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

Authors: Mauricio Fernández; Felix Fritzen
Journal: Journal of Applied Mathematics
 (2020)

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## Abstract

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.


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## Body

## 1. Introduction

The Löwner partial order introduced by [1] is connected to several matrix partial orders. It implies several matrix inequalities and has been widely studied (see, e.g., [2–6]). This matrix partial order is specifically important for linear and nonlinear optimization problems in semidefinite programming (see, e.g., [7–14]) since it describes an essential part of the constraints in many real-world optimization problems.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. The 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∈ℝn×n with n=6 for three-dimensional linear elasticity. A zeroth-order bound of the effective linear material behavior of the N-phase solid is a symmetric matrix B¯¯∈ℝn×n which satisfies 0≤x¯TB¯¯−A¯¯ix¯,∀x¯∈ℝn,∀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 may not be identical but can at least be fixed. Then, an optimal zeroth-order bound is chosen through the minimization of a, in general, nonlinear function φβ¯ depending on parameters β¯ of B¯¯=B¯¯β¯, cf. [17] for a discussion. This yields the minimization problem(1)minβ¯φβ¯such that0≤x¯TB¯¯β¯−A¯¯ix¯,∀x¯∈ℝn∀i∈1,…,N,which is a classical, in general, nonlinear semidefinite programming problem. Of course, this optimization 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 thermo-elasticity, cf. [19], where the number of parameters is four. The 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. This would significantly reduce the computational time of the optimization problem defining the zeroth-order bounds. This 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. This 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.The 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 formsB¯¯β¯. This parameter constraint condensation has not been conducted in either [15–18] 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 matrix—and of a finite matrix set—corresponding to the set of material stiffnesses of a composite material—is introduced. The 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 ON for the two-parametric case. Furthermore, geometric interpretations of the stationary condition are provided. The 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.The manuscript is structured as follows: In Section2, Löwner majorants of a single symmetric matrix and finite set of symmetric matrices are defined. The 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. Then, 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/LoewnerMajorantopen-source MATLAB software containing all programs and data required for the reproduction of all shown examples of the present work.Notation. Throughout this manuscript, the set of real numbers is denoted by ℝ. Column vectors over ℝn,n∈2,3,…, are denoted by single-underlined characters, e.g., x¯,y¯. Rectangular matrices over ℝm×n are denoted by double-underlined characters, e.g., A¯¯,B¯¯. Square matrices over ℝn×n are referred to as matrices of order n. The set of symmetric matrices of order n with finite eigenvalues is denoted by Sn. The transposition is denoted by the superscript ⋅T, e.g., a¯Tb¯ equals the scalar product of the vectors, and a¯b¯T yields the outer product of the vectors. The eigenvalues of a symmetric matrix A¯¯ are denoted by λiA¯¯∈ℝ with i=1,…,n. The multiplicity of an eigenvalue λ is denoted as mλ. The l2 norm of a vector a¯ and the Frobenius norm of a matrix A¯¯ are simply noted as a¯ and A¯¯. The 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. The Moore–Penrose inverse of a matrix is denoted by A¯¯+.


## 2. Löwner Order and Majorants for Symmetric Matrices

In this work, we consider the Löwner partial order of symmetric matricesA¯¯,B¯¯∈Sn (see, e.g., [6]):(2)A¯¯⪯B¯¯⟺0≤x¯TB¯¯−A¯¯x¯,∀x¯∈ℝn⟺B¯¯−A¯¯∈Sn+,where Sn+⊂Sn denotes the positive semidefinite cone. Any B¯¯∈Sn is referred to, in this work, as a (Löwner) majorant of a given A¯¯∈Sn if A¯¯⪯B¯¯ holds. Furthermore, any B¯¯∈Sn is referred to as a sharp (Löwner) majorant of a given A¯¯∈Sn if A¯¯⪯B¯¯ and detB¯¯−A¯¯=0 hold, i.e., if B¯¯−A¯¯∈∂Sn+ holds. For given A¯¯∈Sn, the corresponding majorant set ℬ⊂ℝn×n and sharp majorant set ∂ℬ are denoted as(3)ℬ=B¯¯∈Sn:A¯¯⪯B¯¯,∂ℬ=B¯¯∈Sn:A¯¯⪯B¯¯∧detB¯¯−A¯¯=0.Of course, ℬ is nonempty, unbounded, and convex. The trivial majorant T¯¯∈∂ℬ of a given A¯¯ is defined as(4)T¯¯=Λ1I¯¯,Λ1=maxiλiA¯¯.For a finite set A=A¯¯1,…,A¯¯N of symmetric matrices, we define the analogous majorant set(5)ℬA=∩i=1NB¯¯∈Sn:A¯¯i⪯B¯¯=B¯¯∈Sn:A¯¯⪯B¯¯∀A¯¯∈A,whose boundary ∂ℬA represents the set of all sharp majorants of A:(6)∂ℬA=B¯¯∈Sn:A¯¯⪯B¯¯∀A¯¯∈A∧∃A¯¯∈A:detB¯¯−A¯¯=0.The trivial element on ∂ℬA is(7)T¯¯A=Λ1AI¯¯,Λ1A=maxA¯¯∈AmaxiλiA¯¯.


## 3. Two-Parametric Majorant

### 3.1. Construction of Two-Parametric Majorants for a Single Matrix

For givenA¯¯∈Sn, from the corresponding majorant set ℬ, we investigate a two-parametric form with given (i.e., fixed) normalized vector p¯0∈ℝn, p¯0=1, defined as a rank-one perturbation of a scaled identity via(8)B¯¯20=β1I¯¯+β2p¯0p¯0T∈Sn,β¯=β1,β2T∈ℝ2.In the context of zeroth-order bounds of the effective linear elastic behavior of solid materials, p¯0=1/3,1/3,1/3,0,0,0 is given for isotropic zeroth-order bounds if a specific parametrization is chosen, cf., e.g., [16] or [17] for details.The goal of this section is the derivation of condensed conditions for the parametersβ¯ such that B¯¯20∈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(9)Λ1=Λ11≥Λ2=Λ22≥⋯≥Λn=Λnn,and corresponding orthogonal eigenvector matrix U¯¯, i.e., U¯¯U¯¯T=U¯¯TU¯¯=I¯¯. We denote the set of eigenvectors of Λ¯¯ as ℰ and the space of eigenvectors corresponding to a particular eigenvalue as ℰi, i.e.,(10)ℰ=x¯∈ℝn:Λ¯¯x¯=Λx¯,Λ∈Λ1,…,ℰi=x¯∈ℝn:Λ¯¯x¯=Λix¯,i=1,2,…,n.A change of basis does not alter the Löwner partial order, i.e., A¯¯⪯B¯¯20⟺Λ¯¯⪯U¯¯TB¯¯20U¯¯. We, therefore, define(11)B¯¯2=U¯¯TB¯¯20U¯¯=β1I¯¯+β2p¯p¯T,p¯=U¯¯Tp¯0,and note p¯=p¯0=1. For the remainder of this section, we seek the conditions on the parameters β¯=β1,β2 describing the parameter set(12)ℬ2=β¯∈ℝ2:Λ¯¯⪯B¯¯2.Due to linear dependency of B¯¯2 on β¯ and the general properties of ℬ⊂ℝn×n, the parameter set ℬ2⊂ℝ2 is nonempty, unbounded, and convex. Note that β¯∈∂ℬ2⟺B¯¯20β¯∈∂ℬ 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 ofℬ2 for p¯∈ℰ(iii)
Remark 1: remarks for cases on∂ℬ2 for Lemma 1(iv)
Lemma 2: closed form description ofℬ2 for p¯∉ℰ and β1 bounded from below by Λ1 or Λ2(v)
Remark 2: remarks for cases on∂ℬ2 for Lemma 2(vi)
Lemma 3: closed form description ofℬ2 for p¯∈ℰ and β1 bounded from below by a constant between Λ1 and Λ2(vii)
Corollary 1: recapitulation of some topological properties ofℬ2 relevant for semidefinite programming problems(viii)
Remark 3: remarks for cases on∂ℬ2 for Lemma 3(ix)
Remark 4: summarizing remarks and perspective for corresponding semidefinite programming problemsPreparations. Before we derive the conditions describing ℬ2 in (12), we introduce some quantities and relations needed throughout this section. The condition Λ¯¯≺B¯¯2 is equivalent to the positive semidefinitness of the difference matrix C¯¯:(13)C¯¯=B¯¯2−Λ¯¯=D¯¯+β2p¯p¯T,D¯¯=β1I¯¯−Λ¯¯,i.e.,(14)0≤x¯TC¯¯x¯∀x¯∈ℝn.We denote the dimension of the kernel of the difference matrix C¯¯ as(15)κ=dimkerC¯¯.We define the mutually orthogonal vector spaces(16)P=spanp¯,Q=x¯∈ℝn:x¯Tp¯=0,and introduce a matrix Q¯¯∈ℝn×n−1 such that the concatenated matrix p¯,Q¯¯∈ℝn×n is orthogonal, i.e., the columns of Q¯¯ form an orthonormal basis of Q. We define(17)μP=p¯TΛ¯¯p¯,μQ=maxx¯∈Q∧x¯=1x¯TΛ¯¯x¯=max1≤i≤n−1λiQ¯¯TΛ¯¯Q¯¯,cQ=p¯TμQI¯¯−Λ¯¯+p¯,where μQ corresponds to the largest eigenvalue of the reduced matrix Q¯¯TΛ¯¯Q¯¯∈Sn−1. Inserting vectors from P and Q into (14), respectively, yields the two elementary necessary conditions:(18)β1+β2≥μP,(19)β1≥μQ.Note that μQ is bounded by Λ1 due to(20)Λ1=maxx¯=1x¯TΛ¯¯x¯≥maxx¯∈Q∧x¯=1x¯TΛ¯¯x¯=μQ.Furthermore, the relation(21)μQ≥Λ2,is obtained based on the following arguments:(1)
Ifp¯Te¯1=0, then e¯1∈Q and μQ=Λ1≥Λ2 holds.(2)
Ifp¯Te¯1≠0, then using α=p¯Te¯2/p¯Te¯1, one can define q¯=e¯2−αe¯1∈Q with q¯Tq¯=1+α2. Substituting e¯2=αe¯1+q¯ implies(22)Λ2=e¯2TΛ¯¯e¯2=αe¯1+q¯TαΛ1e¯1+Λ¯¯q¯=−α2Λ1+1+α2q¯TΛ¯¯q¯q¯Tq¯≤17μQ−α2Λ1−μQ≤20μQ.Therefore,(23)Λ2≤μQ≤Λ1,generally holds. Denote the algebraic multiplicity of the maximum eigenvalue Λ1 by mΛ1. It is noted that(24)μQ=Λ1⟺P⊥ℰ1∨mΛ1≥2,and based on (22),(25)μQ=Λ2<Λ1⟹mΛ1=1∧P⊥ℰ1∧P⊥ℰ2,holds. Based on these preparations, we now proceed to the description of the parameter set ℬ2 through three lemmas.Lemma 1.
Ifp¯∈ℰ holds, then the parameter set ℬ2 is described by(26)ℬ2=β¯∈ℝ2:β1≥μQ∧β2≥μP−β1.Proof.
Ifp¯∈ℰ, corresponding to Λi for some i∈1,…,n, then (18) and (19) simplify to(27)β1+β2≥μP=Λi,β1≥μQ=maxj≠iΛj.For this case, (27) is equivalent to Λ¯¯⪯B¯¯2 such that ℬ2 defined in (12) simplifies to (26).Remark 1.
For Lemma1, all β¯∈∂ℬ2 (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 kerC¯¯ of dimension κ, cf. (15). The special cases for β¯∈∂ℬ2(28)p¯∈ℰ∧mμQ=1∧β1=μQ∧β2>μP−μQ⟹κ=1∧kerC¯¯⊥P,(29)p¯∈ℰ∧β1>μQ∧β2=μP−β1⟹κ=1∧kerC¯¯⊥P,deliver a one-dimensional kernel of C¯¯. It is noted that only (29) ensures κ=1, independent of the multiplicity mμQ of μQ, and, more importantly, that kerC¯¯⊥P holds. This property is of interest for Section 4.Lemma 2.
Ifp¯∉ℰ and μQ∈Λ1,Λ2 hold, then the parameter set ℬ2 is described:(1)
ForμQ=Λ1 and eigenspace ℰ1 corresponding to Λ1 as(30)ℬ2=β¯∈ℝ2:μQ≤β1∧β2min≤β2,β2min=−p¯TD¯¯−1p¯−1,β1≠Λ1,−cQ−1,β1=Λ1∧P⊥ℰ1,0,β1=Λ1∧P⊥ℰ1.(2)
ForμQ=Λ2<Λ1 and cQ≠0, cf. (17), as(31)ℬ2=β¯∈ℝ2:μQ≤β1∧β2min≤β2,β2min=−p¯TD¯¯−1p¯−1,β1≠Λ1∧β1≠Λ2,0,β1=Λ1,−cQ−1,β1=Λ2.(3)
ForμQ=Λ2<Λ1 and cQ=0, cf. (17), as(32)ℬ2=β¯∈ℝ2:μQ<β1∧β2min≤β2,β2min=−p¯TD¯¯−1p¯−1,β1≠Λ1,0,β1=Λ1.Proof.
We investigate the intervalμQ,∞ for β1 in ℬ2 with μQ∈Λ1,Λ2 based on the following cases:(1)
β1∈Λ1,∞: In this case, sharp majorants require a singular difference matrix C¯¯=B¯¯2−Λ¯¯=D¯¯+β2p¯p¯T. Due to the restriction Λ1<β1 of this case, the diagonal matrix D¯¯=β1I¯¯−Λ¯¯ is positive definite such that, based on the determinant lemma, cf., e.g., [23] or [24], we consider(33)0=detC¯¯=detD¯¯+β2p¯p¯T=detD¯¯1+β2p¯TD¯¯−1p¯,which is a single-valued constraint in terms of β2 with p¯TD¯¯−1p¯>0. This is fulfilled iff(34)β2=−1p¯TD¯¯−1p¯,meaning that (34) describes for every β1∈Λ1,∞ the corresponding β2 yielding a sharp majorant, i.e., (34) specifies a portion of ∂ℬ2. Due to the convexity of ℬ2, β2≥−p¯TD¯¯−1p¯−1 delivers trivially nonsharp majorants for β1∈Λ1,∞.(2)
β1=Λ1: the matrix D¯¯=Λ1I¯¯−Λ¯¯ is singular and positive semidefinite. We consider the matrices(35)E¯¯1=e¯1…e¯mΛ1,E¯¯1⊥=e¯mΛ1+1…e¯n,which contain in the respective columns a basis of ℰ1 for Λ1 and of the corresponding orthogonal space ℰ1⊥. We examine the following cases:(a)
IfP⊥ℰ1 holds, then C¯¯ is already singular, independently of β2, since(36)e¯1TC¯¯e¯1=e¯1TΛ1I¯¯−Λ¯¯+β2p¯p¯Te¯1=e¯1TΛ1e¯1−Λ¯¯e¯1=o¯,holds, meaning thatkerC¯¯ is at least one-dimensional. Based on (35) and x¯=E¯¯1⊥ξ¯, ξ¯∈ℝn−mΛ1, (14) can be further reduced to(37)0≤ξ¯TC¯¯˜ξ¯,∀ξ¯∈ℝn−mΛ1,C¯¯˜=D¯¯˜+β2p¯˜p¯˜T,D¯¯˜=E¯¯1⊥TD¯¯E¯¯1⊥,p¯˜=E¯¯1⊥Tp¯.The matrixD¯¯˜ 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(38)β2=−1p¯˜TD¯¯˜−1p¯˜=−1p¯TD¯¯+p¯,whereD¯¯+ denotes the Moore–Penrose inverse of D¯¯. The choice (38) induces a second vanishing eigenvalue in C¯¯ such that its kernel is then at least two-dimensional.(b)
IfP⊥ℰ1 holds, then using (35) and x¯=E¯¯1ξ¯, ξ¯∈ℝmΛ1 in (14) yields the necessary condition(39)β2≥0.This condition is also sufficient forΛ¯¯⪯B¯¯2. For β2=0, one retrieves for the current scenario the trivial majorant β¯=Λ1,0∈∂ℬ2, cf. (4).If μQ=Λ1 holds, cf. (24), then cases 1 and 2 of this proof describe β2min, at what cQ≠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 μQ=Λ1. If Λ2<Λ1 and μQ=Λ2 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:(2′)
β1=Λ1: Due to P⊥ℰ1, (39) must hold for this case.(3′)
β1∈Λ2,Λ1: For the present case, D¯¯=β1I¯¯−Λ¯¯ is 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¯TD¯¯−1p¯ may vanish for such indefinite D¯¯. More precisely, the term p¯TD¯¯−1p¯ may vanish, iff D¯¯−1p¯∈Q holds. We search now for a vector q¯=Q¯¯ξ¯∈Q such that p¯=D¯¯q¯ holds, with the necessary condition(40)o¯=Q¯¯TD¯¯q¯=Q¯¯Tβ1I¯¯−Λ¯¯Q¯¯ξ¯=β1ξ¯−Q¯¯TΛ¯¯Q¯¯ξ¯.This means thatξ¯ would be required to correspond to the eigenvector of the reduced matrix Q¯¯TΛ¯¯Q¯¯ for eigenvalue β1. But, we consider β1 larger than the maximum eigenvalue μQ=Λ2 of Q¯¯TΛ¯¯Q¯¯ such that no such ξ¯ exists, and therefore, the term p¯TD¯¯−1p¯ cannot vanish. Solving (33) for β2 yields again (34).(4′)
β1=Λ2: Due to P⊥ℰ2, analogous reasoning as in 2(a) is applied to this case with corresponding E¯¯2 and E¯¯2⊥ up to the corresponding equation(41)0≤ξ¯TC¯¯˜ξ¯,∀ξ¯∈ℝn−mΛ2,C¯¯˜=D¯¯˜+β2p¯˜p¯˜T,D¯¯˜=E¯¯2⊥TD¯¯E¯¯2⊥,p¯˜=E¯¯2⊥Tp¯.The difference to 2(a) here is thatD¯¯˜ 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(42)0=detC¯¯˜=detD¯¯˜1+β2p¯˜TD¯¯˜−1p¯˜.Here, the termp¯˜TD¯¯˜−1p¯˜=p¯TD¯¯+p¯ equals the quantity cQ for μQ=Λ2, see (17). The quantity cQ may or may not vanish, depending solely on Λ¯¯ and p¯. This means that if cQ≠0, then (42) can be solved for β2 as in (38), yielding β1∈μQ,∞ and (31)—cf. Lemma 2.2 as depicted in Figure 1. But, if cQ=0 holds, then there exists no β2 for β1=Λ2 such that β1,β2∈ℬ2 holds since detC¯¯˜ cannot vanish. Therefore, cQ=0 yields (32) and, more importantly, excludes μQ=Λ2 from the range of β1 for the current scenario, i.e., β1∈μQ,∞, cf. Lemma 2.3 illustrated in Figure 1.Figure 1
Illustration of Lemma2.Remark 2.
It is noted that, for Lemma2, 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 ℬ2 is depicted in Figure 2.
In cases 1 and 3′ of the proof of Lemma 2, the rank-one perturbationβ2p¯p¯T can only induce a one-dimensional kernel on the difference matrix C¯¯=D¯¯+β2p¯p¯T with regular D¯¯ for β2 fulfilling (34). This corresponds to the majority of points described through β2min in Lemma 2, only excluding the special case β1=Λ1. The corresponding kernel vector n¯ with kerC¯¯=spann¯ is obtained as(43)n¯=D¯¯−1p¯⟹C¯¯n¯=D¯¯n¯+−11p¯Tn¯p¯p¯Tn¯=o¯,such that(44)p¯∉ℰ∧μQ<β1∧β1≠Λ1∧34⟹κ=1∧kerC¯¯⊥P,holds since 0≠p¯Tn¯=p¯TD¯¯−1p¯ holds. The corresponding points β¯∈∂ℬ2 based on β2min inducing (44) are indicated in Figure 2 at the lower border of ℬ2 with κ=1, and more importantly, due to the consequences of Section 4, kerC¯¯⊥P.
ForμQ=Λ1, i.e., Lemma 2.1 is considered, the point at β1=μQ=Λ1 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⊥ℰ1, C¯¯ is already singular (C¯¯e¯1=o¯, i.e., e¯1∈kerC¯¯) and for β2 fulfilling (38), the rank-one perturbation β2p¯p¯T induces a further vanishing eigenvalue of C¯¯ such that kerC¯¯ is at least two-dimensional. This special case corresponds to the black point in Figure 2. For β2 above the critical value given in (38), kerC¯¯ is then at least one-dimensional such that κ=1 is only possible at β1=μQ=Λ1 under the current assumptions only for mΛ1=1, i.e.,(45)p¯∉ℰ∧mΛ1=1∧P⊥ℰ1∧β1=Λ1∧β2>−cQ−1⟹κ=1∧kerC¯¯⊥P.In case 2(b), i.e., P⊥ℰ1, p¯ is not perpendicular to the complete eigenspace of Λ1, but for mΛ1≥2, there always exists at least one eigenvector to Λ1 which is perpendicular to p¯. Therefore, κ=1 is only possible at β1=Λ1 under the current assumptions for mΛ1=2, i.e.,(46)p¯∉ℰ∧mΛ1=2∧P⊥ℰ1∧β1=Λ1∧β2>0⟹κ=1∧kerC¯¯⊥P.These arguments imply that, for Lemma 2.1, based on (44)–(46), κ=1 and kerC¯¯⊥P are assured only for points β¯∈∂ℬ2 corresponding to μQ=Λ1<β1 and β2=β2minβ1.
ForμQ=Λ2<Λ1, cf. (25), we need to examine the points at β1=Λ1 and β1=Λ2=μQ for Lemma 2.2 and Lemma 2.3. For case 2′, addressing β=Λ1,(47)p¯∉ℰ∧mΛ1=1∧P⊥ℰ1∧β1=Λ1∧β2=0⟹κ=1∧kerC¯¯⊥P,follows such that, for μQ<β1, the lower boundary of ℬ2 depicted in Figure 2 yields κ=1 and kerC¯¯⊥P. Analogous reasoning as for (45) yields for case 4′, addressing β1=Λ2=μQ,(48)p¯∉ℰ∧μQ=Λ2<Λ1∧cQ≠0∧mΛ2=1∧β1=Λ2∧β2>−cQ−1⟹κ=1∧kerC¯¯⊥P.
It can be concluded for Lemma2 that only for μQ<β1 and β¯∈∂ℬ2, the kernel of C¯¯ stays one-dimensional (indicated in Figure 2 by the border with κ=1) and, more importantly, that kerC¯¯⊥P holds. This property is of interest for Section 4.Figure 2
Dimension of kernelκ=dimkerC¯¯ of difference matrix C¯¯ on ∂ℬ2 for Lemma 2.1 and Lemma 2.2 with μQ∈Λ1,Λ2 and cQ≠0; in case of cQ=0, corresponding to Lemma 2.3, ∂ℬ2 is described by the curve β2minβ1 (κ=1), which then diverges to +∞ at β1=μQ.Lemma 3.
Ifp¯∉ℰ and Λ2<μQ<Λ1 hold, then the parameter set ℬ2 is described by(49)ℬ2=β¯∈ℝ2:μQ<β1∧β2min≤β2,β2min=−p¯TD¯¯−1p¯−1,β1≠Λ1,0,β1=Λ1.Proof.
We investigate the intervalμQ,∞ for β1 in ℬ2 based on the following cases:(1)
β1∈Λ1,∞: the results for this case are identical to case 1 of the proof of Lemma 2.(2)
β1=Λ1: due to μQ≠Λ1⟺mΛ1=1∧P⊥ℰ1, this case follows the reasoning of case 2(b) of the proof of Lemma 2.(3)
β1∈μQ,Λ1: the results for this case are identical to case 3′ of the proof of Lemma 2.(4)
β1=μQ: due to Λ2<μQ<Λ1, D=μQI¯¯−Λ¯¯ is regular, and the determinant lemma can be applied as in (33). The vector q¯=D¯¯−1p¯ can be defined based on the regularity of D¯¯ such that only one vector q¯ exists fulfilling p¯=D¯¯q¯. The vector p¯ then fulfills(50)o¯=Q¯¯Tp¯=Q¯¯TD¯¯q¯=μQQ¯¯Tq¯−Q¯¯TΛ¯¯q¯,which can only be achieved by one specific eigenvector of the reduced matrix Q¯¯TΛ¯¯Q¯¯ corresponding to its maximum eigenvalue μQ, i.e., q¯=Q¯¯ξ¯ with Q¯¯TΛ¯¯Q¯¯ξ¯=μQξ¯. This implies q¯=D¯¯−1p¯=Q¯¯ξ¯∈Q and, consequently, 0=p¯Tq¯=p¯TD¯¯−1p¯. Since the term p¯TD¯¯−1p¯ vanishes, then, regarding (33), β2 cannot render the determinant of C¯¯ to zero, i.e., for β1=μQ, there exists no β2 delivering a point on ∂ℬ2. More explicitly, μQ is excluded from the range of β1.Corollary 1.
The setℬ2 is always an unbounded closed convex set. In particular, it is never compact, which has implications for optimization problems over ℬ2.Remark 3.
For Lemma3, κ=1 and kerC¯¯⊥P hold for all β¯∈∂ℬ2. This is concluded following Remark 2, see reasoning for (44) and (47). This 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 μQ∈Λ1,Λ2 ⟶ Lemma 2(iii)
Case 3:p¯ is not an eigenvector of Λ¯¯ and Λ2<μQ<Λ1 ⟶ Lemma 3It should be noted that if a function is to be minimized over ℬ2, then the spectral factorization of given A¯¯ and the transformation of given p¯0 to p¯=U¯¯Tp¯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 ℬ2. The minimization can then be carried out at its peak efficiency over ℬ2, if a minimum exists. A short discussion of the existence of minima is given in Section 5.


### 3.2. Majorization of a Set of Matrices

Consider the following finite set ofN given symmetric matrices(51)A=A¯¯1,…,A¯¯N,A¯¯∈Sn∀A¯¯∈A,and a given vector(52)p¯0∈ℝn.The corresponding convex sets(53)ℬ2i=β1,β2∈ℝ2:A¯¯i⪯B¯¯20,are the majorant sets for each of the matrices of A. Denote the respective spectral factorizations as(54)A¯¯i=U¯¯iΛ¯¯iU¯¯iT,and define the corresponding vectors(55)p¯i=U¯¯iTp¯0.Since A¯¯i⪯B¯¯20⟺Λ¯¯i⪯B¯¯2p¯=p¯i holds, the results of the previous section describe the corresponding sets. The intersection of all sets ℬ2i delivers the majorant set for the set of matrices A, denoted as(56)ℬ2A=∩i=1Nℬ2i.Note that, due to β1,β2⟶∞ being admissible for any bounded matrix, the set ℬ2A is always nonempty and, due to the intersection of convex sets, also convex.


### 3.3. Examples: Majorization of a Single Matrix

In the following, majorants for the matrixΛ¯¯ are sought-after for three different vectors p¯1, p¯2, and p¯3(57)Λ¯¯=5/200010001/2,p¯1=1,0,0T,p¯2=0,1/2,1/2T,p¯3=3/5,4/5,0T,leading to the aforementioned cases, seedemo1.min the provided software [22].Consideration ofp¯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 ℬ2. The boundary of ℬ2 is indicated by the black line in Figure 3(a) defined by β2=μP−β1,μP=Λ1 and by the vertical line corresponding to β1=μQ=Λ2. The value of the quadratic forms x¯TΛ¯¯x¯ and x¯TB¯¯2x¯ 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 β1,β2 on the boundary of ℬ2 imply the existence of tangential contact points of the contours of the majorant and of the original matrix. The shown contours correspond to the black points in Figure 3(a).Figure 3
(a) Parameter regionℬ2 for Λ¯¯ and p¯1; (b) isolines of the quadratic forms x¯TΛ¯¯x¯ and x¯TB¯¯2x¯ in the (1,2)-plane for all normalized x¯ for the marked black points of (a).
(a)(b)Consideration ofp¯2 yields an instance of Lemma 2 (p¯ is not an eigenvector of Λ¯¯, mΛ1=1, and p¯Te¯1=0, i.e., μQ=Λ1). The corresponding results are depicted in Figure 4. It should be noted that, for β1=μQ=Λ1, the pseudoinverse of D¯¯ is required and cQ is evaluated, while for β1>μQ, the inverse of D¯¯ is computed. In Figure 4(b), the region around β1=Λ1 is shown more clearly. The contour plots in Figures 4(c) and 4(d) indicate the difference between the majorant and the original matrix, with the second plot showing that all contours of Figure 4(c) in fact have a contact point with the original hypersurface (drawn as dashed line) outside of the (1,2)-plane, cf. Figure 4(d).Figure 4
(a) Parameter regionℬ2 for Λ¯¯ and p¯2; (b) zoomed region around β1=Λ1=5/2 marked with the black rectangle in (a); (c) isolines of the quadratic forms x¯TΛ¯¯x¯ and x¯TB¯¯2x¯ in the (1,2)-plane for all normalized x¯ for the marked black points of (a); (d) evaluation of the quadratic forms in the (2,3)-plane.
(a)(b)(c)(d)Lastly,p¯3 is an instance of Lemma 3 (p¯ is not an eigenvector of Λ¯¯ and Λ2<μQ<Λ1), with corresponding results depicted in Figure 5. Most notably, the value β1=μQ marks the asymptote of the boundary of ℬ2 since for β1=μQ, no β2 exists yielding a point on ∂ℬ2, cf. case 4 of the proof of Lemma 3.Figure 5
(a) Parameter regionℬ2 for Λ¯¯ and p¯3; (b) isolines of the quadratic forms x¯TΛ¯¯x¯ and x¯TB¯¯2x¯ in the (1,2)-plane for all normalized x¯ for the marked black points of (a) for Λ1=5/2≤β1; (c) evaluation of corresponding points for μQ<β1≤Λ1=5/2.
(a)(b)(c)


### 3.4. Example: Majorization of a Set of Matrices

Consider the finite set of symmetric matrices(58)A=A¯¯1,A¯¯2,A¯¯1=20000000−3,A¯¯2=19−414−161452−162−10,and the vector(59)p¯0=1,0,0T.The border of the corresponding set ℬ21 (an instance of Lemma 1) is depicted in Figure 6 by the straight lines. The border of the corresponding ℬ22 (an instance of Lemma 3) is depicted by the curved black line in Figure 6. The majorant set ℬ2A defined in (56) is depicted in Figure 6 by the gray region. The depicted region can be reproduced with the filedemo2.musing the provided MATLAB software, cf. [22].Figure 6
The straight lines depict the border of the setℬ21, the curved line represents the border of the set ℬ22, and the gray region illustrates the set ℬ2A.Next, a set ofN=10 random symmetric matrices of dimension n=100 is generated. The 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 ℬ2A.Figure 7
The gray region indicates the admissible domainℬ2A of a set of 10 random matrices of dimension 100. Individual lines represent β2minβ1 values for each matrix.


## 3.1. Construction of Two-Parametric Majorants for a Single Matrix

For givenA¯¯∈Sn, from the corresponding majorant set ℬ, we investigate a two-parametric form with given (i.e., fixed) normalized vector p¯0∈ℝn, p¯0=1, defined as a rank-one perturbation of a scaled identity via(8)B¯¯20=β1I¯¯+β2p¯0p¯0T∈Sn,β¯=β1,β2T∈ℝ2.In the context of zeroth-order bounds of the effective linear elastic behavior of solid materials, p¯0=1/3,1/3,1/3,0,0,0 is given for isotropic zeroth-order bounds if a specific parametrization is chosen, cf., e.g., [16] or [17] for details.The goal of this section is the derivation of condensed conditions for the parametersβ¯ such that B¯¯20∈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(9)Λ1=Λ11≥Λ2=Λ22≥⋯≥Λn=Λnn,and corresponding orthogonal eigenvector matrix U¯¯, i.e., U¯¯U¯¯T=U¯¯TU¯¯=I¯¯. We denote the set of eigenvectors of Λ¯¯ as ℰ and the space of eigenvectors corresponding to a particular eigenvalue as ℰi, i.e.,(10)ℰ=x¯∈ℝn:Λ¯¯x¯=Λx¯,Λ∈Λ1,…,ℰi=x¯∈ℝn:Λ¯¯x¯=Λix¯,i=1,2,…,n.A change of basis does not alter the Löwner partial order, i.e., A¯¯⪯B¯¯20⟺Λ¯¯⪯U¯¯TB¯¯20U¯¯. We, therefore, define(11)B¯¯2=U¯¯TB¯¯20U¯¯=β1I¯¯+β2p¯p¯T,p¯=U¯¯Tp¯0,and note p¯=p¯0=1. For the remainder of this section, we seek the conditions on the parameters β¯=β1,β2 describing the parameter set(12)ℬ2=β¯∈ℝ2:Λ¯¯⪯B¯¯2.Due to linear dependency of B¯¯2 on β¯ and the general properties of ℬ⊂ℝn×n, the parameter set ℬ2⊂ℝ2 is nonempty, unbounded, and convex. Note that β¯∈∂ℬ2⟺B¯¯20β¯∈∂ℬ 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 ofℬ2 for p¯∈ℰ(iii)
Remark 1: remarks for cases on∂ℬ2 for Lemma 1(iv)
Lemma 2: closed form description ofℬ2 for p¯∉ℰ and β1 bounded from below by Λ1 or Λ2(v)
Remark 2: remarks for cases on∂ℬ2 for Lemma 2(vi)
Lemma 3: closed form description ofℬ2 for p¯∈ℰ and β1 bounded from below by a constant between Λ1 and Λ2(vii)
Corollary 1: recapitulation of some topological properties ofℬ2 relevant for semidefinite programming problems(viii)
Remark 3: remarks for cases on∂ℬ2 for Lemma 3(ix)
Remark 4: summarizing remarks and perspective for corresponding semidefinite programming problemsPreparations. Before we derive the conditions describing ℬ2 in (12), we introduce some quantities and relations needed throughout this section. The condition Λ¯¯≺B¯¯2 is equivalent to the positive semidefinitness of the difference matrix C¯¯:(13)C¯¯=B¯¯2−Λ¯¯=D¯¯+β2p¯p¯T,D¯¯=β1I¯¯−Λ¯¯,i.e.,(14)0≤x¯TC¯¯x¯∀x¯∈ℝn.We denote the dimension of the kernel of the difference matrix C¯¯ as(15)κ=dimkerC¯¯.We define the mutually orthogonal vector spaces(16)P=spanp¯,Q=x¯∈ℝn:x¯Tp¯=0,and introduce a matrix Q¯¯∈ℝn×n−1 such that the concatenated matrix p¯,Q¯¯∈ℝn×n is orthogonal, i.e., the columns of Q¯¯ form an orthonormal basis of Q. We define(17)μP=p¯TΛ¯¯p¯,μQ=maxx¯∈Q∧x¯=1x¯TΛ¯¯x¯=max1≤i≤n−1λiQ¯¯TΛ¯¯Q¯¯,cQ=p¯TμQI¯¯−Λ¯¯+p¯,where μQ corresponds to the largest eigenvalue of the reduced matrix Q¯¯TΛ¯¯Q¯¯∈Sn−1. Inserting vectors from P and Q into (14), respectively, yields the two elementary necessary conditions:(18)β1+β2≥μP,(19)β1≥μQ.Note that μQ is bounded by Λ1 due to(20)Λ1=maxx¯=1x¯TΛ¯¯x¯≥maxx¯∈Q∧x¯=1x¯TΛ¯¯x¯=μQ.Furthermore, the relation(21)μQ≥Λ2,is obtained based on the following arguments:(1)
Ifp¯Te¯1=0, then e¯1∈Q and μQ=Λ1≥Λ2 holds.(2)
Ifp¯Te¯1≠0, then using α=p¯Te¯2/p¯Te¯1, one can define q¯=e¯2−αe¯1∈Q with q¯Tq¯=1+α2. Substituting e¯2=αe¯1+q¯ implies(22)Λ2=e¯2TΛ¯¯e¯2=αe¯1+q¯TαΛ1e¯1+Λ¯¯q¯=−α2Λ1+1+α2q¯TΛ¯¯q¯q¯Tq¯≤17μQ−α2Λ1−μQ≤20μQ.Therefore,(23)Λ2≤μQ≤Λ1,generally holds. Denote the algebraic multiplicity of the maximum eigenvalue Λ1 by mΛ1. It is noted that(24)μQ=Λ1⟺P⊥ℰ1∨mΛ1≥2,and based on (22),(25)μQ=Λ2<Λ1⟹mΛ1=1∧P⊥ℰ1∧P⊥ℰ2,holds. Based on these preparations, we now proceed to the description of the parameter set ℬ2 through three lemmas.Lemma 1.
Ifp¯∈ℰ holds, then the parameter set ℬ2 is described by(26)ℬ2=β¯∈ℝ2:β1≥μQ∧β2≥μP−β1.Proof.
Ifp¯∈ℰ, corresponding to Λi for some i∈1,…,n, then (18) and (19) simplify to(27)β1+β2≥μP=Λi,β1≥μQ=maxj≠iΛj.For this case, (27) is equivalent to Λ¯¯⪯B¯¯2 such that ℬ2 defined in (12) simplifies to (26).Remark 1.
For Lemma1, all β¯∈∂ℬ2 (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 kerC¯¯ of dimension κ, cf. (15). The special cases for β¯∈∂ℬ2(28)p¯∈ℰ∧mμQ=1∧β1=μQ∧β2>μP−μQ⟹κ=1∧kerC¯¯⊥P,(29)p¯∈ℰ∧β1>μQ∧β2=μP−β1⟹κ=1∧kerC¯¯⊥P,deliver a one-dimensional kernel of C¯¯. It is noted that only (29) ensures κ=1, independent of the multiplicity mμQ of μQ, and, more importantly, that kerC¯¯⊥P holds. This property is of interest for Section 4.Lemma 2.
Ifp¯∉ℰ and μQ∈Λ1,Λ2 hold, then the parameter set ℬ2 is described:(1)
ForμQ=Λ1 and eigenspace ℰ1 corresponding to Λ1 as(30)ℬ2=β¯∈ℝ2:μQ≤β1∧β2min≤β2,β2min=−p¯TD¯¯−1p¯−1,β1≠Λ1,−cQ−1,β1=Λ1∧P⊥ℰ1,0,β1=Λ1∧P⊥ℰ1.(2)
ForμQ=Λ2<Λ1 and cQ≠0, cf. (17), as(31)ℬ2=β¯∈ℝ2:μQ≤β1∧β2min≤β2,β2min=−p¯TD¯¯−1p¯−1,β1≠Λ1∧β1≠Λ2,0,β1=Λ1,−cQ−1,β1=Λ2.(3)
ForμQ=Λ2<Λ1 and cQ=0, cf. (17), as(32)ℬ2=β¯∈ℝ2:μQ<β1∧β2min≤β2,β2min=−p¯TD¯¯−1p¯−1,β1≠Λ1,0,β1=Λ1.Proof.
We investigate the intervalμQ,∞ for β1 in ℬ2 with μQ∈Λ1,Λ2 based on the following cases:(1)
β1∈Λ1,∞: In this case, sharp majorants require a singular difference matrix C¯¯=B¯¯2−Λ¯¯=D¯¯+β2p¯p¯T. Due to the restriction Λ1<β1 of this case, the diagonal matrix D¯¯=β1I¯¯−Λ¯¯ is positive definite such that, based on the determinant lemma, cf., e.g., [23] or [24], we consider(33)0=detC¯¯=detD¯¯+β2p¯p¯T=detD¯¯1+β2p¯TD¯¯−1p¯,which is a single-valued constraint in terms of β2 with p¯TD¯¯−1p¯>0. This is fulfilled iff(34)β2=−1p¯TD¯¯−1p¯,meaning that (34) describes for every β1∈Λ1,∞ the corresponding β2 yielding a sharp majorant, i.e., (34) specifies a portion of ∂ℬ2. Due to the convexity of ℬ2, β2≥−p¯TD¯¯−1p¯−1 delivers trivially nonsharp majorants for β1∈Λ1,∞.(2)
β1=Λ1: the matrix D¯¯=Λ1I¯¯−Λ¯¯ is singular and positive semidefinite. We consider the matrices(35)E¯¯1=e¯1…e¯mΛ1,E¯¯1⊥=e¯mΛ1+1…e¯n,which contain in the respective columns a basis of ℰ1 for Λ1 and of the corresponding orthogonal space ℰ1⊥. We examine the following cases:(a)
IfP⊥ℰ1 holds, then C¯¯ is already singular, independently of β2, since(36)e¯1TC¯¯e¯1=e¯1TΛ1I¯¯−Λ¯¯+β2p¯p¯Te¯1=e¯1TΛ1e¯1−Λ¯¯e¯1=o¯,holds, meaning thatkerC¯¯ is at least one-dimensional. Based on (35) and x¯=E¯¯1⊥ξ¯, ξ¯∈ℝn−mΛ1, (14) can be further reduced to(37)0≤ξ¯TC¯¯˜ξ¯,∀ξ¯∈ℝn−mΛ1,C¯¯˜=D¯¯˜+β2p¯˜p¯˜T,D¯¯˜=E¯¯1⊥TD¯¯E¯¯1⊥,p¯˜=E¯¯1⊥Tp¯.The matrixD¯¯˜ 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(38)β2=−1p¯˜TD¯¯˜−1p¯˜=−1p¯TD¯¯+p¯,whereD¯¯+ denotes the Moore–Penrose inverse of D¯¯. The choice (38) induces a second vanishing eigenvalue in C¯¯ such that its kernel is then at least two-dimensional.(b)
IfP⊥ℰ1 holds, then using (35) and x¯=E¯¯1ξ¯, ξ¯∈ℝmΛ1 in (14) yields the necessary condition(39)β2≥0.This condition is also sufficient forΛ¯¯⪯B¯¯2. For β2=0, one retrieves for the current scenario the trivial majorant β¯=Λ1,0∈∂ℬ2, cf. (4).If μQ=Λ1 holds, cf. (24), then cases 1 and 2 of this proof describe β2min, at what cQ≠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 μQ=Λ1. If Λ2<Λ1 and μQ=Λ2 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:(2′)
β1=Λ1: Due to P⊥ℰ1, (39) must hold for this case.(3′)
β1∈Λ2,Λ1: For the present case, D¯¯=β1I¯¯−Λ¯¯ is 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¯TD¯¯−1p¯ may vanish for such indefinite D¯¯. More precisely, the term p¯TD¯¯−1p¯ may vanish, iff D¯¯−1p¯∈Q holds. We search now for a vector q¯=Q¯¯ξ¯∈Q such that p¯=D¯¯q¯ holds, with the necessary condition(40)o¯=Q¯¯TD¯¯q¯=Q¯¯Tβ1I¯¯−Λ¯¯Q¯¯ξ¯=β1ξ¯−Q¯¯TΛ¯¯Q¯¯ξ¯.This means thatξ¯ would be required to correspond to the eigenvector of the reduced matrix Q¯¯TΛ¯¯Q¯¯ for eigenvalue β1. But, we consider β1 larger than the maximum eigenvalue μQ=Λ2 of Q¯¯TΛ¯¯Q¯¯ such that no such ξ¯ exists, and therefore, the term p¯TD¯¯−1p¯ cannot vanish. Solving (33) for β2 yields again (34).(4′)
β1=Λ2: Due to P⊥ℰ2, analogous reasoning as in 2(a) is applied to this case with corresponding E¯¯2 and E¯¯2⊥ up to the corresponding equation(41)0≤ξ¯TC¯¯˜ξ¯,∀ξ¯∈ℝn−mΛ2,C¯¯˜=D¯¯˜+β2p¯˜p¯˜T,D¯¯˜=E¯¯2⊥TD¯¯E¯¯2⊥,p¯˜=E¯¯2⊥Tp¯.The difference to 2(a) here is thatD¯¯˜ 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(42)0=detC¯¯˜=detD¯¯˜1+β2p¯˜TD¯¯˜−1p¯˜.Here, the termp¯˜TD¯¯˜−1p¯˜=p¯TD¯¯+p¯ equals the quantity cQ for μQ=Λ2, see (17). The quantity cQ may or may not vanish, depending solely on Λ¯¯ and p¯. This means that if cQ≠0, then (42) can be solved for β2 as in (38), yielding β1∈μQ,∞ and (31)—cf. Lemma 2.2 as depicted in Figure 1. But, if cQ=0 holds, then there exists no β2 for β1=Λ2 such that β1,β2∈ℬ2 holds since detC¯¯˜ cannot vanish. Therefore, cQ=0 yields (32) and, more importantly, excludes μQ=Λ2 from the range of β1 for the current scenario, i.e., β1∈μQ,∞, cf. Lemma 2.3 illustrated in Figure 1.Figure 1
Illustration of Lemma2.Remark 2.
It is noted that, for Lemma2, 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 ℬ2 is depicted in Figure 2.
In cases 1 and 3′ of the proof of Lemma 2, the rank-one perturbationβ2p¯p¯T can only induce a one-dimensional kernel on the difference matrix C¯¯=D¯¯+β2p¯p¯T with regular D¯¯ for β2 fulfilling (34). This corresponds to the majority of points described through β2min in Lemma 2, only excluding the special case β1=Λ1. The corresponding kernel vector n¯ with kerC¯¯=spann¯ is obtained as(43)n¯=D¯¯−1p¯⟹C¯¯n¯=D¯¯n¯+−11p¯Tn¯p¯p¯Tn¯=o¯,such that(44)p¯∉ℰ∧μQ<β1∧β1≠Λ1∧34⟹κ=1∧kerC¯¯⊥P,holds since 0≠p¯Tn¯=p¯TD¯¯−1p¯ holds. The corresponding points β¯∈∂ℬ2 based on β2min inducing (44) are indicated in Figure 2 at the lower border of ℬ2 with κ=1, and more importantly, due to the consequences of Section 4, kerC¯¯⊥P.
ForμQ=Λ1, i.e., Lemma 2.1 is considered, the point at β1=μQ=Λ1 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⊥ℰ1, C¯¯ is already singular (C¯¯e¯1=o¯, i.e., e¯1∈kerC¯¯) and for β2 fulfilling (38), the rank-one perturbation β2p¯p¯T induces a further vanishing eigenvalue of C¯¯ such that kerC¯¯ is at least two-dimensional. This special case corresponds to the black point in Figure 2. For β2 above the critical value given in (38), kerC¯¯ is then at least one-dimensional such that κ=1 is only possible at β1=μQ=Λ1 under the current assumptions only for mΛ1=1, i.e.,(45)p¯∉ℰ∧mΛ1=1∧P⊥ℰ1∧β1=Λ1∧β2>−cQ−1⟹κ=1∧kerC¯¯⊥P.In case 2(b), i.e., P⊥ℰ1, p¯ is not perpendicular to the complete eigenspace of Λ1, but for mΛ1≥2, there always exists at least one eigenvector to Λ1 which is perpendicular to p¯. Therefore, κ=1 is only possible at β1=Λ1 under the current assumptions for mΛ1=2, i.e.,(46)p¯∉ℰ∧mΛ1=2∧P⊥ℰ1∧β1=Λ1∧β2>0⟹κ=1∧kerC¯¯⊥P.These arguments imply that, for Lemma 2.1, based on (44)–(46), κ=1 and kerC¯¯⊥P are assured only for points β¯∈∂ℬ2 corresponding to μQ=Λ1<β1 and β2=β2minβ1.
ForμQ=Λ2<Λ1, cf. (25), we need to examine the points at β1=Λ1 and β1=Λ2=μQ for Lemma 2.2 and Lemma 2.3. For case 2′, addressing β=Λ1,(47)p¯∉ℰ∧mΛ1=1∧P⊥ℰ1∧β1=Λ1∧β2=0⟹κ=1∧kerC¯¯⊥P,follows such that, for μQ<β1, the lower boundary of ℬ2 depicted in Figure 2 yields κ=1 and kerC¯¯⊥P. Analogous reasoning as for (45) yields for case 4′, addressing β1=Λ2=μQ,(48)p¯∉ℰ∧μQ=Λ2<Λ1∧cQ≠0∧mΛ2=1∧β1=Λ2∧β2>−cQ−1⟹κ=1∧kerC¯¯⊥P.
It can be concluded for Lemma2 that only for μQ<β1 and β¯∈∂ℬ2, the kernel of C¯¯ stays one-dimensional (indicated in Figure 2 by the border with κ=1) and, more importantly, that kerC¯¯⊥P holds. This property is of interest for Section 4.Figure 2
Dimension of kernelκ=dimkerC¯¯ of difference matrix C¯¯ on ∂ℬ2 for Lemma 2.1 and Lemma 2.2 with μQ∈Λ1,Λ2 and cQ≠0; in case of cQ=0, corresponding to Lemma 2.3, ∂ℬ2 is described by the curve β2minβ1 (κ=1), which then diverges to +∞ at β1=μQ.Lemma 3.
Ifp¯∉ℰ and Λ2<μQ<Λ1 hold, then the parameter set ℬ2 is described by(49)ℬ2=β¯∈ℝ2:μQ<β1∧β2min≤β2,β2min=−p¯TD¯¯−1p¯−1,β1≠Λ1,0,β1=Λ1.Proof.
We investigate the intervalμQ,∞ for β1 in ℬ2 based on the following cases:(1)
β1∈Λ1,∞: the results for this case are identical to case 1 of the proof of Lemma 2.(2)
β1=Λ1: due to μQ≠Λ1⟺mΛ1=1∧P⊥ℰ1, this case follows the reasoning of case 2(b) of the proof of Lemma 2.(3)
β1∈μQ,Λ1: the results for this case are identical to case 3′ of the proof of Lemma 2.(4)
β1=μQ: due to Λ2<μQ<Λ1, D=μQI¯¯−Λ¯¯ is regular, and the determinant lemma can be applied as in (33). The vector q¯=D¯¯−1p¯ can be defined based on the regularity of D¯¯ such that only one vector q¯ exists fulfilling p¯=D¯¯q¯. The vector p¯ then fulfills(50)o¯=Q¯¯Tp¯=Q¯¯TD¯¯q¯=μQQ¯¯Tq¯−Q¯¯TΛ¯¯q¯,which can only be achieved by one specific eigenvector of the reduced matrix Q¯¯TΛ¯¯Q¯¯ corresponding to its maximum eigenvalue μQ, i.e., q¯=Q¯¯ξ¯ with Q¯¯TΛ¯¯Q¯¯ξ¯=μQξ¯. This implies q¯=D¯¯−1p¯=Q¯¯ξ¯∈Q and, consequently, 0=p¯Tq¯=p¯TD¯¯−1p¯. Since the term p¯TD¯¯−1p¯ vanishes, then, regarding (33), β2 cannot render the determinant of C¯¯ to zero, i.e., for β1=μQ, there exists no β2 delivering a point on ∂ℬ2. More explicitly, μQ is excluded from the range of β1.Corollary 1.
The setℬ2 is always an unbounded closed convex set. In particular, it is never compact, which has implications for optimization problems over ℬ2.Remark 3.
For Lemma3, κ=1 and kerC¯¯⊥P hold for all β¯∈∂ℬ2. This is concluded following Remark 2, see reasoning for (44) and (47). This 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 μQ∈Λ1,Λ2 ⟶ Lemma 2(iii)
Case 3:p¯ is not an eigenvector of Λ¯¯ and Λ2<μQ<Λ1 ⟶ Lemma 3It should be noted that if a function is to be minimized over ℬ2, then the spectral factorization of given A¯¯ and the transformation of given p¯0 to p¯=U¯¯Tp¯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 ℬ2. The minimization can then be carried out at its peak efficiency over ℬ2, if a minimum exists. A short discussion of the existence of minima is given in Section 5.


## 3.2. Majorization of a Set of Matrices

Consider the following finite set ofN given symmetric matrices(51)A=A¯¯1,…,A¯¯N,A¯¯∈Sn∀A¯¯∈A,and a given vector(52)p¯0∈ℝn.The corresponding convex sets(53)ℬ2i=β1,β2∈ℝ2:A¯¯i⪯B¯¯20,are the majorant sets for each of the matrices of A. Denote the respective spectral factorizations as(54)A¯¯i=U¯¯iΛ¯¯iU¯¯iT,and define the corresponding vectors(55)p¯i=U¯¯iTp¯0.Since A¯¯i⪯B¯¯20⟺Λ¯¯i⪯B¯¯2p¯=p¯i holds, the results of the previous section describe the corresponding sets. The intersection of all sets ℬ2i delivers the majorant set for the set of matrices A, denoted as(56)ℬ2A=∩i=1Nℬ2i.Note that, due to β1,β2⟶∞ being admissible for any bounded matrix, the set ℬ2A is always nonempty and, due to the intersection of convex sets, also convex.


## 3.3. Examples: Majorization of a Single Matrix

In the following, majorants for the matrixΛ¯¯ are sought-after for three different vectors p¯1, p¯2, and p¯3(57)Λ¯¯=5/200010001/2,p¯1=1,0,0T,p¯2=0,1/2,1/2T,p¯3=3/5,4/5,0T,leading to the aforementioned cases, seedemo1.min the provided software [22].Consideration ofp¯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 ℬ2. The boundary of ℬ2 is indicated by the black line in Figure 3(a) defined by β2=μP−β1,μP=Λ1 and by the vertical line corresponding to β1=μQ=Λ2. The value of the quadratic forms x¯TΛ¯¯x¯ and x¯TB¯¯2x¯ 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 β1,β2 on the boundary of ℬ2 imply the existence of tangential contact points of the contours of the majorant and of the original matrix. The shown contours correspond to the black points in Figure 3(a).Figure 3
(a) Parameter regionℬ2 for Λ¯¯ and p¯1; (b) isolines of the quadratic forms x¯TΛ¯¯x¯ and x¯TB¯¯2x¯ in the (1,2)-plane for all normalized x¯ for the marked black points of (a).
(a)(b)Consideration ofp¯2 yields an instance of Lemma 2 (p¯ is not an eigenvector of Λ¯¯, mΛ1=1, and p¯Te¯1=0, i.e., μQ=Λ1). The corresponding results are depicted in Figure 4. It should be noted that, for β1=μQ=Λ1, the pseudoinverse of D¯¯ is required and cQ is evaluated, while for β1>μQ, the inverse of D¯¯ is computed. In Figure 4(b), the region around β1=Λ1 is shown more clearly. The contour plots in Figures 4(c) and 4(d) indicate the difference between the majorant and the original matrix, with the second plot showing that all contours of Figure 4(c) in fact have a contact point with the original hypersurface (drawn as dashed line) outside of the (1,2)-plane, cf. Figure 4(d).Figure 4
(a) Parameter regionℬ2 for Λ¯¯ and p¯2; (b) zoomed region around β1=Λ1=5/2 marked with the black rectangle in (a); (c) isolines of the quadratic forms x¯TΛ¯¯x¯ and x¯TB¯¯2x¯ in the (1,2)-plane for all normalized x¯ for the marked black points of (a); (d) evaluation of the quadratic forms in the (2,3)-plane.
(a)(b)(c)(d)Lastly,p¯3 is an instance of Lemma 3 (p¯ is not an eigenvector of Λ¯¯ and Λ2<μQ<Λ1), with corresponding results depicted in Figure 5. Most notably, the value β1=μQ marks the asymptote of the boundary of ℬ2 since for β1=μQ, no β2 exists yielding a point on ∂ℬ2, cf. case 4 of the proof of Lemma 3.Figure 5
(a) Parameter regionℬ2 for Λ¯¯ and p¯3; (b) isolines of the quadratic forms x¯TΛ¯¯x¯ and x¯TB¯¯2x¯ in the (1,2)-plane for all normalized x¯ for the marked black points of (a) for Λ1=5/2≤β1; (c) evaluation of corresponding points for μQ<β1≤Λ1=5/2.
(a)(b)(c)


## 3.4. Example: Majorization of a Set of Matrices

Consider the finite set of symmetric matrices(58)A=A¯¯1,A¯¯2,A¯¯1=20000000−3,A¯¯2=19−414−161452−162−10,and the vector(59)p¯0=1,0,0T.The border of the corresponding set ℬ21 (an instance of Lemma 1) is depicted in Figure 6 by the straight lines. The border of the corresponding ℬ22 (an instance of Lemma 3) is depicted by the curved black line in Figure 6. The majorant set ℬ2A defined in (56) is depicted in Figure 6 by the gray region. The depicted region can be reproduced with the filedemo2.musing the provided MATLAB software, cf. [22].Figure 6
The straight lines depict the border of the setℬ21, the curved line represents the border of the set ℬ22, and the gray region illustrates the set ℬ2A.Next, a set ofN=10 random symmetric matrices of dimension n=100 is generated. The 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 ℬ2A.Figure 7
The gray region indicates the admissible domainℬ2A of a set of 10 random matrices of dimension 100. Individual lines represent β2minβ1 values for each matrix.


## 4. Four-Parametric Majorant

### 4.1. Construction of Four-Parametric Majorant for a Single Matrix

In addition to the two-parametric majorant of Section3, a four-parametric majorant is examined in this section. A given A¯¯4∈Sn+1 is considered and partitioned as follows:(60)A¯¯4=A¯¯a¯a¯Ta0∈Sn+1,A¯¯∈Sna¯∈ℝn,a0∈ℝ.From all majorants of A¯¯4, we are interested in this section in the parametrization(61)B¯¯40=β1I¯¯+β2p¯0p¯0Tβ4p¯0β4p¯0Tβ3∈Sn+1,with given normalized p¯0. This four-parametric form arises in linear thermo-elasticity and corresponding isotropic zeroth-order bounds, cf. [19]. The upper left block of B¯¯40 corresponds to B¯¯20, cf. (8). A short inspection of B¯¯4 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(62)U¯¯4=U¯¯o¯o¯T1,diagonalizing the upper left block of A¯¯4, i.e., we define(63)Λ¯¯4=U¯¯4TA¯¯4U¯¯4=Λ¯¯l¯l¯Tl0,l¯=U¯¯Ta¯,l0=a0,(64)B¯¯4=U¯¯4TB¯¯40U¯¯4=B¯¯2β4p¯β4p¯Tβ3,p¯=U¯¯Tp¯0.In this section, B¯¯4 is examined, i.e., we seek the parameter conditions describing the set(65)ℬ4=β¯∈ℝ4:Λ¯¯4⪯B¯¯4.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 ofℬ4 based on β1,β2∈∂ℬ2(iii)
Lemma 5: description ofℬ4 based on β1,β2∈ℬ2\∂ℬ2(iv)
Corollary 2: recapitulation of the admissible regions forβ3,β4 for β1,β2∈ℬ2 and μQ<β1 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 Corollary2Preparations. For the sake of a compact notation, we define the difference matrix(66)C¯¯4=B¯¯4−Λ¯¯4=C¯¯c¯c¯Tc0,C¯¯=B¯¯2−Λ¯¯,c¯=β4p¯−l¯,c0=β3−l0,at what positive semidefinitness of C¯¯4(67)0≤y¯TC¯¯4y¯∀y¯=x¯x0∈ℝn+1,is equivalent to Λ¯¯4⪯B¯¯4. More explicitly, (67) can also be expressed as(68)0≤x¯TC¯¯x¯+2x0x¯Tc¯+c0x02,∀x¯∈ℝn,x0∈ℝ,which allows to obtain the equivalence relation(69)O¯¯⪯C¯¯4⟺O¯¯⪯C¯¯∧c¯⊥kerC¯¯∧c¯TC¯¯+c¯≤c0,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 c0 necessarily has to be nonnegative. The condition O¯¯⪯C¯¯ is fulfilled for the corresponding β1,β2∈ℬ2 of Section 3. We consider, therefore, the following cases for singular positive semidefinite and positive definite C¯¯, i.e., based on Section 3, for β1,β2∈∂ℬ2 and β1,β2∈ℬ2\∂ℬ2, respectively.Lemma 4.
Ifβ1,β2∈∂ℬ2 holds, then the parameter set ℬ4 is described by(70)ℬ4=β¯∈ℝ4:β1,β2∈∂ℬ2∧β4p¯−l¯⊥kerB¯¯2−Λ¯¯∧β3≥l0+β4p¯−l¯TB¯¯2−Λ¯¯+β4p¯−l¯.Proof.
For a better overview of the following proof, the reader should consider Figure8 along the following arguments. For β1,β2∈∂ℬ2, C¯¯ is positive semidefinite and detC¯¯=0 holds such that the kernel kerC¯¯ is nonempty. For sharp majorants B¯¯4 to exist, c¯⊥kerC¯¯, cf. (69), describes the critical condition. We notice the following cases:(1)
κ=1: the kernel of C¯¯ is one-dimensional, i.e., kerC¯¯=spann¯(i)
In Lemma1, cf. Remark 1, for(a)
(28) with n¯⊥p¯ (left border without corner of ℬ2) or(b)
(29) with n¯⊥p¯ (lower border without corner of ℬ2)(ii)
In Lemma2, cf. Remark 2 and Figure 2, for(a)
(45) and (46) with n¯⊥p¯ (left border without corner of ℬ2 for μQ=Λ1),(b)
(48) with n¯⊥p¯ (left border without corner of ℬ2 for μQ=Λ2<Λ1), or(c)
(44) and (47) with n¯⊥p¯ (lower border without corner of ℬ2)(iii)
In Lemma3, cf. Remark 3, with n¯⊥p¯ on the whole border of ℬ2IfkerC¯¯ is one-dimensional, for majorants B¯¯4 to exist(71)0=n¯Tc¯=β4n¯Tp¯−n¯Tl¯,needs to hold, cf. (69). The condition (71) may or may not be fulfilled in a number of exotic cases since some portions of ∂ℬ2, as, e.g., in (28), have a one-dimensional kernel but with p¯⊥n¯.(a)
Ifp¯⊥n¯ holds, cf. Figure 8, then β4 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¯¯Tp¯0 and l¯=U¯¯Ta¯. If additionally n¯⊥l¯ holds, then (71) is fulfilled for all β4 such that, according to (69), majorants B¯¯4 exist if for given β1,β2∈∂ℬ2, the remaining parameters β3 and β4 are chosen such that(72)c¯TC¯¯+c¯=β4p¯−l¯TB¯¯2−Λ¯¯+β4p¯−l¯≤c0=β3−l0,holds. For instance, the left-hand side termc¯TC¯¯+c¯ of (72) is quadratic in β4 and is minimized for(73)∂c¯TC¯¯+c¯∂β4=2p¯TC¯¯+c¯=0⟹β4=p¯TC¯¯+l¯p¯TC¯¯+p¯,which then yields in (72) the lowest possible c0 and corresponding β3, cf. Figure 8. Naturally, if n¯⊥l¯ holds, then (71) is not fulfilled and no majorants exist under the current assumptions for β1,β2∈∂ℬ2, cf. Figure 8.(b)
Ifp¯⊥n¯ holds, which is the case, e.g., on the lower border of ℬ2 in Lemma 1 and Lemma 2 (up to β1=μQ) and on the whole border of ℬ2 in Lemma 3, then (71) can be solved uniquely for β4, yielding(74)β4=n¯Tl¯n¯Tp¯.From (74), one can obtain based on (69), i.e., c0=β3−l0≥c¯TC¯¯+c¯, the minimum β3(75)β3≥l0+n¯Tl¯n¯Tp¯p¯−l¯TC¯¯+n¯Tl¯n¯Tp¯p¯−l¯,cf. Figure8.(2)
κ≥2: if the dimension of kerC¯¯ is greater than one, then the kernel is described as(76)kerC¯¯=spann¯1,…,n¯κ,at what we choose, without loss of generality, the kernel basis vectorsn¯i such than at maximum only one, say n¯1, is not perpendicular to p¯, i.e.,(77)n¯iTp¯=0,∀i=2,…,κ.The conditionc¯⊥kerC¯¯ is then equivalent to(78)0=β4n¯1Tp¯−n¯1Tl¯∧0=n¯iTl¯,∀i=2,…,κ.Therefore, (78) makes clear that, for β1,β2∈∂ℬ2 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=β4n¯1Tp¯−n¯1Tl¯ in (78) can be analyzed as in case 1, yielding analogous results.Figure 8
Illustration of cases occurring in Lemma4.Lemma 5.
Ifβ1,β2∈ℬ2\∂ℬ2 holds, then the parameter set ℬ4 is described by(79)ℬ4=β¯∈ℝ4:β1,β2∈ℬ2\∂ℬ2∧β3≥l0+β4p¯−l¯TB¯¯2−Λ¯¯−1β4p¯−l¯.Proof.
Forβ1 and β2 chosen according to the results of Section 3 truly inside of ℬ2, 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(80)C¯¯4=C¯¯0+U¯¯V¯¯T,C¯¯0=C¯¯o¯o¯T1,U¯¯=c¯o¯c0−11,V¯¯=o¯c¯10,the matrix determinant lemma yields(81)0=detC¯¯4=detC¯¯0+U¯¯V¯¯T=detC¯¯0detI¯¯2+V¯¯TC¯¯0−1U¯¯=detC¯¯c0−c¯TC¯¯−1c¯,which, for c0=β3−l0 is fulfilled iff(82)β3=l0+c¯TC¯¯−1c¯.This result is in accordance with (69) and delivers a singular positive semidefinite C¯¯4. The parameter β4 is free and coupled to the minimum β3 given in (82) through the positive quadratic term c¯TC¯¯−1c¯, where c¯=β4p¯−l¯, cf. (66). By demanding stationarity of c¯TC¯¯−1c¯ in respect to β4, one obtains(83)∂c¯TC¯¯−1c¯∂β4=2p¯TC¯¯−1c¯=0⟹β4=p¯TC¯¯−1l¯p¯TC¯¯−1p¯.Due to the positive definiteness of C¯¯, the choice (83) p¯TC¯¯−1c¯=0 induces a minimum of β3 in (82):(84)β3=l0+β4p¯−l¯TC¯¯−1c¯=l0−l¯TC¯¯−1c¯,cf. Figure 9.Figure 9
Illustration of Lemma5.Corollary 2.
For any pointβ^1,β^2∈∂ℬ2 with μQ<β^1, the family of admissible β3,β4 points generated with β^1,β2∈ℬ2\∂ℬ2 and β2>β^2 is described by parabolas with common minimum and argument of it. The argument of the minimum of all parabolas is given by the unique admissible value β4 at β^1,β^2∈∂ℬ2.Proof.
Consider Figure10. For any β^1,β^2∈∂ℬ2, e.g., the point depicted in Figure 10 marked with a triangle, based on Remark 1, Remark 2, and Remark 3, we know that, for β1>μQ on the lower boundary of ℬ2, the relations κ=1, kerC¯¯=spann¯, n¯=D¯¯−1p¯, and n¯⊥p¯ hold. Based on the proof of Lemma 4, admissible β3,β4 are described by the unique admissible β4Lemma4 given in (74) and inequality (75), cf. the branch for κ=1 and n¯⊥p¯ in Figure 8. For points β^1,β2∈ℬ2\∂ℬ2 with β2>β^2, the corresponding admissible domains for β3,β4 are described through Lemma 5, at what the stationary point of these parabolas is given at β4Lemma5 fulfilling (83). The unique admissible β4Lemma4 given in (74) and the stationary point β4Lemma5 given in (83) are identical for β^1,β^2∈∂ℬ2 and β^1,β2∈ℬ2\∂ℬ2 with β2>β^2. This is shown as follows. Based on the kernel vector n¯=D¯¯−1p¯ for the regular D¯¯=β^1I¯¯−Λ¯¯, the inverse of the difference matrix C¯¯ is expressed with the Sherman–Morrison formula as(85)C¯¯−1=D¯¯+β2p¯p¯T−1=D¯¯−1−β2D¯¯−1p¯p¯TD¯¯−11+β2p¯TD¯¯−1p¯=D¯¯−1−β2n¯n¯T1+β2p¯Tn¯.It should be emphasized again that, in (85), β2 is chosen truly in the interior of ℬ2, but C¯¯−1 can always be represented with the kernel vector n¯=D¯¯−1p¯ obtained at β^1,β^2∈∂ℬ2. The relation (85) implies for the stationary point β4Lemma5 given in (83)(86)β4Lemma5=l¯TC¯¯−1p¯p¯TC¯¯−1p¯=l¯Tn¯1−β2p¯Tn¯/1+β2p¯Tn¯p¯Tn¯1−β2p¯Tn¯/1+β2p¯Tn¯=l¯Tn¯p¯Tn¯=β4Lemma4,such that β4Lemma4, cf. (74), and β4Lemma5 are identical, as depicted in Figure 10. The corresponding values of the minima β3Lemma5=l0−l¯TC¯¯−1c¯ given in (84) of the parabolas are identical to the minimum β3Lemma4=l0+c¯TC¯¯+c¯ of (75). This is shown as follows. First, the minimum of the parabolas β3Lemma5=l0−l¯TC¯¯−1c¯ requires the computation of C¯¯−1c¯. Hereby, it is useful to note that since β4Lemma4=β4Lemma5 holds, the vector c¯=β4Lemma5p¯−l¯=β4Lemma4p¯−l¯ is orthogonal to n¯, i.e., c¯Tn¯=0, cf. (71) corresponding to Lemma 4. Based on the Sherman–Morrison formula for C¯¯−1 (85) and the property c¯⊥n¯, the minimum β3Lemma5, cf. (84), is simplified as follows:(87)β3Lemma5=l0−l¯TC¯¯−1c¯=l0−l¯TD¯¯−1c¯.Now, concerning the minimum β3Lemma4=l0+c¯TC¯¯+c¯ for β^1,β^2∈∂ℬ2 and β4Lemma4, cf. (75), the Moore–Penrose inverse C¯¯+=D¯¯+β2p¯p¯T+ is required. Based on [26] (cf. Theorem 6 therein), C¯¯+ can be expressed as(88)D¯¯+β2p¯p¯T+=D¯¯−1−1n¯Tn¯n¯n¯TD¯¯−1+D¯¯−1n¯n¯T−n¯Tp¯n¯n¯T.Consideration of c¯⊥n¯, cf. Lemma 4, yields(89)β3Lemma4=l0+c¯TC¯¯+c¯=l0+c¯TD¯¯−1c¯=l0+c¯Tβ4Lemma4n¯−D¯¯−1l¯=l0−l¯TD¯¯−1c¯,such that the minima of the parabolas (87) and (89) are identical, as depicted in Figure 10.Figure 10
Illustration of Corollary2.Remark 5.
It should be noted that in view of future optimization problems overℬ4, from a numerical point of view, all tedious cases on the left border of ℬ2 with P⊥kerC¯¯ and κ≥2 for Lemma 1 and Lemma 2 can be avoided easily. This is achieved through Corollary 2 and by imposing the additional constraint(90)μQs≤β1,μQs=μQ+δ,0<δ≪1,in standard numerical optimization procedures as a limit constraint for β1 with a shifted constant μQs slightly greater than μQ for some user-defined δ. The shifted condition (90) and Corollary 2 allow for a straightforward and simple implementation of the corresponding results of Lemma 4 and Lemma 5. This approach has been considered for the implementation offered in [22].


### 4.2. Majorization of a Set of Matrices

As in the two-parametric case, for a given vectorp¯0∈ℝn and a given matrix set(91)A=A¯¯41,…,A¯¯4N,A¯¯4∈Sn+1∀A¯¯4∈A,with as in (60) partitioned matrices, we define the sets(92)ℬ4i=β¯∈ℝ4:A¯¯4i⪯B¯¯40.The majorant set of A is the corresponding intersection of all ℬ4i:(93)ℬ4A=∩i=1Nℬ4i.


## 4.1. Construction of Four-Parametric Majorant for a Single Matrix

In addition to the two-parametric majorant of Section3, a four-parametric majorant is examined in this section. A given A¯¯4∈Sn+1 is considered and partitioned as follows:(60)A¯¯4=A¯¯a¯a¯Ta0∈Sn+1,A¯¯∈Sna¯∈ℝn,a0∈ℝ.From all majorants of A¯¯4, we are interested in this section in the parametrization(61)B¯¯40=β1I¯¯+β2p¯0p¯0Tβ4p¯0β4p¯0Tβ3∈Sn+1,with given normalized p¯0. This four-parametric form arises in linear thermo-elasticity and corresponding isotropic zeroth-order bounds, cf. [19]. The upper left block of B¯¯40 corresponds to B¯¯20, cf. (8). A short inspection of B¯¯4 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(62)U¯¯4=U¯¯o¯o¯T1,diagonalizing the upper left block of A¯¯4, i.e., we define(63)Λ¯¯4=U¯¯4TA¯¯4U¯¯4=Λ¯¯l¯l¯Tl0,l¯=U¯¯Ta¯,l0=a0,(64)B¯¯4=U¯¯4TB¯¯40U¯¯4=B¯¯2β4p¯β4p¯Tβ3,p¯=U¯¯Tp¯0.In this section, B¯¯4 is examined, i.e., we seek the parameter conditions describing the set(65)ℬ4=β¯∈ℝ4:Λ¯¯4⪯B¯¯4.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 ofℬ4 based on β1,β2∈∂ℬ2(iii)
Lemma 5: description ofℬ4 based on β1,β2∈ℬ2\∂ℬ2(iv)
Corollary 2: recapitulation of the admissible regions forβ3,β4 for β1,β2∈ℬ2 and μQ<β1 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 Corollary2Preparations. For the sake of a compact notation, we define the difference matrix(66)C¯¯4=B¯¯4−Λ¯¯4=C¯¯c¯c¯Tc0,C¯¯=B¯¯2−Λ¯¯,c¯=β4p¯−l¯,c0=β3−l0,at what positive semidefinitness of C¯¯4(67)0≤y¯TC¯¯4y¯∀y¯=x¯x0∈ℝn+1,is equivalent to Λ¯¯4⪯B¯¯4. More explicitly, (67) can also be expressed as(68)0≤x¯TC¯¯x¯+2x0x¯Tc¯+c0x02,∀x¯∈ℝn,x0∈ℝ,which allows to obtain the equivalence relation(69)O¯¯⪯C¯¯4⟺O¯¯⪯C¯¯∧c¯⊥kerC¯¯∧c¯TC¯¯+c¯≤c0,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 c0 necessarily has to be nonnegative. The condition O¯¯⪯C¯¯ is fulfilled for the corresponding β1,β2∈ℬ2 of Section 3. We consider, therefore, the following cases for singular positive semidefinite and positive definite C¯¯, i.e., based on Section 3, for β1,β2∈∂ℬ2 and β1,β2∈ℬ2\∂ℬ2, respectively.Lemma 4.
Ifβ1,β2∈∂ℬ2 holds, then the parameter set ℬ4 is described by(70)ℬ4=β¯∈ℝ4:β1,β2∈∂ℬ2∧β4p¯−l¯⊥kerB¯¯2−Λ¯¯∧β3≥l0+β4p¯−l¯TB¯¯2−Λ¯¯+β4p¯−l¯.Proof.
For a better overview of the following proof, the reader should consider Figure8 along the following arguments. For β1,β2∈∂ℬ2, C¯¯ is positive semidefinite and detC¯¯=0 holds such that the kernel kerC¯¯ is nonempty. For sharp majorants B¯¯4 to exist, c¯⊥kerC¯¯, cf. (69), describes the critical condition. We notice the following cases:(1)
κ=1: the kernel of C¯¯ is one-dimensional, i.e., kerC¯¯=spann¯(i)
In Lemma1, cf. Remark 1, for(a)
(28) with n¯⊥p¯ (left border without corner of ℬ2) or(b)
(29) with n¯⊥p¯ (lower border without corner of ℬ2)(ii)
In Lemma2, cf. Remark 2 and Figure 2, for(a)
(45) and (46) with n¯⊥p¯ (left border without corner of ℬ2 for μQ=Λ1),(b)
(48) with n¯⊥p¯ (left border without corner of ℬ2 for μQ=Λ2<Λ1), or(c)
(44) and (47) with n¯⊥p¯ (lower border without corner of ℬ2)(iii)
In Lemma3, cf. Remark 3, with n¯⊥p¯ on the whole border of ℬ2IfkerC¯¯ is one-dimensional, for majorants B¯¯4 to exist(71)0=n¯Tc¯=β4n¯Tp¯−n¯Tl¯,needs to hold, cf. (69). The condition (71) may or may not be fulfilled in a number of exotic cases since some portions of ∂ℬ2, as, e.g., in (28), have a one-dimensional kernel but with p¯⊥n¯.(a)
Ifp¯⊥n¯ holds, cf. Figure 8, then β4 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¯¯Tp¯0 and l¯=U¯¯Ta¯. If additionally n¯⊥l¯ holds, then (71) is fulfilled for all β4 such that, according to (69), majorants B¯¯4 exist if for given β1,β2∈∂ℬ2, the remaining parameters β3 and β4 are chosen such that(72)c¯TC¯¯+c¯=β4p¯−l¯TB¯¯2−Λ¯¯+β4p¯−l¯≤c0=β3−l0,holds. For instance, the left-hand side termc¯TC¯¯+c¯ of (72) is quadratic in β4 and is minimized for(73)∂c¯TC¯¯+c¯∂β4=2p¯TC¯¯+c¯=0⟹β4=p¯TC¯¯+l¯p¯TC¯¯+p¯,which then yields in (72) the lowest possible c0 and corresponding β3, cf. Figure 8. Naturally, if n¯⊥l¯ holds, then (71) is not fulfilled and no majorants exist under the current assumptions for β1,β2∈∂ℬ2, cf. Figure 8.(b)
Ifp¯⊥n¯ holds, which is the case, e.g., on the lower border of ℬ2 in Lemma 1 and Lemma 2 (up to β1=μQ) and on the whole border of ℬ2 in Lemma 3, then (71) can be solved uniquely for β4, yielding(74)β4=n¯Tl¯n¯Tp¯.From (74), one can obtain based on (69), i.e., c0=β3−l0≥c¯TC¯¯+c¯, the minimum β3(75)β3≥l0+n¯Tl¯n¯Tp¯p¯−l¯TC¯¯+n¯Tl¯n¯Tp¯p¯−l¯,cf. Figure8.(2)
κ≥2: if the dimension of kerC¯¯ is greater than one, then the kernel is described as(76)kerC¯¯=spann¯1,…,n¯κ,at what we choose, without loss of generality, the kernel basis vectorsn¯i such than at maximum only one, say n¯1, is not perpendicular to p¯, i.e.,(77)n¯iTp¯=0,∀i=2,…,κ.The conditionc¯⊥kerC¯¯ is then equivalent to(78)0=β4n¯1Tp¯−n¯1Tl¯∧0=n¯iTl¯,∀i=2,…,κ.Therefore, (78) makes clear that, for β1,β2∈∂ℬ2 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=β4n¯1Tp¯−n¯1Tl¯ in (78) can be analyzed as in case 1, yielding analogous results.Figure 8
Illustration of cases occurring in Lemma4.Lemma 5.
Ifβ1,β2∈ℬ2\∂ℬ2 holds, then the parameter set ℬ4 is described by(79)ℬ4=β¯∈ℝ4:β1,β2∈ℬ2\∂ℬ2∧β3≥l0+β4p¯−l¯TB¯¯2−Λ¯¯−1β4p¯−l¯.Proof.
Forβ1 and β2 chosen according to the results of Section 3 truly inside of ℬ2, 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(80)C¯¯4=C¯¯0+U¯¯V¯¯T,C¯¯0=C¯¯o¯o¯T1,U¯¯=c¯o¯c0−11,V¯¯=o¯c¯10,the matrix determinant lemma yields(81)0=detC¯¯4=detC¯¯0+U¯¯V¯¯T=detC¯¯0detI¯¯2+V¯¯TC¯¯0−1U¯¯=detC¯¯c0−c¯TC¯¯−1c¯,which, for c0=β3−l0 is fulfilled iff(82)β3=l0+c¯TC¯¯−1c¯.This result is in accordance with (69) and delivers a singular positive semidefinite C¯¯4. The parameter β4 is free and coupled to the minimum β3 given in (82) through the positive quadratic term c¯TC¯¯−1c¯, where c¯=β4p¯−l¯, cf. (66). By demanding stationarity of c¯TC¯¯−1c¯ in respect to β4, one obtains(83)∂c¯TC¯¯−1c¯∂β4=2p¯TC¯¯−1c¯=0⟹β4=p¯TC¯¯−1l¯p¯TC¯¯−1p¯.Due to the positive definiteness of C¯¯, the choice (83) p¯TC¯¯−1c¯=0 induces a minimum of β3 in (82):(84)β3=l0+β4p¯−l¯TC¯¯−1c¯=l0−l¯TC¯¯−1c¯,cf. Figure 9.Figure 9
Illustration of Lemma5.Corollary 2.
For any pointβ^1,β^2∈∂ℬ2 with μQ<β^1, the family of admissible β3,β4 points generated with β^1,β2∈ℬ2\∂ℬ2 and β2>β^2 is described by parabolas with common minimum and argument of it. The argument of the minimum of all parabolas is given by the unique admissible value β4 at β^1,β^2∈∂ℬ2.Proof.
Consider Figure10. For any β^1,β^2∈∂ℬ2, e.g., the point depicted in Figure 10 marked with a triangle, based on Remark 1, Remark 2, and Remark 3, we know that, for β1>μQ on the lower boundary of ℬ2, the relations κ=1, kerC¯¯=spann¯, n¯=D¯¯−1p¯, and n¯⊥p¯ hold. Based on the proof of Lemma 4, admissible β3,β4 are described by the unique admissible β4Lemma4 given in (74) and inequality (75), cf. the branch for κ=1 and n¯⊥p¯ in Figure 8. For points β^1,β2∈ℬ2\∂ℬ2 with β2>β^2, the corresponding admissible domains for β3,β4 are described through Lemma 5, at what the stationary point of these parabolas is given at β4Lemma5 fulfilling (83). The unique admissible β4Lemma4 given in (74) and the stationary point β4Lemma5 given in (83) are identical for β^1,β^2∈∂ℬ2 and β^1,β2∈ℬ2\∂ℬ2 with β2>β^2. This is shown as follows. Based on the kernel vector n¯=D¯¯−1p¯ for the regular D¯¯=β^1I¯¯−Λ¯¯, the inverse of the difference matrix C¯¯ is expressed with the Sherman–Morrison formula as(85)C¯¯−1=D¯¯+β2p¯p¯T−1=D¯¯−1−β2D¯¯−1p¯p¯TD¯¯−11+β2p¯TD¯¯−1p¯=D¯¯−1−β2n¯n¯T1+β2p¯Tn¯.It should be emphasized again that, in (85), β2 is chosen truly in the interior of ℬ2, but C¯¯−1 can always be represented with the kernel vector n¯=D¯¯−1p¯ obtained at β^1,β^2∈∂ℬ2. The relation (85) implies for the stationary point β4Lemma5 given in (83)(86)β4Lemma5=l¯TC¯¯−1p¯p¯TC¯¯−1p¯=l¯Tn¯1−β2p¯Tn¯/1+β2p¯Tn¯p¯Tn¯1−β2p¯Tn¯/1+β2p¯Tn¯=l¯Tn¯p¯Tn¯=β4Lemma4,such that β4Lemma4, cf. (74), and β4Lemma5 are identical, as depicted in Figure 10. The corresponding values of the minima β3Lemma5=l0−l¯TC¯¯−1c¯ given in (84) of the parabolas are identical to the minimum β3Lemma4=l0+c¯TC¯¯+c¯ of (75). This is shown as follows. First, the minimum of the parabolas β3Lemma5=l0−l¯TC¯¯−1c¯ requires the computation of C¯¯−1c¯. Hereby, it is useful to note that since β4Lemma4=β4Lemma5 holds, the vector c¯=β4Lemma5p¯−l¯=β4Lemma4p¯−l¯ is orthogonal to n¯, i.e., c¯Tn¯=0, cf. (71) corresponding to Lemma 4. Based on the Sherman–Morrison formula for C¯¯−1 (85) and the property c¯⊥n¯, the minimum β3Lemma5, cf. (84), is simplified as follows:(87)β3Lemma5=l0−l¯TC¯¯−1c¯=l0−l¯TD¯¯−1c¯.Now, concerning the minimum β3Lemma4=l0+c¯TC¯¯+c¯ for β^1,β^2∈∂ℬ2 and β4Lemma4, cf. (75), the Moore–Penrose inverse C¯¯+=D¯¯+β2p¯p¯T+ is required. Based on [26] (cf. Theorem 6 therein), C¯¯+ can be expressed as(88)D¯¯+β2p¯p¯T+=D¯¯−1−1n¯Tn¯n¯n¯TD¯¯−1+D¯¯−1n¯n¯T−n¯Tp¯n¯n¯T.Consideration of c¯⊥n¯, cf. Lemma 4, yields(89)β3Lemma4=l0+c¯TC¯¯+c¯=l0+c¯TD¯¯−1c¯=l0+c¯Tβ4Lemma4n¯−D¯¯−1l¯=l0−l¯TD¯¯−1c¯,such that the minima of the parabolas (87) and (89) are identical, as depicted in Figure 10.Figure 10
Illustration of Corollary2.Remark 5.
It should be noted that in view of future optimization problems overℬ4, from a numerical point of view, all tedious cases on the left border of ℬ2 with P⊥kerC¯¯ and κ≥2 for Lemma 1 and Lemma 2 can be avoided easily. This is achieved through Corollary 2 and by imposing the additional constraint(90)μQs≤β1,μQs=μQ+δ,0<δ≪1,in standard numerical optimization procedures as a limit constraint for β1 with a shifted constant μQs slightly greater than μQ for some user-defined δ. The shifted condition (90) and Corollary 2 allow for a straightforward and simple implementation of the corresponding results of Lemma 4 and Lemma 5. This approach has been considered for the implementation offered in [22].


## 4.2. Majorization of a Set of Matrices

As in the two-parametric case, for a given vectorp¯0∈ℝn and a given matrix set(91)A=A¯¯41,…,A¯¯4N,A¯¯4∈Sn+1∀A¯¯4∈A,with as in (60) partitioned matrices, we define the sets(92)ℬ4i=β¯∈ℝ4:A¯¯4i⪯B¯¯40.The majorant set of A is the corresponding intersection of all ℬ4i:(93)ℬ4A=∩i=1Nℬ4i.


## 5. Application to Semidefinite Programming

For a given finite matrix set(94)A=A¯¯1,…,A¯¯N,A¯¯∈Sν∀A¯¯∈A,consider the majorant set ℬdA of A for some parametrized B¯¯β¯∈Sν with parameter vector β¯∈ℝd(95)ℬdA=β¯∈ℝd:A¯¯⪯B¯¯β¯∀A¯¯∈A.The objective is now the minimization of a function φ:ℝd⟼ℝ over ℬdA:(96)minβ¯∈ℬdAφβ¯.If the function B¯¯β¯ is linear in the parameter β¯, which is assumed from this point on, then ℬdA is nonempty, unbounded, and convex, and the minimization problem (96) is a classical semidefinite programming problem (see, e.g., [7]) such that it may be tackled with several numerical approaches from literature (see, e.g., [8, 10, 12]). For continuous and convex φ, the minimum, if it exists, may be located in the interior of ℬdA or on its border ∂ℬdA. Therefore, an unconstrained optimization should always be performed first. If its solution is not an element of the interior of ℬdA, then a constrained optimization problem may be considered to find the optimum on ∂ℬdA, i.e., among the sharp Löwner majorants. It should be remarked that convex functions φ can be formulated for corresponding convex ℬdA such that no minimum exists. A clear example for such a case is constructed based on the results of Section 3 for a single matrix. Assume the two-parametric form B¯¯=B¯¯20 of Section 3 and choose A¯¯=A¯¯2 and p¯0 as given in (58) and (59) such that p¯0 is not an eigenvector of A¯¯ and Λ2<μQ<Λ2 holds. This implies an instance of Lemma 3, yielding the region ℬdA=ℬ2 as depicted in Figure 6 for ℬ22. Now, consider the linear function φβ¯=g¯Tβ¯ with constant gradient g¯∈ℝ2. An example is displayed in Figure 11(a) with g¯=3,1T.Figure 11
Examples for linear and nonlinear functionφ over ℬ2: φ=3β1+β2 (a); φ=3β12+β22 (b). The gradient of φ and the tangent of the curve β1,β2minβ1 at the global minimum (black point) are depicted by the corresponding black arrows, while the dashed line represents φβ¯=minβ¯∈ℬ2φ.Regarding Figure11(a), it becomes immediately clear that a minimum of the convex φβ¯ exists in the current ℬ2. But, this is not always the case for arbitrary g¯ since, e.g., for g¯=0,1T values on the lower boundary of ℬ2 continuously decrease due to the unboundedness of ℬ2 emphasized in Corollary 1 such that no minimum exists for g¯=0,1T. Therefore, in general, depending on the properties of the convex set ℬdA and on the properties of the convex φβ¯, a minimum may exist. For the rest of this section, we do not consider such cases, i.e., we assume that the convex φβ¯ is formulated such that a global minimum exists in ℬdA, as in the examples shown in Figure 11. These examples can be reproduced withdemo3.min [22]. The global minimum for instances of Lemma 3, as the examples depicted in Figure 11, is obtained by solving the single nonlinear equation for β1(97)g¯Tt¯β2=β2minβ1=0,g¯=∂φ∂β1,∂φ∂β2T,t¯=1,∂β2min∂β1T,based on the orthogonality of the gradient g¯ of φ and the tangent t¯ of the curve β1,β2min for β1>μQ. An initial guess can be generated with the trivial majorant, i.e., β1=Λ1, such that standard numerical approaches solving the nonlinear equation for β1 can be used, independently of the dimension of the given matrix A¯¯.As mentioned in Section1, the presented results can also be used in the context of the zeroth-order bounds of elasticity. The upper zeroth-order bound of, e.g, graphite, see [16], with stiffness matrix given in GPa (109N/m2) in normalized Voigt notation, see, e.g., [27],(98)A¯¯=106018015000180106015000151573200000090000009000000880GPa∈ℝ6×6,and dimensionless vector p¯0 given by(99)p¯0=1/3,1/3,1/3,0,0,0∈ℝ6,were defined in [16] through the minimization of φ=trB¯¯20=6β1+β2, where β1 and β2 are also given in GPa. In engineering, the bulk modulus K and the shear modulus G are related to β1 and β2 by K=β1+β2/3 and G=β1/2. The case for graphite yields(100)Λ1,Λ2,Λ3,Λ4,Λ5,Λ6=145800849+2553,880,880,142553−5800849,9,9GPa,μQ=880GPa,cQ=−13879123301GPa,i.e., an instance of Lemma 2.2. This shows that, in general, Lemma 2.1 and Lemma 2.2 are also relevant for real-world problems and practical applications. In Lemma 2.2, β1∈μQ,∞ is considered, and the corner point β1,β2=μQ,−cQ−1∈∂ℬ2 yields the global minimum for φ, see Figure 12. This 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=β1+β2/3=2132890/4161GPa≈512.591GPa and shear modulus G=β1/2=440GPa). 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 ℬ2 since the condensed explicit conditions describing ∂ℬ2 are now available for evaluation.Figure 12
Computation of upper zeroth-order bound of [16] based on φ=trB¯¯20=6β1+β2 for graphite (instance of Lemma 2.2): ∂ℬ2 (black line), minimum of φ over ℬ2 at β1,β2=μQ,−cQ−1∈∂ℬ2 (black point), and φ=minβ1,β2∈ℬ2φ (dashed lined).The results of the present work may also be applied to more complex semidefinite programming problems in higher dimensions as follows. For a given vectorp¯0∈ℝn, either the case for Sν=Sn in (94) with(101)B¯¯=B¯¯20⟹ℬdA=ℬ2A,or the case Sν=Sn+1 with(102)B¯¯=B¯¯40⟹ℬdA=ℬ4A,can be considered. For (101), minimizers on the border of the convex set ℬdA=ℬ2A 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. The definition of the majorant of a set of symmetric matrices is nontrivial in the four-parametric version. First, the set ℬ2A should be identified. Then, three options exist: (i) pick critical β1,β2∈∂ℬ2A and choose β4 carefully (see Lemma 4) or (ii) select subcritical β1,β2∈ℬ2A\∂ℬ2A leading to unconstrained β4∈ℝ and β3 constrained by the lower estimate (82) (see Lemma 5). These 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).


### 5.1. Example: Four-Parametric Optimization Majorizing a Single Matrix

We consider the semidefinite programming problem (96) for a single matrix A¯¯4 and the four-parametric form B¯¯4 of Section 4. A matrix A¯¯4∈S11 and a vector p¯0∈ℝ10 with Λ11≈−11.3075, Λ1≈4.7144, and μQ≈4.206 are considered in order to optimize the Frobenius norm(103)φβ¯=∥B¯¯4β¯∥F=nβ12+β22+β32+2β42,n=10.Note that the unconstrained minimum of this function is obtained at β¯=o¯∉ℬ4. Hence, the constrained minimum will be found on the boundary ∂ℬ4. The explicit data of the current problem can recovered based ondemo4.min the provided software [22].The results of Lemma1–Lemma 4 for the explicit construction of sharp majorants may be applied as follows for the generation of an initial guess for the future optimization over ℬ4 for a single matrix:(1)
Generate a discrete point setI1h from the interval I1=μQs,β1max with μQs as in (90) (e.g., through an equidistant discretization of I1)(2)
Compute the correspondingβ˜2β1=β2minβ1 for each β1∈I1h(3)
Use Lemma4 noting κ=1∧P⊥kerC¯¯∀β1∈I1h:(a)
Compute the correspondingβ˜4β1=β4β1,β˜2β1 with (74)(b)
Compute the correspondingβ˜3β1 = minimum β3 in (75)(4)
Evaluate objective functionφ˜β1=φβ¯β1 accordingly, and return critical β1c yielding the minimum value for β1∈I1hExecution of the just described simple approach yields a critical β1c=4.6028 implying(104)β¯c=4.6028,−4.1089,4.5598,−2.4422∈∂ℬ4,φ˜β1c=φβ¯c=16.1697.The straightforward implementation of the results of Lemma 4 and Lemma 5 simplified through the additional constraint of Remark 5 yields for the numerical minimization(105)β¯∗=4.6302,0.3030,4.6269,−1.1840∈∂ℬ4,φβ¯∗=15.4495<φβ¯c.Notably, β¯c induces two zero eigenvalues in C¯¯4, while β¯∗ leads to a one-dimensional kernel. Algorithmically, the optimization gets away with two nonlinear inequality constraints independently of the dimension n of the problem, seedemo4.min [22] for details and reproduction of the results of this example.


### 5.2. Example: Four-Parametric Optimizations Majorizing a Set of Matrices

Next, we consider the semidefinite programming problem (96) for a finite matrix set A=A¯¯41,…,A¯¯4N with N=5 and n=10, B¯¯4 and the Frobenius norm given in (103). The explicit data of the current problem can be recovered based ondemo5.min the provided software [22].The results of Lemma1–Lemma 5 for the explicit construction of sharp majorants may be used in order to generate an initial guess as follows:(1)
Generate a discrete point setI1h from the interval I1=μQs,β1max with μQs>maxi=1,…,NμQcorresponding toA¯¯4i (e.g., through an equidistant discretization of I1)(2)
Computeβ˜2β1=maxi=1,…,Nβ2minβ1corresponding toA¯¯4i for β1∈I1h and build discretized curve C=β1,β2∈ℬ2A:β1∈I1h,β2=β˜2β1(3)
ShiftC by some small positive Δ into the interior of ℬ2A and rewrite C=β1,β2∈ℬ2A∖∂ℬ2A:β1,β2=β1′+Δ,β2′+Δ,β1′,β2′∈C(4)
Use Lemma5:(a)
Setβ˜4β1=0∀β1∈I1h(b)
Compute correspondingβ˜3β1=maxi=1,…,Nβ3according to82forA¯¯4iwithβ1,β2∈C(5)
Evaluateφβ1,β˜2β1,β˜3β1,β˜4β1 for all β1∈I1h and determine minimum(6)
Go to 3 (i.e., shiftC again by Δ), execute 4 and 5, and if the new minimum is lower, repeat 3–5; otherwise, return minimumExecution of this simple approach yields for the considered data, the critical value β1c=8.2100, and accordingly,(106)β¯c=8.2100,1.9358,3.8857,0,φβ¯c=26.3229.The straightforward implementation of the results of Lemma 4 and Lemma 5 simplified through the additional constraint(107)μQs≤β1,μQs=δ+maxi=1,…,NμQcorresponding toA¯¯4i,as motivated in Remark 5, yields the minimum of the numerical minimization(108)β¯∗=7.3810,0.7335,6.5763,−2.1962,φβ¯∗=24.4586<φβ¯c.Both β¯c and β¯∗ yielded a difference matrix C¯¯ with only one vanishing eigenvalue. As in the previous example, the optimization gets away with two nonlinear inequality constraints per matrix A¯¯∈A independently of the dimension n of the matrices A¯¯∈A, seedemo5.min [22] for details and reproduction of the results of this example.


## 5.1. Example: Four-Parametric Optimization Majorizing a Single Matrix

We consider the semidefinite programming problem (96) for a single matrix A¯¯4 and the four-parametric form B¯¯4 of Section 4. A matrix A¯¯4∈S11 and a vector p¯0∈ℝ10 with Λ11≈−11.3075, Λ1≈4.7144, and μQ≈4.206 are considered in order to optimize the Frobenius norm(103)φβ¯=∥B¯¯4β¯∥F=nβ12+β22+β32+2β42,n=10.Note that the unconstrained minimum of this function is obtained at β¯=o¯∉ℬ4. Hence, the constrained minimum will be found on the boundary ∂ℬ4. The explicit data of the current problem can recovered based ondemo4.min the provided software [22].The results of Lemma1–Lemma 4 for the explicit construction of sharp majorants may be applied as follows for the generation of an initial guess for the future optimization over ℬ4 for a single matrix:(1)
Generate a discrete point setI1h from the interval I1=μQs,β1max with μQs as in (90) (e.g., through an equidistant discretization of I1)(2)
Compute the correspondingβ˜2β1=β2minβ1 for each β1∈I1h(3)
Use Lemma4 noting κ=1∧P⊥kerC¯¯∀β1∈I1h:(a)
Compute the correspondingβ˜4β1=β4β1,β˜2β1 with (74)(b)
Compute the correspondingβ˜3β1 = minimum β3 in (75)(4)
Evaluate objective functionφ˜β1=φβ¯β1 accordingly, and return critical β1c yielding the minimum value for β1∈I1hExecution of the just described simple approach yields a critical β1c=4.6028 implying(104)β¯c=4.6028,−4.1089,4.5598,−2.4422∈∂ℬ4,φ˜β1c=φβ¯c=16.1697.The straightforward implementation of the results of Lemma 4 and Lemma 5 simplified through the additional constraint of Remark 5 yields for the numerical minimization(105)β¯∗=4.6302,0.3030,4.6269,−1.1840∈∂ℬ4,φβ¯∗=15.4495<φβ¯c.Notably, β¯c induces two zero eigenvalues in C¯¯4, while β¯∗ leads to a one-dimensional kernel. Algorithmically, the optimization gets away with two nonlinear inequality constraints independently of the dimension n of the problem, seedemo4.min [22] for details and reproduction of the results of this example.


## 5.2. Example: Four-Parametric Optimizations Majorizing a Set of Matrices

Next, we consider the semidefinite programming problem (96) for a finite matrix set A=A¯¯41,…,A¯¯4N with N=5 and n=10, B¯¯4 and the Frobenius norm given in (103). The explicit data of the current problem can be recovered based ondemo5.min the provided software [22].The results of Lemma1–Lemma 5 for the explicit construction of sharp majorants may be used in order to generate an initial guess as follows:(1)
Generate a discrete point setI1h from the interval I1=μQs,β1max with μQs>maxi=1,…,NμQcorresponding toA¯¯4i (e.g., through an equidistant discretization of I1)(2)
Computeβ˜2β1=maxi=1,…,Nβ2minβ1corresponding toA¯¯4i for β1∈I1h and build discretized curve C=β1,β2∈ℬ2A:β1∈I1h,β2=β˜2β1(3)
ShiftC by some small positive Δ into the interior of ℬ2A and rewrite C=β1,β2∈ℬ2A∖∂ℬ2A:β1,β2=β1′+Δ,β2′+Δ,β1′,β2′∈C(4)
Use Lemma5:(a)
Setβ˜4β1=0∀β1∈I1h(b)
Compute correspondingβ˜3β1=maxi=1,…,Nβ3according to82forA¯¯4iwithβ1,β2∈C(5)
Evaluateφβ1,β˜2β1,β˜3β1,β˜4β1 for all β1∈I1h and determine minimum(6)
Go to 3 (i.e., shiftC again by Δ), execute 4 and 5, and if the new minimum is lower, repeat 3–5; otherwise, return minimumExecution of this simple approach yields for the considered data, the critical value β1c=8.2100, and accordingly,(106)β¯c=8.2100,1.9358,3.8857,0,φβ¯c=26.3229.The straightforward implementation of the results of Lemma 4 and Lemma 5 simplified through the additional constraint(107)μQs≤β1,μQs=δ+maxi=1,…,NμQcorresponding toA¯¯4i,as motivated in Remark 5, yields the minimum of the numerical minimization(108)β¯∗=7.3810,0.7335,6.5763,−2.1962,φβ¯∗=24.4586<φβ¯c.Both β¯c and β¯∗ yielded a difference matrix C¯¯ with only one vanishing eigenvalue. As in the previous example, the optimization gets away with two nonlinear inequality constraints per matrix A¯¯∈A independently of the dimension n of the matrices A¯¯∈A, seedemo5.min [22] for details and reproduction of the results of this example.


## 6. Conclusions

Closed form expressions for the representation of two-parametric Löwener majorants based on a rank-one perturbed scaled identity matrix are presented in Section3. Different scenarios accounting for the multiplicity of eigenvalues and for the inclination of the vector p¯ with respect to the eigenbasis of the matrix under consideration are formulated. The majorization of a set of symmetric matrices A, yielding the global majorant set ℬ2A, is then obtained by an intersection of the individual admissibility domains. Most importantly, the intersection is always unbounded, and thus, nonempty. Analogous results are obtained for the four-parametric case illustrated in Section 4, at what the explicit conditions defining the majorant set, with β3 and β4 chosen accordingly, allow for efficient algorithms, cf. Corollary 2 and Remark 5. Examples have been presented for the two- and four-parametric cases. The application of these results in semidefinite programming problems has been sketched in Section 5. The results of the present work also find application in materials science and engineering in the field of materials design of linear elastic and linear thermo-elastic properties. So-called zeroth-order bounds of linear material properties are determined through optimization problems as the one described in Section 5, see, e.g., [17] or [19], such that the results of the present work substantially simplify the computation of the zeroth-order bounds and the related automated selection of material of large material databases. The data and the software for reproduction of the results are available via GitHub, see [22].



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*Source: 9091387-2020-06-11.xml*