B. Craven and B. Mond. Linear programming with matrix variables. Linear Algebra and Appl., 38:73--80, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the problem reduces to minimizing the maximum eigenvalue 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 ....

B. Craven and B. Mond. Linear programming with matrix variables. Linear Algebra and Appl., 38:73--80, 1981.

....SDP papers indeed indicates an explosion of research effort, starting around 1991. A closer look reveals that interest in this class of problems is somewhat older, and dates back to the 1960 s (see e.g. 6] A paper on SDP from 1981 is descriptively named Linear Programming with Matrix Variables [11], and this apt title may be the best way to introduce the problem. The goal is to minimize the inner product hC; Xi : Tr (CX) of two n Theta n symmetric matrices, a constant matrix C and a variable matrix X, subject to a set of constraints, where Tr denotes the trace (sum of diagonal ....

B. Craven and B. Mond. Linear programming with matrix variables. Linear Algebra Appl., 38:73--80, 1981.

.... other variants of Farkas lemma in a similar vain, all of which are extensions of lemmas for the component wise inequalities, as given for example in Schrijver s text [Sch86] Related extensions for infinite programs have been studied in [Hur58] and [CK77] and in the case of matrix variables in [CM81]. In all of these extensions we need to assume either some closedness criteria, or the dual problem must be modified by cones other than P (as in [Wol81] for instance. We mention a few more: Lemma 4 Let A 2 n 2 Thetam be a matrix whose columns are linearly independent and are of the form ....

B. D. Craven and B. Mond. Linear Programming with Matrix Variables. Linear Algebra and its Applications, 38:73--80, 1981.

....example of 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, ....

B. Craven and B. Mond. Linear programming with matrix variables. Linear Algebra and Appl., 38:73--80, 1981.

.... 100] and continued into the 1980 s (see e.g. the historical outline in [18] In addition, SDP is a special case of optimization over cone constraints (generalized linear programming) which dates back more than 30 years to e.g. Bellman and Fan [10] and was an ongoing active area of research, e.g. [11, 33, 21, 22, 47, 108, 69, 17]. The last ten years has seen an enormous interest in the SDP area, due to many new and important applications in, e.g. combinatorial optimization, engineering (systems and control) statistics, etc. This interest increased greatly due to the fact that SDP problems can be solved efficiently (are ....

B. CRAVEN and B. MOND. Linear programming with matrix variables. Linear Algebra Appl., 38:73--80, 1981.

....makes it possible to convert any SDP and any monotone SDLCP in a subalgebra of M n ( 0 C; IR) or M n (IH; IR) into some SDP and some monotone SDLCP in a subalgebra T of M 2n (IR) or M 4n (IR) respectively. The homomorphism ae from M n (IK; IR) into M dn (IR) was utilized in the paper [3] where duality of general linear programs with real, complex and quaternion matrix variables was discussed. In Section 5.2, we present a classification of subalgebras of M n (IR) The main results are roughly summarized as follows: ffl There are exactly three types of irreducible ....

B. D. Craven and B. Mond, "Linear programming with matrix variables," Linear Algebra and Its Applications 38 (1981) 73--80.

.... several other variants of Farkas lemma in a similar vein, all of which are extensions of lemmas for the component wise inequalities, as given for example in Schrijver s text [60] Related extensions for infinite programs have been studied in [34] and [13] and in the case of matrix variables in [14]. In all 1 Alternative extensions without closedness assumption are treated in [11, 12, 64] 8 F. Alizadeh of these extensions we need to assume either some closedness criteria, or the lemma must be modified by using cones other than P (as in [64] for instance. We mention a few more: Lemma ....

B. D. Craven and B. Mond, Linear Programming with Matrix Variables, Linear Algebra Appl., 38 (1981), pp. 73--80.

.... matrices predates LP, e.g. by Bohnenblust in 1948, 9] and further in [4] In fact, SDP is a special case of cone programming, studied by e.g. Duffin in 1956, 17] The more general topic of cone programming was a major area of study both in finite dimensional and abstract spaces e.g. [20, 8, 13, 14]. Observations about LP like duality using a Slater like constraint qualification appear in 1963 by Bellman and Fan, 6] while a strong duality without a constraint qualification is developed in [10, 72] with a regularization algorithm in [11] Semidefinite programming itself appeared in the ....

B. CRAVEN and B. MOND. Linear programming with matrix variables. Linear Algebra and Its Applications, 38:73--80, 1981.

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B. Craven and B. Mond. Linear programming with matrix variables. Linear Algebra and Appl., 38:73--80, 1981.

No context found.

B. CRAVEN and B. MOND. Linear programming with matrix variables. Linear Algebra Appl., 38:73--80, 1981.

No context found.

B. D. Craven and B. Mond (1981), "Linear programming with matrix variables ", Linear Algebra and its Applications 38, pp. 73-80.

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