J.M. BORWEIN and H. WOLKOWICZ. Characterization of optimality for the abstract convex program with nite-dimensional range. J. Austral. Math. Soc. Ser. A, 30(4):390{ 411, 1980/81.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the original Hoffman lemma applies. In the rest of this section we assume that K is a general convex cone. In addition to this we assume that the Slater condition is satisfied, i.e. Assumption 2 (b L) int K 6= The following lemma is well known; see e.g. Duffin [5] Borwein and Wolkowicz [2], Luo, Sturm and Zhang [11] Nesterov and Nemirovskii [14] and Sturm [21] For completeness we provide here a short proof. Lemma 2.2 Suppose that Assumption 2 holds. Then, for any y 2 L K with y 6= 0 it must follow that b T y 0. 4 Proof. Suppose, for the sake of deriving a ....

J.M. Borwein and H. Wolkowicz, Characterizations of optimality for the abstract convex program with finite dimensional range, Journal of the Australian Mathematical Analysis and Applications 83 (1981) 495-530.

....condition characterizing optimality for the convex semidefinite programming problems (SDP ) holds without a constraint qualification. Various characterizations of optimality without constraint qualifications have been given in the literature for the standard convex programming problems (see [3, 4, 7, 12, 13, 15]) Borwein and Wolkowicz [3] presented necessary and su#cient (nonasymptotic) conditions in terms of a modified Lagrangian function and the so called minimal cones. This approach was used recently in [19] to obtain strong duality for semidefinite programming problems. Our approach is to derive an ....

Borwein, J. M. and Wolkowicz, H. (1981), Characterizations of optimality for the abstract convex program with finite dimensional range, J. Austr. Math. Soc. Ser. A, 30, 390-411.

....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] and for alternative extensions refer ....

....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] and for alternative extensions refer to [BW81b, BW81a]. It is worth mentioning that [AN87] study the duality theory from the point of view of basic feasible solutions and extend the tableau based proofs of LP duality. The latest version of Nesterov and Nemirovskii s text [NN92] also treats cone duality for the general convex programs. Papers of ....

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J. Borwein and H. Wolkowicz. Characterization of optimality for the abstract convex program with finite dimensional range. J. Austral. Math. Soc. series A, 30:390--411, 1981.

....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 alternative extensions refer to [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 alternative extensions refer to [11, 12]. It is worth mentioning that Anderson and Nash in [3] study the duality theory from the point of view of basic feasible solutions 6 F. Alizadeh and extend the tableau based proofs of LP duality. The latest version of Nesterov and Nemirovskii s text [50] also treats cone duality for the ....

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J. M. Borwein and H. Wolkowicz, Characterization of optimality for the abstract convex program with finite dimensional range, J. Austral. Math. Soc. Ser. A, 30 (1981), pp. 390-- 411.

....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 denoted ....

J. Borwein, H. Wolkowicz, Characterization of Optimality for the Abstract Convex Program with Finite Dimensional Range, J. Austral. Math. Soc., Series A, 30(1981), pp. 390-411.

.... Slater s constraint qualification, are generic, see [8] But there are classes of problems where strong duality fails, e.g. relaxations that arise from hard combinatorial problems, e.g. 13] One can regularize semidefinite programs and guarantee that Slater s constraint qualification holds, e.g. [3, 12, 2]. This involves finding the minimal face of the semidefinite cone that contains the feasible set, i.e. the so called minimal cone. A numerical procedure for regularization is presented in [3] However, this process is not computationally tractable. An equivalent approach is the extended ....

....is still an open question, see e.g. 7] Extensions. The SDP s that we consider have all contained linear objectives and constraints. There is no reason to restrict SDPs to this special case. Duality for general cone programs with possible nonlinear objectives and constraints is considered in [3, 2, 11]. Applications for quadratic objectives SDP appear in e.g. 6, 1] ....

BORWEIN, J.M., and WOLKOWICZ, H.: `Characterizations of optimality for the abstract convex program with finite dimensional range', J. Austra. Math. Soc. Series A 30 (1981), 390--411.

....dual optimal solution set follows from Theorem 5. 2 For the case that K is closed and solid, the strong duality theorem with Slater condition is well known; see for example [1, 15, 31] among others. The result of Theorem 7, which holds for general convex cones, is due to Borwein and Wolkowicz [9]. Theorem 7 implies the following well known fact: if K is closed and o FP Theta o FD 6= then F P Theta F D 6= and (x ) T s = c T x b T s = 0; for all (x ; s ) 2 F P Theta F D : For the above case, we say that a conic convex programming problem ....

.... conic convex program CP(b; c; A;K) however, we may try to replace K by a lower dimensional face, say face (K; s) for a certain s 2 K , such that (b A) face (K; s) b A) K; b A) rel face (K; s) 6= If we succeed in finding such a face (which is then known as the minimal cone [10, 9, 46]) then we can regularize CP(b; c; A; K) to CP(b; c; A; face(K; s) which satisfies the generalized Slater condition. Such a regularization approach, which we call primal regularization, was proposed by Borwein and Wolkowicz [10, 9] and Wolkowicz [46] In this section, we propose a dual ....

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J.M. Borwein and H. Wolkowicz. Characterizations of optimality for the abstract convex program with finite dimensional range. Journal of the Australian Mathematical Society, 30:390-- 411, 1981.

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J.M. BORWEIN and H. WOLKOWICZ. Characterization of optimality for the abstract convex program with nite-dimensional range. J. Austral. Math. Soc. Ser. A, 30(4):390{ 411, 1980/81.

No context found.

J.M. BORWEIN and H. WOLKOWICZ. Characterization of optimality for the abstract convex program with finite-dimensional range. J. Austral. Math. Soc. Ser. A, 30(4):390-- 411, 1980/81.

No context found.

J.M. BORWEIN and H. WOLKOWICZ. Characterization of optimality for the abstract convex program with finite-dimensional range. J. Austral. Math. Soc. Ser. A, 30(4):390--411, 1980/81.

....14:36; p.5 6 Henry Wolkowicz a feasible point in the interior of P : Difficulties in the duality theory can arise due to the following property of the faces, 83] P F is always closed; P span(F) is never closed. REMARK 1.3. Modified optimality conditions without any CQ were given in [16] and also in [85, 83] Now, consider the typical primal SDP (P ) min C ffl X s.t. AX = b X 0; where C 2 S n ; b 2 IR m ; and A : S n : IR m ; is a linear operator, i.e. the components are defined as (AX) i = Trace A i X = b i ; for some given symmetric matrices A i : We can ....

J.M. BORWEIN and H. WOLKOWICZ. Characterization of optimality for the abstract convex program with finite-dimensional range. J. Austral. Math. Soc. Ser. A, 30(4):390--411, 1980/81.

....for the primal (dual, respectively) then there is a zero duality gap, and the dual (primal, respectively) is attained. If the Slater constraint qualification fails, then one can work with the minimal cones and obtain characterizations of optimality and strong duality theorems, see [9]. For (PSDP) this is the minimal face of P containing the feasible set. For (DSDP) this is the minimal face of P which contains C Gamma A F ; where F denotes the feasible set. One can also obtain a strong dual program whose size is polynomial in the data of the original problem [53] The ....

J.M. BORWEIN and H. WOLKOWICZ. Characterizations of optimality for the abstract convex program with finite dimensional range. J. Austra. Math. Soc. Series A, 30:390--411, 1981.

....question is the formulation of duals that close the duality gap. Infinite dimensional linear programming is also studied in the books by Glasho# and Gustafson [18] and Anderson and Nash [2] More recently, duals that guarantee strong duality for general abstract convex programs have been given in [13, 12, 11, 10]. The special case of a linear program with cone constraints is treated in [38] 1.3. Outline. This paper is motivated by the recent paper of Ramana [33] A dual program, called an extended Lagrange Slater dual program and denoted (ELSD) is presented therein. Strong duality holds for this dual ....

....by the recent paper of Ramana [33] A dual program, called an extended Lagrange Slater dual program and denoted (ELSD) is presented therein. Strong duality holds for this dual and, in addition, it can be written down in polynomial time. Previous work on general (convex) cone constrained programs [13, 38, 11, 10] also presented dual programs for which strong duality holds. The results were based on regularization and on finding the so called minimal cone of the program (P) We denote these duals by (DRP) A procedure for defining the STRONG DUALITY FOR SEMIDEFINITE PROGRAMMING 643 minimal cone was ....

<F4.08e+05> J. M. Borwein and H.<F4.039e+05> Wolkowicz,<F4.112e+05> Characterizations of optimality for the abstract convex program with finite dimensional<F4.039e+05> range, J. Austral. Math. Soc. Ser. A, 30 (1981), pp. 390-- 411.

.... 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 literature in many places, specifically statistics (covariance matrices, clustering) 51] and control theory (Lyapunov stability, Ricatti equations) e.g. 51, 67, 66] where interior point methods such as ....

.... 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 following form. P) p : ....

[Article contains additional citation context not shown here]

J.M. BORWEIN and H. WOLKOWICZ. Characterizations of optimality for the abstract convex program with finite dimensional range. J. Austra. Math. Soc. Series A, 30:390--411, 1981.

....Note that the results in this paper still hold in the more general setting when X is a vector in a given vector space V and A is a linear operator from V onto the space of m Theta m matrices. Therefore these problems fall into the class of programs with finite dimensional range studied in [5]. Our main result is the explicit representation of the general solution of (IVP) presented as Theorem 2.2. Section 3 discusses two matlab programs that find these explicit solutions. 2 THEORETICAL RESULTS We let A Gamma denote a generalized inverse of A, i.e. a linear operator that ....

J.M. BORWEIN and H. WOLKOWICZ. Characterizations of optimality for the abstract convex program with finite dimensional range. J. Austra. Math. Soc. Series A, 30:390--411, 1981.

....question is the formulation of duals that close the duality gap. Infinite dimensional linear programming is also studied in the books by Glashoff and Gustafson [18] and Anderson and Nash [2] More recently, duals that guarantee strong duality for general abstract convex programs have been given in [13, 12, 11, 10]. The special case of a linear program with cone constraints is treated in [38] 1.3 Outline This note is motivated by the recent paper of Ramana [34] Therein a dual program, called an extended Lagrange Slater dual program and denoted ELSD, is presented for which strong duality holds and more ....

....the recent paper of Ramana [34] Therein a dual program, called an extended Lagrange Slater dual program and denoted ELSD, is presented for which strong duality holds and more importantly it can be written down in polynomial time. Previous work on general (convex) cone constrained programs, see [13, 38, 11, 10], presented dual programs for which strong duality holds and was based on regularization and on finding the so called minimal cone of P : We denote this dual by DRP. A procedure for defining the minimal cone was presented in [11] This procedure started with an initial feasible point and reduced ....

J.M. BORWEIN and H. WOLKOWICZ. Characterizations of optimality for the abstract convex program with finite dimensional range. J. Austra. Math. Soc. Series A, 30:390--411, 1981.

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