R.D.C. Monteiro and P. Zanjacomo. General Interior-Point Maps and Existence of Weighted Paths for Nonlinear Semide nite Complementarity Problems. Working paper, School of ISyE, Georgia Tech, USA, April 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....K ffi Theta K ]ffi V (x; s) 7 Gamma x 1 s 1 : x n s n Delta T : We will now generalize this notion to self scaled conic programming and prove bounds on kfl(x; s)k in terms of the duality gap Omega x; s ff (see Theorem 4.1. 3) Independently from our work Monteiro and Zanjacomo [29] found transformations with similar properties to our target maps for the case of semidefinite programming. In the present section we will show that optimal solutions Gamma x ; s Delta to the optimization problems (1.2) correspond to fl(x; s) 0. This suggests the intuitive ....

.... et al. 24] and Jansen et al. 19] For a complete treatment see also [36] Considerable interest has arisen recently in generalizing weighted centers to semidefinite programming, and several competing approaches have been developed by Monteiro Pang [31] Sturm Zhang [39] Monteiro Zanjacomo [29], Burer Monteiro [3] and Burer Monteiro Zhang [4, 5] In this section we present a generalization of 155 weighted centers to the more general framework of self scaled conic programming. Lemma 4.3.1. Let (X; S) be a congruent square root field for the self scaled barrier functional F 2 C 3 ....

R.D.C. Monteiro and P. Zanjacomo. General Interior-Point Maps and Existence of Weighted Paths for Nonlinear Semidefinite Complementarity Problems. Working paper, School of ISyE, Georgia Tech, USA, April 1998.

....V space approach has been developed by Kojima Mizuno Yoshise [10] Monteiro Adler [16] Mizuno [12] Jansen Roos Terlaky Vial [7, 8] Todd [25] and others. Various generalization of V space have been developed in the context of semidefinite programming by Sturm Zhang [23, 24] Monteiro Zanjacomo [15] and Burer Monteiro [1] in the more general context of self scaled conic programming by Tuncel [28] and a related technique has been developed by Tuncel [29] for general convex conic programming problems. The notion of square root fields we present in this article is most closely related to ....

R.D.C. Monteiro and P. Zanjacomo. General Interior-Point Maps and Existence of Weighted Paths for Nonlinear Semidefinite Complementarity Problems. Working paper, School of ISyE, Georgia Tech, USA, April 1998.

.... of primal dual interior point methods for linear programming (see e.g. Kojima Mizuno Yoshise [12] Monteiro Adler [18] Mizuno [14] Jansen Roos Terlaky Vial [9, 10] Todd [27] Related generalizations to semidefinite programming have been analyzed by Sturm Zhang [25, 26] Monteiro Zanjacomo [17] and Burer Monteiro [1] Primal dual interior point methods for convex optimization problems are designed to solve a problem and its dual jointly by making use of convex duality theory. The paradigm for such algorithms usually is to reduce the duality gap between primal and dual approximate ....

.... Kojima et al. 13] and Jansen et al. 9] For a complete treatment see also [23] Considerable interest has arisen recently to generalize weighted centers to semidefinite programming, and several competing approaches have been developed by Monteiro Pang [19] Sturm Zhang [26] and MonteiroZanjacomo [17]. In the present section we present yet another generalization of weighted centers to SDP that extends to the more general framework of self scaled conic programming. Self scaled conic programming is extensively discussed in the two seminal papers [21, 22] of Nesterov Todd. See also Guler [5] and ....

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R.D.C. Monteiro and P. Zanjacomo. General Interior-Point Maps and Existence of Weighted Paths for Nonlinear Semidefinite Complementarity Problems. Working paper, School of ISyE, Georgia Tech, USA, April 1998.

....primal dual interior point methods. see e.g. Kojima Mizuno Yoshise [18] Monteiro Adler [25] Mizuno [21] JansenRoos Terlaky Vial [15, 16] For various generalizations of some aspects of this theory and for other related material see also Monteiro Pang [26] Sturm Zhang [33] MonteiroZanjacomo [24], Tuncel [37, 38] Todd [34] and Burer Monteiro [1] target directions are thus a promising family of search directions for use in a unifying theory of primal dual interior point methods for a large class of convex optimization problems including linear, semidefinite and second order cone ....

R.D.C. Monteiro and P. Zanjacomo. General Interior-Point Maps and Existence of Weighted Paths for Nonlinear Semidefinite Complementarity Problems. Working paper, School of ISyE, Georgia Tech, USA, April 1998.

....centers to the framework of self scaled conic programming. Weighted centers were introduced to the linear programming literature by Kojima, Mizuno and Yoshise [11] and have been generalized to the framework of semidefinite programming by Monteiro Pang [18] Sturm Zhang [25] and Monteiro Zanjacomo [16]. Let E be a n dimensional real vector space and E ] its dual. Throughout this article we will reserve super or subscript ] for duals, superscript for adjoints and superscript ffi for topological interiors. Let K ae E be a regular cone in E, i.e. convex, pointed and with non empty interior ....

R.D.C. Monteiro and P. Zanjacomo. General Interior-Point Maps and Existence of Weighted Paths for Nonlinear Semidefinite Complementarity Problems. Working paper, School of ISyE, Georgia Tech, USA, April 1998.

....rederive the polynomial convergence of these two algorithms in a unified way; more specifically, we obtain the same iteration complexity of O(n p L) for the first map and a slightly worse iteration complexity of O(n 2 L) for the second map. The third and fourth maps have been studied in [15] [16] and [23] yet no polynomial convergence analysis of long step algorithms based on these maps have been established. We show that the third and fourth maps also fit nicely into our general framework and hence obtain for the first time polynomially convergent feasible long step algorithms for these ....

....based on the other two maps, namely the L T x SL x map and the V 2 map, where L x j chol (X) and V j W 1=2 XW 1=2 = W Gamma1=2 SW Gamma1=2 with W being the unique symmetric matrix such that S = WXW , are studied here for the first time. The two last maps have been introduced in [15] [16] and [23] and were studied there from different points of view. We now state a technical lemma which will be used in the upcoming subsections. Lemma 4.1 Let (X; S) 2 C, G j G (X;S) and R j R (X;S) Then, for all A 2 S n , kG(A)k 2 F k( Phi s ) Gamma1 (A)k 2 F kR 2 k ; kG ....

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R. D. C. Monteiro and P. R. Zanj'acomo. General interior-point maps and existence of weighted paths for nonlinear semidefinite complementarity problems. Technical Report, School of ISyE, Georgia Institute of Technology, Atlanta, GA 30332, USA, April 1998. To appear in Mathematics of Operations Research.

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R.D.C. Monteiro and P. Zanjacomo. General Interior-Point Maps and Existence of Weighted Paths for Nonlinear Semide nite Complementarity Problems. Working paper, School of ISyE, Georgia Tech, USA, April 1998.

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