S. J. Wright and F. Jarre. The role of linear objective functions in barrier methods. Mathematical Programming, Series A, 84:357--373, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....paths which will be useful later on. It is shown in [6, 16] that under AS3, AS4 and AS5, rw (w ; 0) is nonsingular, and a continuity argument yields that it remains nonsingular in a small neighborhood of w . In the following technical lemma, which is a simple extension of that proved in [20] to the case of linear equality constraints, we now verify that the central path is well de ned in the intersection of this neighborhood and E , and show that it has useful continuity properties. Superlinear Convergence of Primal Dual Interior Point Algorithms for NLP 7 Lemma 3.1 Under AS2 AS5, ....

S. J. Wright and F. Jarre. The role of linear objective functions in barrier methods. Preprint MCS-P485-1294, Argonne National Laboratory, Argonne, Illinois, USA, April 1998.

....z Department of Computational and Applied Mathematics, Rice University, Houston, TX, 77005. This author was supported in part by DOE Grant DE FG03 97ER25331, DOE LANL Contract 03891 99 23, and NSF Grant DMS 9973339. zhang caam.rice.edu) 1 VILLALOBOS, TAPIA, ZHANG 2 Recent work, such as [9, 14, 15, 17], has focused on studying the performance of Newton logbarrier methods for the inequality constrained optimization problem. Since Newton primal dual methods have been a success for solving the linear program, current investigation [3, 5, 11, 12] has focused on extending these methods to solve the ....

....can be written in the following manner F 0 B (x; Gamma1 = Theta U B (x) UN (x) H 11 (x; H 12 (x; H T 12 (x; H 22 (x; U B T (x) UN T (x) 19) where H 11 (x; O( H 12 (x; O( and H 22 (x; O(1) VILLALOBOS, TAPIA, ZHANG 11 S. J. Wright and Jarre [17] use the implicit function theorem to describe a continuous trajectory of solutions for the system given in (3) with the right hand side modified, that is, F (x; z; e : where 2 IR n and 2 IR. We make use of their lemma, modifying it with specific choices of and to suit our ....

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S. J. Wright and F. Jarre. The role of linear objective functions in barrier methods. Technical Report P485-1294, Mathematics and Computer Science Division, Argonne National Laboratory, 1997.

....k superlinearly to zero (that is, lim k 1 k 1 = k = 0) while taking no more than a fixed number of Newton steps at each value of k . In the case of LICQ, rapid convergence of this type has been investigated by Conn, Gould, and Toint [7] Benchakroun, Dussault, and Mansouri [4] Wright and Jarre [29], and Wright [26] We anticipate that similar results will continue to hold when LICQ is replaced by MFCQ, because the central path continues to be smooth and the convergence domain (46) for Newton s method is similar in both cases. A detailed investigation of this claim and an analysis of the ....

S. J. Wright and F. Jarre. The role of linear objective functions in barrier methods. Mathematical Programming, Series A, 84:357--373, 1998.

....National Laboratory Stephen Wright Delta Florian Jarre The role of linear objective functions in barrier methods: Corrigenda September 22, 2000 Abstract. The published paper contains a number of typographical errors and an incomplete proof. We indicate the corrections here. Our paper [1] contains the following typographical errors. Page 364, statement of Proposition 1. Replace Assume that (30) is satisfied: by Assume that the conditions of Theorem 1 hold and that (30) is satisfied: Equation (31) Replace the exponent oe Gamma 1 by oe . Equation (37) second ....

....and replacing with the following. For the term in brackets, we have 1 O( oe Gamma1 ) 1 Gamma ) O( oe Gamma1 ) Gamma 1 = Gamma (1 Gamma ) Gamma 1) O( oe Gamma1 ) 1 Gamma ) Gamma 1) 1 O( oe Gamma1 ) 1) Note from equations (23) and (38) of [1] that 1 ae max 1 ae min 1 Gamma oe : 2) Let C and 0 be positive constants such that all the O( oe Gamma1 ) and O( oe ) terms in (48) of [1] are bounded by C oe Gamma1 and C oe , respectively, for all 2 Stephen Wright: Mathematics and Computer Science Division, Argonne ....

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S. J. Wright and F. Jarre. The role of linear objective functions in barrier methods. Mathematical Programming, Series A, 84:357--373, 1998.

....to P ( Delta; it is well known that the first Newton step for each value of usually is a poor search direction, and a step length ff considerably smaller than 1 usually is needed to remain feasible at this iteration (see Conn, Gould, and Toint [5] M. Wright [26] and S. Wright and Jarre [29]) Often, however, subsequent iterations of Newton s method converge rapidly to x( Although the Hessian P xx (x; is positive definite near x = x( see the proof of [10, Theorem 12] the observed rate of convergence of Newton s method is better than we might expect from a naive application of ....

....conditions that are required of the line search parameter. We show that when the iterates lie in neighborhoods of the type discussed in Section 4, the unit step length satisfies these conditions for all sufficiently small values of . In Section 6, we use earlier results of S. Wright and Jarre [29] to show that, when the objective function f is linear and the line search and stopping criteria for the Newton iterations are defined in a certain reasonable way, then the first Newton step taken after each substantial reduction of produces an iterate that is within the quadratic convergence ....

[Article contains additional citation context not shown here]

S. J. Wright and F. Jarre. The role of linear objective functions in barrier methods. Mathematical Programming, Series A, 84:357--373, 1998.

....superlinearly to zero (that is, lim k 1 k 1 = k = 0) while taking no more than a fixed number of Newton steps at each value of k . In the case of LICQ, rapid convergence of this type has been investigated by Conn, Gould, and Toint [7] Benchakroun, Dussault, and Mansouri [4] Wright and Jarre [29], and Wright [26] We anticipate that similar results will continue to hold when LICQ is replaced by MFCQ, because the central path continues to be smooth and the convergence domain (46) for Newton s method is similar in both cases. A detailed investigation of this claim and an analysis of the ....

S. J. Wright and F. Jarre. The role of linear objective functions in barrier methods. Mathematical Programming, Series A, 84:357--373, 1998.

.... k 1 = k = 0) while taking no more than a fixed number of Newton steps at each value of k . In the case of LICQ, rapid convergence of this type has been investigated by Conn, Gould, and Toint [7] Benchakroun, Dussault, 28 Stephen J. Wright, Dominique Orban and Mansouri [4] Wright and Jarre [29], and Wright [26] We anticipate that similar results will continue to hold when LICQ is replaced by MFCQ, because the central path continues to be smooth and the convergence domain (46) for Newton s method is similar in both cases. A detailed investigation of this claim and an analysis of the ....

S. J. Wright and F. Jarre. The role of linear objective functions in barrier methods. Mathematical Programming, Series A, 84:357--373, 1998.

.... 0 is also discussed there. As for Proposition 2, we believe that the algorithm will in general not stagnate at singular points. When the second order sufficient conditions for (9) are satisfied, Algorithm 1 is locally (near the optimal solution of (9) very similar to the one analyzed in [27, 28], and the local superlinear convergence property shown in [27] carries over to the algorithm of the present paper. The details to show this claim are lengthy and remain to be verified as well. 5. Conclusion We propose an algorithm for nonconvex programming that combines features of trust region ....

S.J. Wright and F. Jarre, "The Role of linear Objective Functions in Barrier Methods" Preprint MCS-P485-1294, MCS Division, Argonne National Laboratory. (Revised, August, 1997).

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S. J. Wright and F. Jarre. The role of linear objective functions in barrier methods. Mathematical Programming, 84(2):357-373, 1999.

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