and of the original matrix. e shown contours correspond to the black points in Figure 3(a).

Journal of Applied Mathematics
Consideration of p 2 yields an instance of Lemma 2 (p is not an eigenvector of Λ, m Λ 1 � 1, and p T e 1 � 0, i.e., μ Q � Λ 1 ). e corresponding results are depicted in Figure 4. It should be noted that, for β 1 � μ Q � Λ 1 , the pseudoinverse of D is required and c Q is evaluated, while for β 1 > μ Q , the inverse of D is computed. In Figure 4 Figure 4(d).
Lastly, p 3 is an instance of Lemma 3 (p is not an eigenvector of Λ and Λ 2 < μ Q < Λ 1 ), with corresponding results depicted in Figure 5. Most notably, the value β 1 � μ Q marks the asymptote of the boundary of B 2 since for β 1 � μ Q , no β 2 exists yielding a point on zB 2 , cf. case 4 of the proof of Lemma 3.

Example: Majorization of a Set of Matrices. Consider the finite set of symmetric matrices
and the vector e border of the corresponding set B (1) 2 (an instance of Lemma 1) is depicted in Figure 6 by the straight lines. e border of the corresponding B (2) 2 (an instance of Lemma 3) is depicted by the curved black line in Figure 6. e majorant set B A 2 defined in (56) is depicted in Figure 6 by the gray region. e depicted region can be reproduced with the file demo2.m using the provided MATLAB software, cf. [22].
Next, a set of N � 10 random symmetric matrices of dimension n � 100 is generated. e intersection of the critical domains is shown in Figure 7 as well as the curves denoting the boundaries of the critical domain for each matrix A (i) of the set. Close inspection of the graph shows that there are multiple intersections of these lines which generate the boundary of the overall critical domain B A 2 .

Construction of Four-Parametric Majorant for a Single
Matrix. In addition to the two-parametric majorant of Section 3, a four-parametric majorant is examined in this section. A given A 4 ∈ S n+1 is considered and partitioned as follows: From all majorants of A 4 , we are interested in this section in the parametrization with given normalized p 0 . is four-parametric form arises in linear thermo-elasticity and corresponding isotropic zerothorder bounds, cf. [19]. e upper left block of B shows that it may be represented as a rank-two perturbed scaled identity. As in Section 3, we perform a change of basis with the orthogonal matrix diagonalizing the upper left block of A 4 , i.e., we define  Figure 5: (a) Parameter region B 2 for Λ and p 3 ; (b) isolines of the quadratic forms x T Λ x and x T B 2 x in the (1,2)-plane for all normalized x for the marked black points of (a) for Λ 1 � 5/2 ≤ β 1 ; (c) evaluation of corresponding points for μ Q < β 1 ≤ Λ 1 � 5/2.
In this section, B 4 is examined, i.e., we seek the parameter conditions describing the set As in Section 3.1, the present section is organized as follows: (i) Preparations: introduction of several expressions and relations needed for all the following results (ii) Lemma 4: description of B 4 based on (β 1 , β 2 ) ∈ zB 2 (iii) Lemma 5: description of B 4 based on (β 1 , β 2 ) ∈ B 2 \zB 2 (iv) Corollary 2: recapitulation of the admissible regions for (β 3 , β 4 ) for (β 1 , β 2 ) ∈ B 2 and μ Q < β 1 based on the results of Lemma 4 and Lemma 5 (v) Remark 5: remarks on an additional constraint for a simplification of implementation based on a numerical point of view and Corollary 2 Preparations. For the sake of a compact notation, we define the difference matrix at what positive semidefinitness of C 4 is equivalent to Λ 4 ⪯ B 4 . More explicitly, (67) can also be expressed as which allows to obtain the equivalence relation where C + denotes the Moore-Penrose inverse of C, see, e.g., [25] for a derivation based on the Schur complement. It should be noted that for positive semidefinite C, C + is also positive semidefinite such that c 0 necessarily has to be nonnegative. e condition O ⪯ C is fulfilled for the corresponding (β 1 , β 2 ) ∈ B 2 of Section 3. We consider, therefore, the following cases for singular positive semidefinite and positive definite C, i.e., based on Section 3, for (β 1 , β 2 ) ∈ zB 2 and (β 1 , β 2 ) ∈ B 2 \zB 2 , respectively.

Lemma 4.
If (β 1 , β 2 ) ∈ zB 2 holds, then the parameter set B 4 is described by Proof. For a better overview of the following proof, the reader should consider Figure 8 along the following arguments. For (β 1 , β 2 ) ∈ zB 2 , C is positive semidefinite and det(C) � 0 holds such that the kernel ker(C) is nonempty. For sharp majorants B 4 to exist, c ⊥ ker(C), cf. (69), describes the critical condition. We notice the following cases: (1) κ � 1: the kernel of C is one-dimensional, i.e., ker(C) � span n  If ker(C) is one-dimensional, for majorants B 4 to exist needs to hold, cf. (69). e condition (71) may or may not be fulfilled in a number of exotic cases since some portions of zB 2 , as, e.g., in (28), have a onedimensional kernel but with p ⊥ n.
(a) If p ⊥ n holds, cf. Figure 8, then β 4 loses influence in (71) such that (71) can be fulfilled iff n ⊥ l holds, which is not controllable by the parameters β but solely dictated by the given data A 4 and p 0 , with resulting p � U T p 0 and l � U T a. If additionally n ⊥ l holds, then (71) is fulfilled for all β 4 such that, according to (69), majorants B 4 exist if for given (β 1 , β 2 ) ∈ zB 2 , the remaining parameters β 3 and β 4 are chosen such that holds. For instance, the left-hand side term c T C + c of (72) is quadratic in β 4 and is minimized for which then yields in (72) the lowest possible c 0 and corresponding β 3 , cf. Figure 8. Naturally, if n ⊥ l holds, then (71) is not fulfilled and no majorants exist under the current assumptions for (β 1 , β 2 ) ∈ zB 2 , cf. Figure 8. (b) If p ⊥ n holds, which is the case, e.g., on the lower border of B 2 in Lemma 1 and Lemma 2 (up to β 1 � μ Q ) and on the whole border of B 2 in Lemma 3, then (71) can be solved uniquely for β 4 , yielding Further possible branches  From (74), one can obtain based on (69), i.e., c 0 � β 3 − l 0 ≥ c T C + c, the minimum β 3 cf. Figure 8.
(2) κ ≥ 2: if the dimension of ker(C) is greater than one, then the kernel is described as at what we choose, without loss of generality, the kernel basis vectors n i such than at maximum only one, say n 1 , is not perpendicular to p, i.e., e condition c ⊥ ker(C) is then equivalent to erefore, (78) makes clear that, for (β 1 , β 2 ) ∈ zB 2 and κ ≥ 2, the provided data A 4 and p 0 inducing p and l decide on the existence of majorants under the current assumptions. If l ⊥ n i for i � 2, . . . , κ holds, then the remaining condition 0 � β 4 n T 1 p − n T 1 l in (78) can be analyzed as in case 1, yielding analogous results. □ Lemma 5. If (β 1 , β 2 ) ∈ B 2 \zB 2 holds, then the parameter set B 4 is described by Proof. For β 1 and β 2 chosen according to the results of Section 3 truly inside of B 2 , cf. Figure 9, the difference matrix C � B 2 − Λ is positive definite. For C 4 to be singular, its determinant must vanish. Based on the decomposition the matrix determinant lemma yields which, for c 0 � β 3 − l 0 is fulfilled iff is result is in accordance with (69) and delivers a singular positive semidefinite C 4 . e parameter β 4 is free and coupled to the minimum β 3 given in (82) through the positive quadratic term c T C − 1 c, where c � β 4 p − l, cf. (66). By demanding stationarity of c T C − 1 c in respect to β 4 , one obtains Due to the positive definiteness of C, the choice (83) p T C − 1 c � 0 induces a minimum of β 3 in (82): cf. Figure 9.
□ Remark 5. It should be noted that in view of future optimization problems over B 4 , from a numerical point of view, all tedious cases on the left border of B 2 with P ⊥ ker(C) and κ ≥ 2 for Lemma 1 and Lemma 2 can be avoided easily. is is achieved through Corollary 2 and by imposing the additional constraint in standard numerical optimization procedures as a limit constraint for β 1 with a shifted constant μ s Q slightly greater than μ Q for some user-defined δ. e shifted condition (90) and Corollary 2 allow for a straightforward and simple implementation of the corresponding results of Lemma 4 and Lemma 5. is approach has been considered for the implementation offered in [22].

Majorization of a Set of Matrices.
As in the two-parametric case, for a given vector p 0 ∈ R n and a given matrix set with as in (60) partitioned matrices, we define the sets e majorant set of A is the corresponding intersection of all B (i) 4 :

Application to Semidefinite Programming
For a given finite matrix set problems and practical applications. In Lemma 2.2, β 1 ∈ [μ Q , ∞) is considered, and the corner point (β 1 , β 2 ) � (μ Q , − (c Q ) − 1 ) ∈ zB 2 yields the global minimum for φ, see Figure 12. is result is the exact same upper zeroth-order bound of graphite computed by [16] (cf. Table 3 of [16], based on the relations for the bulk modulus K � (β 1 + β 2 )/3 � 2132890/4161 GPa ≈ 512.591 GPa and shear modulus G � β 1 /2 � 440 GPa). Compared to [16] and the later development by [17,19], the present work does not require special treatment for the computation of the minimum of φ over B 2 since the condensed explicit conditions describing zB 2 are now available for evaluation. e results of the present work may also be applied to more complex semidefinite programming problems in higher dimensions as follows. For a given vector p 0 ∈ R n , either the case for S ] � S n in (94) with or the case S ] � S n+1 with can be considered. For (101), minimizers on the border of the convex set B A d � B A 2 can be determined without much effort based on the results of Section 3. For (102), minimizers can be searched based on the results presented in Section 4. e definition of the majorant of a set of symmetric matrices is nontrivial in the four-parametric version. First, the set B A 2 should be identified. en, three options exist: (i) pick critical (β 1 , β 2 ) ∈ zB A 2 and choose β 4 carefully (see Lemma 4) or (ii) select subcritical (β 1 , β 2 ) ∈ B A 2 \zB A 2 leading to unconstrained β 4 ∈ R and β 3 constrained by the lower estimate (82) (see Lemma 5). ese constraints can be put into two scalar equations which allow for efficient implementation and fast numerical treatment (see the following examples and accompanying open-source software).