A. Ben-Israel, A. Charnes, and K.O. Kortanek. Duality and asymptotic solvability over cones. Bulletin of the AMS, 75:318-324, 1969.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....2 C ; where C X is a closed convex cone in the ( nite) n dimensional linear vector space X, and b lies in the ( nite) m dimensional vector space Y . This format for convex optimization dates back at least to Dun [2] Strong duality results can be found in [2] as well as in Ben Israel et al. [1]. For 0 and 0, we de ne the and level sets for the primal and dual problems as follows: P : n x j Ax = b; x 2 C; c T x z o and D : n s j A T y s = c; s 2 C for some y 2 Y satisfying b T y v o : We make the following assumption: Assumption A: ....

A. Ben-Israel, A. Charnes, and K.O. Kortanek. Duality and asymptotic solvability over cones. Bulletin of the AMS, 75:318-324, 1969.

....programming can be extended to generalized linear programming problems where K replaces in the primal problem and K replaces in the dual problem. Duffin in [Duf56] was the first one to study such generalized duality theories. Later Hurwicz [Hur58] Ben Israel, Charnes and Kortanek [BICK69], Borwein and Wolkowicz [BW81b, BW81a] and Wolkowicz [Wol81] among others developed more general formulations of the duality theory. For a comprehensive treatment of generalized duality theory from the point of view of infinite dimensional linear programs, see the text of Anderson and Nash [AN87] ....

A. Ben-Israel, A. Charnes, and K. O. Kortanek. Duality and Asymptotic Solvability over Cones. Bulletin of American Mathematical Society, 75(2):318--324, 1969.

....cones In this section we describe the theoretical setting for semidefinite programs. We recall duality of linear programs over closed convex cones in detail, and recall duality results for semidefinite programs. The results of this section can be found in papers from the early sixties, such as [14, 6]. We review here the main concepts without proofs, and formulate the results in a setting which is general enough for the present purposes, but avoids deeper results from Hilbert space theory and functional analysis. The presentation here follows mostly [14] We also mention similar approaches, ....

A. BEN-ISRAEL, A. CHARNES, and K.O. KORTANEK. Duality and asymptotic solvability over cones. Bulletin of the AMS 75, 318--324, 1969.

.... 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 ....

A. BEN-ISRAEL, A. CHARNES, and K. KORTANEK. Duality and asymptotic solvability over cones. Bulletin of American Mathematical Society, 75(2):318--324, 1969.

....linear programming can be extended to generalized linear programming problems where K replaces in the primal problem and K replaces in the dual problem. Duffin in [19] was the first one to study such generalized duality theories. Later Hurwicz [34] Ben Israel, Charnes and Kortanek [9], Borwein and Wolkowicz [11, 12] and Wolkowicz [64] among others developed alternative formulations of the duality theory. For a comprehensive treatment of generalized duality theory from the point of view of infinite dimensional linear programs, see the text of Anderson and Nash [3] and for ....

A. Ben-Israel, A. Charnes, and K. O. Kortanek, Duality and Asymptotic Solvability over Cones, Bull. Amer. Math. Soc., 75 (1969), pp. 318--324.

....to polyhedra. Indeed, spectrahedra may be considered next natural successors to polyhedra, as one moves beyond linear constraints in optimization theory. 1. 1 Background Historically, semidefinite programming has been studied in more general contexts such as convex and cone programming (see [BCK69], BW81] CDW75] and [Wol81] See also [Fle85] and [Ove92] Further references can be found in [Ali94] However, the more recent surge of interest in SDP was primarily inspired by the work of [GLS84] see [GLS88] Chapter 9) In this work, the authors associate with every graph G, a convex set ....

A. Ben-Israel, A. Charnes and K. Kortanek, Duality and Asymptotic Solvability over Cones, Bull. of AMS, 75(1969), pp. 318-324.

.... 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 ....

.... in [40] The essential equivalence between constraint qualifications and stability have been extensively studied, e.g. 59, 60, 58, 19] The classical M F CQ is introduced in [47] Characterizations of optimality without constraint qualifications and discussions about zero duality gaps appear in [62, 63, 8, 38, 31, 39] and [7, 12, 11, 10, 72] for the convex and convex cone case. 3 Strong Duality in SDP We now consider duality for the special cone programming problem SDP. We follow [10, 72, 55, 53] For simplicity we consider linear objective and constraint functions; and, we consider the primal SDP in the ....

A. BEN-ISRAEL, A. CHARNES, and K. KORTANEK. Duality and asymptotic solvability over cones. Bulletin of American Mathematical Society, 75(2):318--324, 1969.

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