A. Shapiro. Extremal problems on the set of nonnegative definite matrices. Linear Algebra and Appl., 67:7--18, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of a symmetric matrix that depends affinely on x. In this case, the problem is in fact convex (but still nondifferentiable) Many researchers have considered this problem. Relevant work includes Cullum et al. [CDW75] Craven and Mond [CM81] Polak and Wardi [PW82] Fletcher [Fle85] Shapiro [Sha85], Friedland et al. FNO87] Goh and Teo [GT88] Panier [Pan89] Allwright [All89] Overton [Ove88, Ove92, OW93, OW92] Ringertz [Rin91] Fan and Nekooie [FN92] and Fan [Fan92] In [BY89] Boyd and Yang use the cutting plane algorithm and Shor s subgradient method [Sho85] to solve eigenvalue ....

A. Shapiro. Extremal problems on the set of nonnegative definite matrices. Linear Algebra and Appl., 67:7--18, 1985.

....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, 20] and characterizations of extreme, exposed and smooth points of unit balls [2, 37, 38, 8, 7, 9, 20] This paper concentrates on convexity ....

A. Shapiro. Extremal problems on the set of nonnegative definite matrices. Linear Algebra and its Applications, 67:7--18, 1985.

....structure; in engineering applications many other types arise (e.g. Toeplitz structure) 1. 2 Historical overview An early paper on the theoretical properties of semidefinite programs is Bellman and Fan [BF63] Other references discussing optimality conditions are Craven and Mond [CM81] Shapiro [Sha85], Fletcher [Fle85] Allwright [All88] Wolkowicz [Wol81] and Kojima, Kojima and Hara [KKH94] Many researchers have worked on the problem of minimizing the maximum eigenvalue of a symmetric matrix, which can be cast as a semidefinite program (see x2) See, for instance, Cullum, Donath and Wolfe ....

A. Shapiro. Extremal problems on the set of nonnegative definite matrices. Linear Algebra and Appl., 67:7--18, 1985.

....x 2 m and U 2 S n Thetan . Now define a dual problem max U : tr U=1; U0 min Cx=b; xu h U ; A(x)i: The following theorem, motivated originally by [Boy87,OWa] is a standard saddle point result and follows from [Roc70, Theorem 36.3] For closely related results, see [Ehr79,Sha85a] Theorem 6 Suppose that A(x) is an affine function, so that A k (x) is constant (independent of x) for all k. If the primal problem has a solution, say defined by (x ; U ) then the same pair solves the dual problem. Note that in the unconstrained affine case the dual problem can have a ....

A. Shapiro. Extremal problems on the set of nonnegative definite matrices. Linear Algebra and its Applications, 67:7--18, 1985.

....conclude, we use the conjugacy formula to study duality relationships for various convex optimization problems posed over the cone of positive semidefinite, real symmetric matrices. Interest in matrix optimization problems (and duality in particular) has been growing in recent years (for instance [27, 23, 1, 33, 35, 28]) The examples we choose are of recent interest in applications of interior point methods (see for example [1, 15, 21, 2] as well as for variational characterizations of certain quasi Newton updates (see for example [9, 34] 2 Conjugates of induced matrix functions We begin with a technical ....

....In this section we will illustrate how the conjugacy formula derived in Section 2 can be used to study duality properties of optimization problems involving real symmetric matrices. In particular we can study analogues of linear programming over the cone of positive semidefinite matrices (see [27, 23, 1, 33, 35, 2]) penalized versions of such problems (see for example [1, 15, 21, 2] and convex optimization problems leading to well known quasi Newton formulae for minimization algorithms [9, 34] Suppose that X and Y are finite dimensional inner product spaces. For functions F : X ( Gamma1; 1] and G : Y ....

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A. Shapiro. Extremal problems on the set of nonnegative definite matrices. Linear Algebra and its Applications, 67:7--18, 1985.

....[23] treat duality theory for the eigenvalue optimization problem from the point of view of subdifferentials. Such an approach is related to the Kuhn Tucker duality theory and relies on derivatives or subgradients. Also Lov asz in [41] Grotschel, Lov asz and Schrijver [30, 31, 32] and Shapiro in [61] study more or less the same duality theory as we do, but their treatment is restricted to a special form of SDP. We now proceed to state and prove duality for the semidefinite programming problem. However observe that the following development in particular the weak duality lemma 2.1, lemma 2.2, ....

....is infeasible and z 1 = 1. Hence the assumption z 2 z 1 results in contradiction. Since by weak duality lemma we have that z 2 z 1 we conclude that z 2 = z 1 . It is also possible to derive a complementary slackness theorem. In fact, Grotschel, Lov asz and Schrijver in [31] and Shapiro in [61] mention the complementary slackness theorem for a more restricted form of SDP. Note that when the strong duality theorem is true and both primal and dual problems are bounded and feasible then the duality gap X ffl S vanishes. However, in SDP, as in linear programming, a stronger form of ....

A. Shapiro, Extremal problems on the set of nonnegative definite matrices, Linear Algebra Appl., 67 (1985), pp. 7--18.

....convex case was covered in [14] here we use an independent approach to develop the nonconvex case. Since the seminal paper [5] the study of matrix optimization problems (and in particular eigenvalue optimization) has become extremely prominent. A typical constraint is positive semidefiniteness [7, 22, 25, 26], and with the modern trend towards interior point methods, it has become popular to incorporate this constraint by a barrier penalty function (involving the eigenvalues) 16, 1, 12] A related objective function is used in [8] to give an elegant variational characterization of certain ....

A. Shapiro. Extremal problems on the set of nonnegative definite matrices. Linear Algebra and its Applications, 67:7--18, 1985.

....of d, and hence v(u) is di#erentiable at u 0 . 5 NOTES Lagrangian duality is a well developed concept in mathematical programming. Its origins go back to von Neumann s game theory. In the context of semidefinite programming particular examples of duality schemes were considered, for example, in [1, 19, 25]. Example 2.2, of a linear semidefinite program with a duality gap, is taken from [24] The parametric approach to duality, by applying convex analysis to the parametric problem (2.17) was developed in Rockafellar [17, 18] A proof of the Fenchel Moreau duality theorem can be found in [17] ....

Shapiro, A., Extremal problems on the set of nonnegative definite matrices. Linear Algebra and Applications 1985; 67:7-18.

....is also true. Consider the linear case, with f(x) c T x and G(x) A 0 P m i=1 x i A i . Then OE( Omega Gamma = Omega ffl A 0 if c i Omega ffl A i = 0, i = 1; m, and OE( Omega Gamma = Gamma1 otherwise. Therefore in that case the dual problem takes the form (cf. 1] [27]) max Omega 2Sn Omega ffl A 0 subject to c i Omega ffl A i = 0; i = 1; m; Omega 0: 2.6) Consider now the following quadratic case, f(x) c T x and the mapping G(x) is given in the form (2.4) Then OE( Omega Gamma = min x2IR m n Omega ffl A 0 b T x 1 2 x T ....

....0 satisfying optimality conditions (2.10) and (2.11) matrices Omega A i , i = 1; m, are linearly independent, then the program (P ) has a unique optimal solution. Such uniqueness of the optimal solution was established for minimum trace factor analysis and some of its extensions (see [27]) Because of the complementarity condition (2.9) we have that rank G(x 0 ) rank Omega n: 2.12) We say that the strict complementarity condition holds if rank G(x 0 ) rank Omega = n: 2.13) Consider now the barrier function j(x) Gamma log det( GammaG(x) if G(x) OE 0; 1; ....

A. Shapiro, "Extremal problems on the set of nonnegative definite matrices", Linear Algebra and Its Applications, 67 (1985), pp. 7-18.

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A. Shapiro. Extremal problems on the set of nonnegative definite matrices. Linear Algebra and Appl., 67:7--18, 1985.

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A. SHAPIRO. Extremal problems on the set of nonnegative definite matrices. Linear Algebra Appl., 67:7--18, 1985.

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A. Shapiro (1985), "Extremal problems on the set of nonnegative definite matrices", Linear Algebra and its Applications 67 pp. 7-18.

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Shapiro, A. (1985). Extremal problems on the set of nonnegative definite matrices. Linear Algebra Appl. 67 7-18

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