Research Article | Open Access
Mauricio Fernández, Felix Fritzen, "Construction of a Class of Sharp Löwner Majorants for a Set of Symmetric Matrices", Journal of Applied Mathematics, vol. 2020, Article ID 9091387, 18 pages, 2020. https://doi.org/10.1155/2020/9091387
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.
The Löwner partial order introduced by  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  in the context of statistical bounds of linear elastic properties, further analyzed by , and corrected by [17, 18]. For instance, an N-phase solid is constituted of N materials with corresponding symmetric positive-definite stiffness matrices, say, with for three-dimensional linear elasticity. A zeroth-order bound of the effective linear material behavior of the N-phase solid is a symmetric matrix which satisfies 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 can vary from realization to realization. For practical reasons, see, e.g.,  or  for details, the zeroth-order bound is chosen as an isotropic stiffness such that the orientation of the eigensystems of the stiffnesses 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 depending on parameters of , cf.  for a discussion. This yields the minimization problemwhich 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 reduces significantly, in linear elasticity to two, and the structure of is highly specific. An analogous problem can be formulated for linear thermo-elasticity, cf. , 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 , 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 , 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 , 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 forms . This parameter constraint condensation has not been conducted in either [15–18] or . 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 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 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 Section 2, 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  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 . Column vectors over , are denoted by single-underlined characters, e.g., . Rectangular matrices over are denoted by double-underlined characters, e.g., . Square matrices over are referred to as matrices of order n. The set of symmetric matrices of order n with finite eigenvalues is denoted by . The transposition is denoted by the superscript , e.g., equals the scalar product of the vectors, and yields the outer product of the vectors. The eigenvalues of a symmetric matrix are denoted by with . The multiplicity of an eigenvalue λ is denoted as . The norm of a vector and the Frobenius norm of a matrix are simply noted as and . The identity matrix is noted as . For a compact notation, the orthogonality of two vectors and or two vector spaces and is denoted simply as and , respectively. The Moore–Penrose inverse of a matrix is denoted by .
2. Löwner Order and Majorants for Symmetric Matrices
In this work, we consider the Löwner partial order of symmetric matrices (see, e.g., ):where denotes the positive semidefinite cone. Any is referred to, in this work, as a (Löwner) majorant of a given if holds. Furthermore, any is referred to as a sharp (Löwner) majorant of a given if and hold, i.e., if holds. For given , the corresponding majorant set and sharp majorant set are denoted asOf course, is nonempty, unbounded, and convex. The trivial majorant of a given is defined asFor a finite set of symmetric matrices, we define the analogous majorant setwhose boundary represents the set of all sharp majorants of :The trivial element on is
3. Two-Parametric Majorant
3.1. Construction of Two-Parametric Majorants for a Single Matrix
For given , from the corresponding majorant set , we investigate a two-parametric form with given (i.e., fixed) normalized vector , , defined as a rank-one perturbation of a scaled identity viaIn the context of zeroth-order bounds of the effective linear elastic behavior of solid materials, is given for isotropic zeroth-order bounds if a specific parametrization is chosen, cf., e.g.,  or  for details.
The goal of this section is the derivation of condensed conditions for the parameters such that holds. For the sake of a clearer perspective in this section, we consider the spectral factorization of the given with ordered eigenvalue diagonal matrix with diagonal entriesand corresponding orthogonal eigenvector matrix , i.e., . We denote the set of eigenvectors of as and the space of eigenvectors corresponding to a particular eigenvalue as , i.e.,A change of basis does not alter the Löwner partial order, i.e., . We, therefore, defineand note . For the remainder of this section, we seek the conditions on the parameters describing the parameter setDue to linear dependency of on and the general properties of , the parameter set is nonempty, unbounded, and convex. Note that 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 for (iii)Remark 1: remarks for cases on for Lemma 1(iv)Lemma 2: closed form description of for and bounded from below by or (v)Remark 2: remarks for cases on for Lemma 2(vi)Lemma 3: closed form description of for and bounded from below by a constant between and (vii)Corollary 1: recapitulation of some topological properties of relevant for semidefinite programming problems(viii)Remark 3: remarks for cases on for Lemma 3(ix)Remark 4: summarizing remarks and perspective for corresponding semidefinite programming problems
Preparations. Before we derive the conditions describing in (12), we introduce some quantities and relations needed throughout this section. The condition is equivalent to the positive semidefinitness of the difference matrix :i.e.,We denote the dimension of the kernel of the difference matrix asWe define the mutually orthogonal vector spacesand introduce a matrix such that the concatenated matrix is orthogonal, i.e., the columns of form an orthonormal basis of . We definewhere corresponds to the largest eigenvalue of the reduced matrix . Inserting vectors from and into (14), respectively, yields the two elementary necessary conditions:Note that is bounded by due toFurthermore, the relationis obtained based on the following arguments:(1)If , then and holds.(2)If , then using , one can define with . Substituting impliesTherefore,generally holds. Denote the algebraic multiplicity of the maximum eigenvalue by . It is noted thatand based on (22),holds. Based on these preparations, we now proceed to the description of the parameter set through three lemmas.
Lemma 1. If holds, then the parameter set is described by
Remark 1. For Lemma 1, all (i.e., at least one of the inequalities in (26) turns into an equality) are sharp majorants and induce a singular with nonempty kernel of dimension κ, cf. (15). The special cases for deliver a one-dimensional kernel of . It is noted that only (29) ensures , independent of the multiplicity of , and, more importantly, that holds. This property is of interest for Section 4.
Proof. We investigate the interval for in with based on the following cases:(1): In this case, sharp majorants require a singular difference matrix . Due to the restriction of this case, the diagonal matrix is positive definite such that, based on the determinant lemma, cf., e.g.,  or , we considerwhich is a single-valued constraint in terms of with . This is fulfilled iffmeaning that (34) describes for every the corresponding yielding a sharp majorant, i.e., (34) specifies a portion of . Due to the convexity of , delivers trivially nonsharp majorants for .(2): the matrix is singular and positive semidefinite. We consider the matriceswhich contain in the respective columns a basis of for and of the corresponding orthogonal space . We examine the following cases:(a)If holds, then is already singular, independently of , since holds, meaning that is at least one-dimensional. Based on (35) and , , (14) can be further reduced to The matrix is positive definite, and since is not an eigenvector of , holds. For to be singular, its determinant has to vanish. Analogous application of the matrix determinant lemma as in (33) yields that is singular iff where denotes the Moore–Penrose inverse of . The choice (38) induces a second vanishing eigenvalue in such that its kernel is then at least two-dimensional.(b)If holds, then using (35) and , in (14) yields the necessary condition This condition is also sufficient for . For , one retrieves for the current scenario the trivial majorant , cf. (4).If holds, cf. (24), then cases 1 and 2 of this proof describe , at what , cf. (17), is identified in (38), yielding (30)—cf. Lemma 2.1 as depicted in Figure 1 in the ramification for and . If and 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′): Due to , (39) must hold for this case.(3′): For the present case, 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 may vanish for such indefinite . More precisely, the term may vanish, iff holds. We search now for a vector such that holds, with the necessary condition This means that would be required to correspond to the eigenvector of the reduced matrix for eigenvalue . But, we consider larger than the maximum eigenvalue of such that no such exists, and therefore, the term cannot vanish. Solving (33) for yields again (34).(4′): Due to , analogous reasoning as in 2(a) is applied to this case with corresponding and up to the corresponding equation The difference to 2(a) here is that is not positive definite but regular and indefinite. For to be singular, its determinant must vanish. Application of the matrix determinant lemma as in (33) yields Here, the term equals the quantity for , see (17). The quantity may or may not vanish, depending solely on and . This means that if , then (42) can be solved for as in (38), yielding and (31)—cf. Lemma 2.2 as depicted in Figure 1. But, if holds, then there exists no for such that holds since cannot vanish. Therefore, yields (32) and, more importantly, excludes from the range of for the current scenario, i.e., , cf. Lemma 2.3 illustrated in Figure 1.
Remark 2. It is noted that, for Lemma 2, , cf. (15), holds in a number of scenarios. In order to visualize the following argument, the position of the upcoming cases in the set is depicted in Figure 2.
In cases 1 and 3′ of the proof of Lemma 2, the rank-one perturbation can only induce a one-dimensional kernel on the difference matrix with regular for fulfilling (34). This corresponds to the majority of points described through in Lemma 2, only excluding the special case . The corresponding kernel vector with is obtained assuch thatholds since holds. The corresponding points based on inducing (44) are indicated in Figure 2 at the lower border of with , and more importantly, due to the consequences of Section 4, .
For , i.e., Lemma 2.1 is considered, the point at 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., , is already singular (, i.e., ) and for fulfilling (38), the rank-one perturbation induces a further vanishing eigenvalue of such that is at least two-dimensional. This special case corresponds to the black point in Figure 2. For above the critical value given in (38), is then at least one-dimensional such that is only possible at under the current assumptions only for , i.e.,In case 2(b), i.e., , is not perpendicular to the complete eigenspace of , but for , there always exists at least one eigenvector to which is perpendicular to . Therefore, is only possible at under the current assumptions for , i.e.,These arguments imply that, for Lemma 2.1, based on (44)–(46), and are assured only for points corresponding to and .
For , cf. (25), we need to examine the points at and for Lemma 2.2 and Lemma 2.3. For case 2′, addressing ,follows such that, for , the lower boundary of depicted in Figure 2 yields and . Analogous reasoning as for (45) yields for case 4′, addressing ,It can be concluded for Lemma 2 that only for and , the kernel of stays one-dimensional (indicated in Figure 2 by the border with ) and, more importantly, that holds. This property is of interest for Section 4.
Lemma 3. If and hold, then the parameter set is described by
Proof. We investigate the interval for in based on the following cases:(1): the results for this case are identical to case 1 of the proof of Lemma 2.(2): due to , this case follows the reasoning of case 2(b) of the proof of Lemma 2.(3): the results for this case are identical to case 3′ of the proof of Lemma 2.(4): due to , is regular, and the determinant lemma can be applied as in (33). The vector can be defined based on the regularity of such that only one vector exists fulfilling . The vector then fulfillswhich can only be achieved by one specific eigenvector of the reduced matrix corresponding to its maximum eigenvalue , i.e., with . This implies and, consequently, . Since the term vanishes, then, regarding (33), cannot render the determinant of to zero, i.e., for , there exists no delivering a point on . More explicitly, is excluded from the range of .
Corollary 1. The set is always an unbounded closed convex set. In particular, it is never compact, which has implications for optimization problems over .
Remark 4. Wrapping this section up, the reader solely needs to differentiate the cases:(i)Case 1: is an eigenvector of Lemma 1(ii)Case 2: is not an eigenvector of and Lemma 2(iii)Case 3: is not an eigenvector of and Lemma 3It should be noted that if a function is to be minimized over , then the spectral factorization of given and the transformation of given to 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 . The minimization can then be carried out at its peak efficiency over , 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 of N given symmetric matricesand a given vectorThe corresponding convex setsare the majorant sets for each of the matrices of . Denote the respective spectral factorizations asand define the corresponding vectorsSince holds, the results of the previous section describe the corresponding sets. The intersection of all sets delivers the majorant set for the set of matrices , denoted asNote that, due to being admissible for any bounded matrix, the set 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 matrix are sought-after for three different vectors , , and leading to the aforementioned cases, see demo1.min the provided software .
Consideration of induces an instance of Lemma 1, in which is contained in the eigenspace of , cf. (27). Figure 3 shows the parameter domain . The boundary of is indicated by the black line in Figure 3(a) defined by and by the vertical line corresponding to . The value of the quadratic forms and for normalized vectors within the (1,2)-plane are shown as contours in Figure 3(b). It is readily seen that all parameters on the boundary of 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).
Consideration of yields an instance of Lemma 2 ( is not an eigenvector of , , and , i.e., ). The corresponding results are depicted in Figure 4. It should be noted that, for , the pseudoinverse of is required and is evaluated, while for , the inverse of is computed. In Figure 4(b), the region around 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).
Lastly, is an instance of Lemma 3 ( is not an eigenvector of and ), with corresponding results depicted in Figure 5. Most notably, the value marks the asymptote of the boundary of since for , no exists yielding a point on , cf. case 4 of the proof of Lemma 3.
3.4. Example: Majorization of a Set of Matrices
Consider the finite set of symmetric matricesand the vectorThe border of the corresponding set (an instance of Lemma 1) is depicted in Figure 6 by the straight lines. The border of the corresponding (an instance of Lemma 3) is depicted by the curved black line in Figure 6. The majorant set defined in (56) is depicted in Figure 6 by the gray region. The depicted region can be reproduced with the file demo2.musing the provided MATLAB software, cf. .
Next, a set of random symmetric matrices of dimension 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 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 .
4. Four-Parametric Majorant
4.1. 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 is considered and partitioned as follows:From all majorants of , we are interested in this section in the parametrizationwith given normalized . This four-parametric form arises in linear thermo-elasticity and corresponding isotropic zeroth-order bounds, cf. . The upper left block of corresponds to , cf. (8). A short inspection of 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 matrixdiagonalizing the upper left block of , i.e., we defineIn this section, is examined, i.e., we seek the parameter conditions describing the setAs 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 based on (iii)Lemma 5: description of based on (iv)Corollary 2: recapitulation of the admissible regions for for and 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 matrixat what positive semidefinitness of is equivalent to . More explicitly, (67) can also be expressed aswhich allows to obtain the equivalence relationwhere denotes the Moore–Penrose inverse of , see, e.g.,  for a derivation based on the Schur complement. It should be noted that for positive semidefinite , is also positive semidefinite such that necessarily has to be nonnegative. The condition is fulfilled for the corresponding of Section 3. We consider, therefore, the following cases for singular positive semidefinite and positive definite , i.e., based on Section 3, for and , respectively.
Lemma 4. If holds, then the parameter set is described by
Proof. For a better overview of the following proof, the reader should consider Figure 8 along the following arguments. For , is positive semidefinite and holds such that the kernel is nonempty. For sharp majorants to exist, , cf. (69), describes the critical condition. We notice the following cases:(1): the kernel of is one-dimensional, i.e., (i)In Lemma 1, cf. Remark 1, for(a)(28) with (left border without corner of ) or(b)(29) with (lower border without corner of )(ii)In Lemma 2, cf. Remark 2 and Figure 2, for(a)(45) and (46) with (left border without corner of for ),(b)(48) with (left border without corner of for ), or(c)(44) and (47) with (lower border without corner of )(iii)In Lemma 3, cf. Remark 3, with on the whole border of If is one-dimensional, for majorants to exist needs to hold, cf. (69). The condition (71) may or may not be fulfilled in a number of exotic cases since some portions of , as, e.g., in (28), have a one-dimensional kernel but with .(a)If holds, cf. Figure 8, then loses influence in (71) such that (71) can be fulfilled iff holds, which is not controllable by the parameters but solely dictated by the given data and , with resulting and . If additionally holds, then (71) is fulfilled for all such that, according to (69), majorants exist if for given , the remaining parameters and are chosen such that holds. For instance, the left-hand side term of (72) is quadratic in and is minimized for which then yields in (72) the lowest possible and corresponding , cf. Figure 8. Naturally, if holds, then (71) is not fulfilled and no majorants exist under the current assumptions for , cf. Figure 8.(b)If holds, which is the case, e.g., on the lower border of in Lemma 1 and Lemma 2 (up to ) and on the whole border of in Lemma 3, then (71) can be solved uniquely for , yielding From (74), one can obtain based on (69), i.e., , the min