P.A. FILLMORE and J.P. WILLIAMS. Some convexity theorems for matrices. Glasgow Math. J., 10:110--117, 1971.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....In this paper we concentrate on outlining how the idea of a normal decomposition system provides a simple yet powerful unifying framework in which to study a wide variety of important results. Examples include Schur convexity (see for example [22] the convexity of eigenvalue functions ([10, 6, 11, 3, 13, 19]) calculations of Fenchel conjugates and subdifferentials of convex eigenvalue functions [24, 5, 12, 30, 28, 25, 26, 27, 15, 16, 1, 17, 19] von Neumann s original result [33] and generalizations (for example [4, 20] subdifferentials of unitarily invariant norms [34, 35, 36, 37, 38, 8, 7, 9, ....

....of symmetric matrices Gamma1 (C) fx 2 S n j x positive semidefinite; tr (x) 1g; is closed and convex, with extreme (exposed) points yy T for unit column vectors y in R n . The fact that the set Gamma1 (C) is convex if and only if C is convex, for a symmetric set C, was proved in [11]. Unitarily invariant norms A function h : R l [ Gamma1; 1] is absolutely symmetric if the value h(ff) at a vector ff in R l is independent of the order and signs of the components ff i : in the notation of Example 7.3, h(ff) h(jffj) for all ff. In particular, if such a function is also ....

P.A. Fillmore and J.P. Williams. Some convexity theorems for matrices. Glasgow Mathematical Journal, 12:110--117, 1971.

.... characterization holds: 1 (A) Delta Delta Delta k (A) max A ffl U s:t: trace U = k 0 U I (40) Proof: See Overton and Womersley [OW91, OW92] It is worth mentioning that this result is based on a beautiful convex hull characterization which was known at least as early as 1971, see [FW71], but unfortunately has remained somewhat obscure. Here is the statement of this result: Lemma 12 Let S 1 : fY Y T : Y 2 n Thetak ; Y T Y = Ig and S 2 : fW : W = W T ; trace W = k; 0 W Ig: 20 Then conv S 1 = S 2 ; and S 1 is exactly the set of extreme points of S 2 . For an ....

P. A. Fillmore and J. P. Williams. Some convexity theorems for matrices. Glasgow Mathematical Journal, 12:110--117, 1971.

....relaxation is the trust region relaxation studied in [63] QAPT : min trace AXBX T s.t. XX T I: 55 The constraints are convex with respect to the L owner partial order and so it is hoped that solving this problem would be useful. The set fX : W = XX T Ig is studied in [96, 28] and is useful in eigenvalue variational principles. Furthermore the problem (4.100) is visually similar to the trust region subproblem so we would like to nd a characterization of optimality. We study the matrix trust region relaxation of QAP: SDPT = min trace AXBX T s.t. XX T I: ....

P.A. FILLMORE and J.P. WILLIAMS. Some convexity theorems for matrices. Glasgow Math. J., 10:110-117, 1971.

....property, and Y Y t e = me together with the fact that rows of Y are unit vectors, and get the following basic semidefinite relaxation: minf 1 2 tr LX : X 0; diag(X) e; Xe = meg: The following result can be used to rewrite the Donath Hoffman bound as a semidefinite program. Lemma 4 ([16, 47]) convfY Y t : Y t Y = I k g = fX : tr X = k; I X 0g = M k : If A and B are symmetric matrices of the same size, then A B stands for A Gamma B 0: A discussion of various ways to prove this result can be found in [47] This relation can be used as follows: min Y t Y =I k tr Y ....

P.A. FILLMORE, and J.P. WILLIAMS. Some convexity theorems for matrices. Glasgow Math. J. 12:110--117, 1971.

....convex with respect to the Lowner partial order and so it is hoped that solving this problem would be useful. Also, this problem is visually similar to the TRS discussed above. And so we would like to find a characterization of optimality. The set fX : W = XX T Ig is studied separately in [70, 25] and is useful in eigenvalue variational principles. We now study the matrix trust region relaxation of QAP: SDPT = min Trace AXBX T s.t. XX T I : The following generalization of the Hoffman Wielandt inequality holds. THEOREM 2.4. For any XX T I, we have P n i=1 minf i n Gammai 1 ; ....

P.A. FILLMORE and J.P. WILLIAMS. Some convexity theorems for matrices. Glasgow Mathematical Journal, 10:110--117, 1971.

....of Y satisfies Sigma = I = Z 1 (1 Gamma )Z 2 , where Z i = U t Y i V; i = 1; 2. Since Z i ; i = 1; 2, are still feasible and so have norm 1, we conclude that Z i = I; i = 1; 2, i.e. Y = Y i ; i = 1; 2. Thus Y is an extreme point. 2 Equivalent formulations of Lemma 3. 1 can be found in [11, 37] for gaining a better insight. Although the formulations are slightly different, the resulting feasible set and its extreme points are the same. If the objective function is concave, then the optimum of (P ) is attained at an extreme point of F and is then orthogonal. It is easy to see that f(X) ....

P.A. FILLMORE and J.P. WILLIAMS. Some convexity theorems for matrices. Glasgow Mathematical Journal, pages 110--117, 1971.

....p is convex if and only if C a is convex. In this case an element x of p is an extreme (exposed) point of C if and only if fl(x) is an extreme (exposed) point of C a. Proof The convexity is an immediate consequence of Kostant s theorem. The symmetric matrix case for closed sets C appeared in [10]. The extremal properties follow from Theorem 5.5 and Corollary 6.3 in [20] Theorem 3.7 (Invariant norms) AdK) invariant norms on p are those functions of the form g ffi fl, where g is a W invariant norm on a. In this case the dual norms satisfy (g ffi fl) D = g D ffi fl; and an ....

P.A. Fillmore and J.P. Williams. Some convexity theorems for matrices. Glasgow Mathematical Journal, 12:110--117, 1971.

....convexity and smoothness in this setting analogous to those in our present development: such results for unitarily invariant norms have appeared in [3, 36] Studying convex matrix functions via their Fenchel conjugates is not a new idea. It is implicit for example in some of the techniques in [7], and was used explicitly in [8] to study the sum of the largest k eigenvalues of a real symmetric matrix, an approach also followed in [12] see also [13] The primary aim of these latter papers is to study sensitivity results via the subdifferential set. Various representations of this set ....

P.A. Fillmore and J.P. Williams. Some convexity theorems for matrices. Glasgow Mathematical Journal, 12:110--117, 1971.

....A the following semidefinite programming characterization holds: 1 (A) Delta Delta Delta k (A) max A ffl U s:t: trace U = k 0 U I: 4. 3) It is worth mentioning that this result is based on a beautiful convex hull characterization which was known at least as early as 1971, see [22], but unfortunately has remained somewhat obscure. Here is the statement of this result: Lemma 4.2. Let S 1 : fY Y T : Y 2 n Thetak ; Y T Y = Ig Interior Point Semidefinite Programming 23 and S 2 : fW : W = W T ; trace W = k; 0 W Ig: Then conv S 1 = S 2 ; and S 1 is exactly ....

P. A. Fillmore and J. P. Williams, Some convexity theorems for matrices, Glasgow Math. J., 12 (1971), pp. 110--117.

....(2.5) since it is then automatically satisfied; it was precisely this constraint which was missing in the formula proposed by Fletcher [10, Appendix] After having finished the present work we discovered that the result of Proposition 2. 1 had been established much earlier by Fillmore and Williams [9] (a paper which would deserve to be better known in the community of numerical analysis and optimization) the proof we propose below is radically different: it relies on techniques from convex analysis. Since oe m is a support function, its Legendre Fenchel transform (2:9) oe m : C 2 E 7 ....

....they represent somehow the shadow of Rm along the line directed by A. Clearly, the bounds of Rm (A) and those of Omega m (A) fhhA; Cii : C 2 Omega m g are the same, i.e. maxRm (A) max Omega m (A) and minRm (A) min Omega m (A) More surprisingly, Rm (A) is itself convex (cf. [9] and references therein) as for example R 1 (A) fhAx; xi : kxk = 1g = n (A) 1 (A) Rn (A) ftr Ag : Numerische Mathematik Electronic Edition page numbers may differ from the printed version page 52 of Numer. Math. 70: 45 72 (1995) Sensitivity analysis of all eigenvalues of a ....

Fillmore, P.A., Williams, J.P. (1971): Some convexity theorems for matrices. Glasgow Mathematical Journal 12, 110--117

....H = Gamma1 (C) fX 2 S(n) j (X) 2 Cg: The condition that C is permutation invariant is in some sense superfluous for this observation, but is crucial for our later development. For example, with this assumption the matrix set H is closed and convex if and only if C is closed and convex (see [7, 10]) If we define the diagonal map Diag : R n S(n) by letting Diag x be a diagonal matrix with diagonal entries x 1 ; x 2 ; x n , then the set of diagonal matrices in H is just Diag C. Our main result describes the exposed faces of H in terms of those of C. Let E be a proper, exposed ....

....matrix P . Suppose in addition that C is closed and convex. Then the matrix set Gamma1 (C) fX 2 S(n) j (X) 2 Cg; is also closed and convex: to see this, note that its indicator function satisfies ffi Gamma1 (C) ffi C ffi ; 4. 3) so we can apply the results in [10] see also [7]) We also see from this equation and the conjugacy formula (4.1) that the support functions satisfy ffi Gamma1 (C) ffi C ffi : 4.4) By assumption, the set C is invariant under the group P(n) and it is easy to see that the matrix set Gamma1 (C) is invariant under the group ....

P.A. Fillmore and J.P. Williams. Some convexity theorems for matrices. Glasgow Mathematical Journal, 12:110--117, 1971.

....cost function by using tr Y t LY = tr L(Y Y t ) and replacing Y Y t by a new variable X . In case of (k Gamma GPDH ) we can optimize over S k : convfY Y t : Y t Y = I k g: In order to do so, we need a nice description of this set. Fortunately, such a description exists. Lemma 3. 1 [25, 11] S k = fX : X = X t ; tr X = k; I X 0g: As observed by Alizadeh [1] the eigenvalue bound z DH can now be reformulated as follows. z DH = max u t e=0 min X2S k m 2 tr (L diag(u) X = min X2S k max u t e=0 m 2 tr LX m 2 u t diag(X) min f 1 2 tr LX : diag(X) e; mI X ....

P.A. FILLMORE, and J.P. WILLIAMS. Some convexity theorems for matrices. Glasgow Math. J. 12:110--117, 1971.

No context found.

P.A. FILLMORE and J.P. WILLIAMS. Some convexity theorems for matrices. Glasgow Math. J., 10:110--117, 1971.

No context found.

P. A. Fillmore and J. P. Williams (1971), "Some convexity theorems for matrices ", Glasgow Mathematical Journal 12, pp. 110-117.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC