P. TSENG. Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP. Optim. Methods Softw., 9:245--268, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... Kojima, Shida and Shindoh [15] Kojima, Shindoh and Hara [16] Lin and Saigal [18] Lou, Sturm and Zhang [19] Monteiro [20, 21] Monteiro and Tsuchiya [23, 24] Monteiro and Zhang [25] Nesterov and Nemirovskii [28] Nesterov and Todd [31, 30] Potra and Sheng [32] Sturm and Zhang [33] Tseng [35], Vandenberghe and Boyd [36] and Zhang [38] Most of these more recent works are concentrated on primal dual methods. One of the main goals of this paper is the implementation of primal dual path following and predictor corrector algorithms based on two pure Newton directions proposed in Monteiro ....

P. Tseng. Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP. manuscript, Department of Mathematics, University of Washington, Seattle, Washington, 98195, USA, June 1996.

.... and Wolkowicz [4] Jarre [5] Kojima, Shida and Shindoh [8] Kojima, Shindoh and Hara [10] Lin and Saigal [11] Luo, Sturm and Zhang [12] Monteiro [14, 15] Monteiro and Zhang [18] Nesterov and Nemirovskii [21] Nesterov and Todd [24, 23] Potra and Sheng [25] Sturm and Zhang [26] Tseng [28], Vandenberghe and Boyd [29] and Zhang [31] Most of these more recent works are concentrated on primal dual methods. The first algorithms for SDP and SDLCP that are extensions of primal dual algorithms for LP, such as the long step path following algorithm of Kojima, Mizuno and Yoshise [7] the ....

P. Tseng. Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP. manuscript, Department of Mathematics, University of Washington, Seattle, Washington, 98195, USA, June 1996.

....the same form as linear programming. Therefore, most interior point methods for linear programming can be extended to SDP. Indeed, most published works on SDP are about interior point methods, for example see [1] 2] 3] 8] 11] 12] The SDP problem (1.1) 1. 3) is also called as SDPLP by [9]. Recently, there are some studies on extensions of the SDP. One extension is the semi definite linear complementarity problems (SDPLCP) For more details on SDPLCP, please see [6] 7] and [9] In this paper we consider a new extension of the SDP problems. It is well known that adding a quadratic ....

....for example see [1] 2] 3] 8] 11] 12] The SDP problem (1.1) 1.3) is also called as SDPLP by [9] Recently, there are some studies on extensions of the SDP. One extension is the semi definite linear complementarity problems (SDPLCP) For more details on SDPLCP, please see [6] 7] and [9]. In this paper we consider a new extension of the SDP problems. It is well known that adding a quadratic term in the objective of a linear programming problem gives a quadratic programming problem. This work is partially supported by Chinese NNSF grants 19731010 It is natural for us to ....

P. Tseng, Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP, Optimization Methods and Software, 9(1998) 245-268.

....time. In the past several years, a major part of the research into SDP has focused on both the theoretical and practical solution of SDP problems using extensions of interior point methods for LP. Many authors have proposed interior point algorithms for solving SDP problems (see for example [1, 2, 4, 6, 7, 8, 9, 10, 11, 13, 14, 17, 18, 19, 20, 21, 22, 24, 25, 26]) Many of the recent works on interior point algorithms for SDP are concentrated on primal dual methods. Feasible primal dual path following algorithms for SDP simultaneously solve the primal and dual SDP problems by maintaining primal feasibility in X and dual feasibility in (S; y) while ....

....however, results in an equation of the form X DeltaS DeltaX S = I Gamma XS; 1) which in general yields nonsymmetric directions. Many authors have investigated alternate yet equivalent equations of the central path for which Newton s method does yield symmetric directions (see for example [2, 4, 7, 10, 11, 13, 14, 17, 20, 24]) The work of these authors was based on research supported by the National Science Foundation under grants INT 9600343, CCR 9700448 and CCR 9902010. y School of Mathematics, Georgia Tech, Atlanta, Georgia 30332, USA. email: burer math.gatech.edu) z School of ISyE, Georgia Tech, Atlanta, ....

P. Tseng. Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP. Optimization Methods and Software, 9:245--268, 1998.

.... programming, interior point methods, path following methods, predictor corrector methods, higher order methods, Newton directions, central path, numerical implementation 1 Introduction Many authors have proposed interior point algorithms for solving semidefinite programming (SDP) problems (see [1, 2, 5, 9, 12, 14, 15, 16, 17, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 32, 33, 34]) Most of these more recent works are concentrated on primal dual methods. One of the main goals of this paper is the implementation of primal dual path following and predictor corrector algorithms based on two pure This work was partly supported by the NSF grants CCR 9700448, CCR 9902010 and ....

P. Tseng. Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP. Optimization Methods and Software, 9:245--268, 1998.

.... [5] Kojima, Shida and Shindoh [8] Kojima, Shindoh and Hara [11] Lin and Saigal [12] Luo, Sturm and Zhang [13] Monteiro [15, 16] Monteiro and Tsuchiya [19] Monteiro and Zhang [21] Nesterov and Nemirovskii [24] Nesterov and Todd [26, 27] Potra and Sheng [28] Sturm and Zhang [30] Tseng [35], Vandenberghe and Boyd [36] and Zhang [38] Most of these more recent works are concentrated on primal dual methods. The first algorithms for SDP which are extensions of well known primal dual LP algorithms, such as the long step path following algorithm of Kojima, Mizuno and Yoshise [7] and ....

P. Tseng, Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP, manuscript, Department of Mathematics, University of Washington, Seattle, Washington, 98195, USA, June 1996.

.... [8] Kojima, Shindoh and Hara [10] Lin and Saigal [11] Luo, Sturm and Zhang [12] Monteiro [14, 15] Monteiro and Zhang [20] Monteiro and Tsuchiya [18] Monteiro and Zanj acomo [19] Nesterov and Nemirovskii [23] Nesterov and Todd [26, 25] Potra and Sheng [27] Sturm and Zhang [28] Tseng [30], Vandenberghe and Boyd [31] and Zhang [33] Most of these more recent works are concentrated on primal dual methods. The first algorithms for SDP and SDLCP that are extensions of primal dual algorithms for LP, such as the long step path following algorithm of Kojima, Mizuno and Yoshise [7] the ....

....version of this paper, is based on a different representation of the central path that is directly related to the centrality measures used in standard path following algorithms. This family also contains the HRVW KSH M and NT directions (but not the AHO direction) Finally, we mention that Tseng [30] also considers a family of search directions parametrized by a single scalar parameter which contains the NT and HRVW KSH M directions. Unified convergence analyses for the MZ family have been given by Monteiro and Zhang [20] and Monteiro [15] In the paper [20] iteration complexity bounds are ....

P. Tseng. Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP. manuscript, Department of Mathematics, University of Washington, Seattle, Washington, 98195, USA, June 1996.

....matrices. These applications yield new interior point methods for solving these problems whose convergence can be established under some mild assumptions. It should be noted that many interior point methods for the linear version of these problems have been proposed in the literature (e.g. see [1, 2, 3, 4, 6, 8, 10, 11, 12, 15, 16, 19, 20, 22, 23, 24, 25, 26, 27, 29, 32, 34, 35, 37]) We explain some terminology and fix the notation used throughout the paper. For a given subset S of n , we let int S, cl S, and bd S denote, respectively, the interior, closure, and boundary of S. If the mapping H is (Fr echet) differentiable at a point x in its domain, the Jacobian ....

P. Tseng, Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP, manuscript, Department of Mathematics, University of Washington, Seattle, Washington, 98195, USA, June 1996.

.... and Wolkowicz [5] Jarre [7] Kojima, Shida and Shindoh [10, 12] Kojima, Shindoh and Hara [13] Lin and Saigal [14] Lou, Sturm and Zhang [15] Monteiro [17] Monteiro and Zhang [21] Nesterov and Nemirovskii [24] Nesterov and Todd [26, 27] Potra and Sheng [28] Sturm and Zhang [29] Tseng [31], Vandenberghe and Boyd [32] and Zhang [34] Most of these more recent works are concentrated on primal dual methods. The first SDP algorithms that are extensions of primal dual LP algorithms, such as the longstep path following algorithm of Kojima, Mizuno and Yoshise [9] the short step ....

P. Tseng, Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP, manuscript, Department of Mathematics, University of Washington, Seattle, Washington, 98195, USA, June 1996.

....that have been proposed in the literature, as well as some others that are suggested here. We are concerned with specific directions, rather than the families that have been introduced by Kojima, Shindoh, and Hara [13] Monteiro and Zhang [19, 38, 22, 20] Monteiro and Tsuchiya [21] and Tseng [34] (and the very general family, including all of these, introduced by Kojima, Shida, and Shindoh [15] However, many of the specific directions we consider do lie in these families; in particular, out of the twenty directions we address, six lie in the Monteiro Zhang and nine in the ....

....to our knowledge. It is motivated by trying to symmetrize (32) or a similar version of the equation defining the Dual direction, or alternatively symmetrizing the Newton system (for not necessarily symmetric matrices) coming from X S #I = 0. This direction was also discussed by Tseng [34]. It has the form of (7) with E = S 1 2 # S 1 2 , F = X 1 2 #X 1 2 , REF = #I 1 2 (X 1 2 SX 1 2 S 1 2 XS 1 2 ) Again there is an equivalent form with E the identity: E = I#I, F = S 1 2 X 1 2 #S 1 2 X 1 2 , REF = #S 1 1 2 (S 1 2 X 1 2 SX 1 2 S 1 2 ....

P. Tseng, Search directions and convergence analysis of some infeasible pathfollowing methods for the monotone semi-definite LCP, Technical Report, Department of Mathematics, University of Washington, Seattle, WA, 1996.

.... by Monteiro [13] via a scaled symmetrization operator, ffl the NT direction given by Nesterov Todd [18] and reformulated via the V space transformation by Sturm Zhang [21] ffl the KSH family of directions given by Kojima Shindoh Hara [9] ffl the Tseng family of directions given by [23], ffl the MZ family of directions given by Monteiro Zhang [17] see also Monteiro [13] Zhang [24] and Monteiro [14] ffl the MT family of directions given by Monteiro Tsuchiya [15] Using these directions, numerous primal dual interior point methods such as potential reduction methods, ....

P Tseng. Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP. Technical report, Department of Mathematics, University of Washington, Seattle, USA, 1996.

....if they are solutions of the following nonlinear system: A i ffl X = b i ; i = 1; m; 1.3a) m X i=1 y i A i S = C; 1.3b) XS = 0; X 0; S 0: 1. 3c) Many interior point algorithms for linear programming have been extended to successfully solve semidefinite programming problems (cf. [1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]) Most of these algorithms use one of the following three search directions: the Alizadeh Haeberly Overton (AHO) direction [1] the KojimaShindoh Hara Helmberg Rendl Vanderbei Wolkowicz Monteiro (KSH HRVW M) direction [9, 5, 13] and the Nesterov Todd (NT) direction [18] The algorithm proposed ....

.... simple, we have used the KSH HRVW M direction in the predictor step in our algorithm since the global and local convergence of the algorithms based on this direction has been intensively investigated [20, 21] In fact, we can use other directions in the Monteiro Zhang family of search directions [16, 22, 25] in the predictor step, including the NT direction. We can establish the corresponding global and superlinear convergence by using similar techniques. See [22] for a unified global and local convergence analysis for a class of predictor corrector algorithms. Our algorithm can be regarded as a ....

P. Tseng. Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP. Technical report, Department of Mathematics, University of Washington, Seattle, WA 98195, USA, June 1996.

....of SDP [MZ96] to QCQP problems. He also studies several neighborhoods and examines the so called short, semi long and long step path following methods. Neither of these works deal with the entire Zhang family, or analyze the pure Newton methods of XS SX of [AHO96] or X 1=2 SX 1=2 of [MT96, Tse96] (see below for definitions) 1 Also known as the second order cone, the Lorenz cone and the ice cream cone. RRR 17 98 Page 3 Our approach is more general than those of Faybusovich and Tsuchiya in that we do not rely solely on the Jordan algebra operations. Rather we assume that the euclidean ....

....from an associative algebra whose properties are used prominently in our development. This approach seems to be necessary in some contexts, for instance in defining the Zhang family and in extending Monteiro s analysis [Mon96] as well as the analysis of Monteiro and Tsuchiya [MT96] and Tseng [Tse96]. We should note that there are Jordan Algebras that are not derived from any associative algebra. In case of euclidean Jordan algebras all except one special class are derivable from such associative algebras, though; the only hold out is the 27 dimensional algebra of 3 Theta 3 hermitian ....

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P. Tseng. Search directions and convergence analysis of some infeasible pathfllowing methods for the monotone semi-definite LCP. Technical Report , Department of Mathemaics, University of Washington, Seatle, WA 98195, 1996.

....semidefinite real matrices. Accordingly, there has been considerable e#ort to extend solution approaches for LP and LCP to SDLP and SDLCP. The main focus has been on extending the interior point approach to SDLP (see [1, 2, 29, 31, 34, 35, 38, 47] and references therein) and, to monotone SDLCP [29, 44] and to semidefinite (nonlinear) complementarity problems (SDCP) 39] Recently, extensions of the merit function approach have also been considered [45, 50] In this paper, we consider extensions of a third approach, that of non interior continuation, which has been extensively studied in the ....

....the following lemma showing that F being monotone is su#cient for #H (z) to be invertible for all z and . Moreover, for # given by (4) F being strongly monotone is su#cient for # #H (z) 1 # to be uniformly bounded. The proof uses a lemma from [46] and is based loosely on ideas from [44] on existence of search directions for interiorpoint methods. Lemma 6 If F is monotone and # is given by (4) with g # CM or (7) then #H (z) is invertible for all z # S S and 0. Moreover, if F is strongly monotone and # is given by (4) with g # CM, then for any bounded set B # S ....

Tseng, P., Search directions and convergence analysis of some infeasible pathfollowing methods for the monotone semi-definite LCP, Optim. Methods Software, 9 (1998), 245--268.

.... For most of the search directions proposed, including ones proposed by Nesterov and Todd, by Monteiro, and the family proposed by Shida, Shindoh, and Kojima, existence and uniqueness require only that the iterates lie in the cone of positive definite matrices (see [8] for a nice survey; also see [5, 10] for subsequent work) An exception is a search direction proposed by Alizadeh, Haeberly, and Overton (AHO) 1, 2] whose existence, as is shown in [7, 8, 9] requires additional assumptions (such as approximate centering) on the iterates that may not be satisfied in practice. Yet, interior point ....

....form: Au Bv = r; Mu Gamma v = Gammas; 1) with M : S 7 S a given monotone linear mapping (so hx; Mxi 0 for all x 2 S) and A; B : S 7 S two user chosen invertible linear mappings, and with (r; s) 2 S Theta S given. For feasible methods, s = 0 always. There are many choices for A; B (see [5, 8, 10, 9, 11]) and, as is mentioned earlier, we will focus on a particular choice of A and B proposed by Alizadeh, Haeberly and Overton [1, 2] Au : 1 2 (yu uy) Bv : 1 2 (xv vx) 2) where (x; y) 2 S Theta S . First, we give a short and constructive proof that AB being strictly monotone is in ....

[Article contains additional citation context not shown here]

P. Tseng, Search directions and convergence analysis of some infeasible pathfollowing methods for the monotone semi-definite LCP, Department of Mathematics, University of Washington, Seattle, WA, June 1996.

No context found.

P. TSENG. Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP. Optim. Methods Softw., 9:245--268, 1998.

No context found.

P. Tseng, Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP, Optimization Methods and Software, 9(1998) 245-268.

No context found.

Tseng, P. (1996). Search directions and convergence analysis of some infeasible path-following methods for the monotone semi-definite LCP. manuscript, Dept. of Mathematics, University of Washington, Seattle.

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