R.D.C. MONTEIRO and J.-S. PANG. On two interior-point mappings for nonlinear semidefinite complementarity problems. Math. Oper. Res., 23:39--60, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... an approach that was developed by Kojima 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 ....

R.D.C. Monteiro and J.-S. Pang. On two interior point mappings for nonlinear semidefinite complementarity problems. Mathematics of Operations Research, 23:39--60, 1998.

.... methods, an approach that was developed by 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, ....

....to the framework of self scaled conic programming. This concept plays an important role in the analysis of interior point methods for linear programming (see e.g. Kojima Mizuno Yoshise [12] Various generalizations to the framework of semidefinite programming have been proposed by Monteiro Pang [19], Sturm Zhang [26] and Monteiro Zanjacomo [17] Let us conclude this section with some remarks regarding notation. The set of primal dual strictly feasible solutions and the set of primal dual feasible solutions respectively are defined as follows: F ffi (PD) i Gamma L x 0 Delta K ....

[Article contains additional citation context not shown here]

R.D.C. Monteiro and J.-S. Pang. On two interior point mappings for nonlinear semidefinite complementarity problems. Mathematics of Operations Research, 23:39--60, 1998.

....analysis of various important classes of 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, ....

R.D.C. Monteiro and J.-S. Pang. On two interior point mappings for nonlinear semidefinite complementarity problems. Mathematics of Operations Research, 23:39--60, 1998.

....generalize the notion of weighted analytic 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, ....

R.D.C. Monteiro and J.-S. Pang. On two interior point mappings for nonlinear semidefinite complementarity problems. Mathematics of Operations Research, 23:39--60, 1998.

.... be attained by a feasible primal dual solution pair (X; Z) Moreover, this notion of weighted center is scale invariant (see Todd, Toh and T ut unc u [16] for a definition) A different generalization of weighted centers towards semidefinite programming was recently proposed by Monteiro and Pang [12]. Their weighted center has the property of uniqueness, but lacks some other desirable properties as we will discuss later. This paper is organized as follows. In Section 2, we review some terminology and we define the concept of weighted center for semidefinite programming. We will also discuss ....

....search direction is estimated in Section 3. We show in Section 4 that for any positive definite matrix W , there exists a W weighted center under the primal dual Slater condition. In Section 5, we discuss the new definition of weighted centers in relation to the Monteiro Pang weighted centers [12]. Notation. Given X and Y in n Thetan , the standard inner product is defined by X ffl Y = trX T Y: The Euclidean norm and its associated operator norm, viz. the spectral norm, are both denoted by k Deltak. The Frobenius norm of a matrix X 2 n Thetan is kXk F = p X ffl X. It is ....

[Article contains additional citation context not shown here]

Monteiro, R.D.C. and Pang, J.-S., "On two interior point mappings for nonlinear semidefinite complementarity problems," Technical Report, School of Industrial and Systems Engineering, Georgia Tech, Atlanta, Georgia, U.S.A., 1996.

....We also give sufficient conditions for the accumulation points of sequences fX(w k ; y k )g and f(z(w k ; y k ) y k ; S(w k ; y k ) g to be optimal solutions of problems (P ) and (D) respectively. The following technical result is a slight variant of Lemma 2 of Monteiro and Pang [14]. Proposition 8. Let a linear map A : S n p be given and consider the induced linear map G : S n Theta S n Theta p S n Theta p defined by G(X; S; y) j A (y) Gamma S A(X) 8 (X; S; y) 2 S n Theta S n Theta p : 28) Then, for any constant ....

R. D. C. Monteiro and J. S. Pang. On two interior-point mappings for nonlinear semidefinite complementarity problems. Mathematics of Operations Research, 23:39--60, 1998.

....also give sufficient conditions for the accumulation points of the sequences fX(w k ; y k )g and f(z(w k ; y k ) y k ; S(w k ; y k ) g to be optimal solutions of problems (P ) and (D) respectively. The following technical result is a slight variant of Lemma 2 of Monteiro and Pang [8]. Proposition 3.1 Let a linear map A : S n p be given and consider the induced linear map G : S n Theta S n Theta p S n Theta p defined by G(X; S; y) j A (y) Gamma S A(X) # ; 8 (X; S; y) 2 S n Theta S n Theta p : 5) Then, for any constant ....

R. D. C. Monteiro and J. S. Pang. On two interior-point mappings for nonlinear semidefinite complementarity problems. Mathematics of Operations Research, 23:39-- 60, 1998.

....problem defined on the cone of symmetric positive semidefinite matrices. The linear version of this problem was introduced by Kojima, Shindoh, and Hara [3] and has received a great deal of research attention recently. In what follows, we consider a nonlinear version of this problem defined in [6]. This reference contains a fairly extensive bibliography on interior point methods for solving optimization and complementarity problems defined on the cone of semidefinite matrices; it will be the source for several results that will be used freely in the subsequent development. 3.1 Implicit ....

....point methods for solving optimization and complementarity problems defined on the cone of semidefinite matrices; it will be the source for several results that will be used freely in the subsequent development. 3. 1 Implicit mixed complementarity problems We recall the framework considered in [6]. Let F : S n ThetaS n Theta m S n Theta m be a given mapping. The mixed complementarity problem in symmetric matrices is to find a triple (X; Y; z) 2 S n Theta S n Theta m satisfying F (X; Y; z) 0; X ffl Y = 0; X; Y ) 2 S n Theta S n : 6) As explained ....

[Article contains additional citation context not shown here]

R. D. C. Monteiro and J.-S. Pang, On two interior-point mappings for nonlinear semidefinite complementarity problems, manuscript, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA, May 1996. Submitted to Mathematics of Operations Research.

No context found.

Monteiro, R.D.C., J.S. Pang (1998). On two interior-point mappings for nonlinear semidefinite complementarity problems. Mathematics of Operations Research 23 39--60.

....problem defined on the cone of symmetric positive semidefinite matrices. The linear version of this problem was introduced by Kojima, Shindoh, and Hara [10] and has received a great deal of research attention recently. In what follows, we consider a nonlinear version of this problem defined in [17]. This reference contains a fairly extensive bibliography on interior point methods for solving optimization and complementarity problems defined on the cone of semidefinite matrices; it will be the source for several results that will be used freely in the subsequent development. 3.1 Implicit ....

....point methods for solving optimization and complementarity problems defined on the cone of semidefinite matrices; it will be the source for several results that will be used freely in the subsequent development. 3. 1 Implicit mixed complementarity problems We recall the framework considered in [17]. Let F : S n Theta S n Theta m S n Theta m be a given mapping. The mixed complementarity problem in symmetric matrices is to find a triple (X; Y; z) 2 S n Theta S n Theta m satisfying F (X; Y; z) 0; X ffl Y = 0; X; Y ) 2 S n Theta S n : 6) As ....

[Article contains additional citation context not shown here]

R. D. C. Monteiro and J.-S. Pang, On two interior-point mappings for nonlinear semidefinite complementarity problems, manuscript, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA, May 1996. To appear in Mathematics of Operations Research.

No context found.

R.D.C. MONTEIRO and J.-S. PANG. On two interior-point mappings for nonlinear semidefinite complementarity problems. Math. Oper. Res., 23:39--60, 1998.

No context found.

R.D.C. Monteiro and J.-S. Pang. On two interior point mappings for nonlinear semidefinite complementarity problems. Mathematics of Operations Research, 23:39--60, 1998. 24

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