R. J. Duffin. Infinite Programs. In H. W. Kuhn and A. W. Tucker, editor, Linear inequalities and Related Systems, pages 157--170. Princeton University Press, Princeton, NJ, 1956.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....matrices induce the component wise and the Lowner partial orders, respectively. The duality theory in 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 [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 ....

R. J. Duffin. Infinite Programs. In H. W. Kuhn and A. W. Tucker, editor, Linear inequalities and Related Systems, pages 157--170. Princeton University Press, Princeton, NJ, 1956.

....theorem povides a dual characterization of closed convex cones, viz. K = fx 2 n j x T y 0 8y 2 K g: Similarly, the interior of a solid convex cone is characterized by int K = fx 2 n j x T y 0 80 6= y 2 K g: 5) 3 Self Duality Self duality has been defined by Duffin [2] for conic convex programs that are formulated in the so called symmetric form. More recently, Ye, Todd and Mizuno [30] formulated a linear program in a different form, and argued that their program is self dual since the dual of the problem is equivalent to the primal . Below, we propose a ....

R.J. Duffin. Infinite programs. In H.W. Kuhn and A.W. Tucker, editors, Linear Inequalities and Related Systems, pages 157--170. Princeton University Press, Princeton, NJ, 1956.

....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, ....

....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, contained in [48, 39, 60] In [40] a self dual formulation is investigated. 3.1 Deriving the Dual Problem We formulate a pair of dual problems (P) and (D) as follows. Let S and T be a pair of closed convex cones. A set S is called a cone if s 2 S implies ....

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R.J. DUFFIN. Infinite Programs, in: Linear inequalities and related systems, (H.W. Kuhn and A.W. Tucker eds.), Annals of Mathematical Studies 38, Princeton University Press, 157--170, 1956.

.... in several different areas, e.g. positive definite completion problems, maximum entropy estimation, and bounds for hard combinatorial problems, see e.g. the survey of Vandenberghe and Boyd [42] Though SDP has been studied in the past, as part of the more general cone programming problem, see e.g. [11, 43], there has been a renewed interest due to the successful applications to discrete optimization [27, 14] and to systems theory [5] In addition, the relaxations are equivalent to the reformulation and linearization technique, see e.g. the survey discussion in [40] which provides further evidence ....

R.J. DUFFIN. Infinite programs. In A.W. Tucker, editor, Linear Equalities and Related Systems, pages 157--170. Princeton University Press, Princeton, NJ, 1956.

....C Y induces an ordering on Y: In this introductory section we assume CX and C Y are closed. Given A 2 L(X; Y ) b 2 Y; c 2 X we define the LP instance d : A; b; c ) by sup x2X c x s:t: Ax b x 0: Many researchers have studied linear programming in this generality (cf. 1] [2], 3] 5] Although linear programming from this vantage point is generally referred to by names such as infinite linear programming we prefer the phrase analytic linear programming because of close connections to functional analysis. Although we use the symbol the reader should note that ....

....b 2 C(A) i.e. if it can be made consistent by an arbitrarily slight perturbation of b. The following proposition relies only on the local convexity of the normed space Y . In the case of finite dimensional polyhedral linear programming the proposition is Farkas lemma. Proposition 2. 1 (Duffin [2]) Assume A 2 L(X; Y ) b 2 Y . Consider the following two systems: Ax b x 0 y A 0 y 0 y b 0 The first system is asymptotically consistent if and only if the second is inconsistent. Proof. Let C(A) denote the set of all functionals y 2 Y satisfying y b 0 for ....

R.J. Duffin, "Infinite programs," in Linear Equalities and Related Systems, H.W. Kuhn and A.W. Tucker, eds., Princeton University Press, Princeton, 1956, 157-170.

....matrices induce the component wise and the Lowner partial orders, respectively. The duality theory in 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 ....

R. J. Duffin, Infinite Programs, in Linear inequalities and Related Systems, H. W. Kuhn and A. W. Tucker, eds., Princeton University Press, Princeton, NJ, 1956, pp. 157--170.

....the most important set of matrices and has applications in many diverse areas. The study of the geometry of the cone of semidefinite 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 ....

....in vol II (1811 12) page 88 93) In fact, Von Neumann had a duality for LP back in the 1940s, which was based on using game theory, 68] Later Gale, Kuhn, Tucker [25] developed a duality theory for general problems. Duality theory for cone programs in abstract spaces followed as well, see e.g. [52, 32, 16, 17] and the references therein. Optimality conditions for nonlinear programs date back to Fritz John [34] Other related references and extensions to cone problems in abstract spaces include [41, 58] The history of constraint qualifications is discussed in [40] The essential equivalence between ....

R.J. DUFFIN. Infinite programs. In A.W. Tucker, editor, Linear Equalities and Related Systems, pages 157--170. Princeton University Press, Princeton, NJ, 1956.

....proving the claim that p is indeed the infimum. Similarly, infeasibility of (P) can be established by using a Farkas type dual solution. To a large extent, duality results for linear programming can be generalized to the setting of conic convex programming, as was pointed out by Duffin [15]. However, certificates in the context of conic convex programming, such as those proposed in [15] can be infinitely long. More recently, Borwein and Wolkowicz [10] proposed a regularization scheme which results in certificates of finite length. We will see however, that checking the feasibility ....

....established by using a Farkas type dual solution. To a large extent, duality results for linear programming can be generalized to the setting of conic convex programming, as was pointed out by Duffin [15] However, certificates in the context of conic convex programming, such as those proposed in [15], can be infinitely long. More recently, Borwein and Wolkowicz [10] proposed a regularization scheme which results in certificates of finite length. We will see however, that checking the feasibility (correctness) of regularized certificates can be a nontrivial task. Fortunately, the structure of ....

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R.J. Duffin. Infinite programs. In H.W. Kuhn and A.W. Tucker, editors, Linear Inequalities and Related Systems, pages 157--170. Princeton University Press, Princeton, NJ, 1956.

....task requires a careful study of duality theory for conic convex programming in the absence of constraint qualifications. There have been several studies of duality theory for conic convex programming, most of which assume a constraint qualification (in particular Slater s condition) see Duffin [3], Nesterov and Nemirovsky [11] and Alizadeh [1] Wolkowicz [20] and Ramana et al. 14] manage to dispense with any constraint qualification by working with a nonstandard primal dual pair. It is currently not known though, whether the interior point methods can cope with this nonstandard duality. ....

....We note the following relations: y 2 (MA) y T (Mx) 0; 8x 2 A ( M T y) T x = 0; 8x 2 A ( M T y 2 A ( y 2 M GammaT A : The proof is complete. 2 3 Relations of the Conic Primal Dual Pair Semidefinite and conic convex programming duality has been studied by Duffin [3], Wolkowicz [20] Ramana et al. 14] Nesterov and Nemirovsky [11] and Alizadeh [1] among others. A Slater condition or similar constraint qualification is usually assumed in order to arrive at duality results that show great similarity with the linear programming duality [3, 11, 1] However, in ....

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Duffin, R.J., "Infinite programs," in: H.W. Kuhn and A.W. Tucker, eds. Linear Inequalities and Related Systems (Princeton University Press, Princeton, NJ, 1956), 157-170.

....[33] 1.2.2 Early Duality Extensions of finite linear programming duality to infinite dimensions and or to optimization problems over cones has been studied in the literature. We do not give a comprehensive survey, since that would probably be impossible. But we mention several early results. In [16], Duffin studies infinite linear programs, i.e. programs for which there are an infinite number of constraints and or an infinite number of variables. Also studied is the notion of optimization with respect to a partial order induced by a cone. Infinite linear programming is closely related to the ....

R.J. DUFFIN. Infinite programs. In A.W. Tucker, editor, Linear Equalities and Related Systems, pages 157--170. Princeton University Press, Princeton, NJ, 1956.

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R.J. Duffin, Infinite programs. In A.W. Tucker, editor, Linear Equalities and Related Systems, pages 157--170. Princeton University Press, Princeton, NJ, 1956.

No context found.

R.J. Duffin, "Infinite programs," in: H.W. Kuhn and A.W. Tucker, eds., Linear Inequalities and Related Systems, Princeton University Press, Princeton, 1956, 157-170.

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