B. Jansen, C. Roos, T. Terlaky, and J.P. Vial, Primal-dual algorithms for linear programming based on the logarithmic barrier method, J. Optim. Theory Appl. 83(1994) 1-26.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....case we have 0 # u T v # 2 # Du # 2 D 1 v 2 # # Du # 2 2 D 1 v 2 2 . 11 The above lemma allows us to establish a relationship in between #(z, #) defined in (3.18) and the proximity measure #(z, #) r # xs r xs # 2 (4. 3) considered in [4] (see also [14] Corollary 4.2 For any z # IR 2n and any # 0 we have # # # #(z, #) # #(z, #) # # # #(z, #) where # is given by (4.2) Now we establish a relation between #(z, #) and #(z, z) Lemma 4.3 The following identity holds for any z # IR 2n and any # 0 # 2 (z, #) ....

B. Jansen, C. Roos, T. Terlaky, and J.-P. Vial. Primal-dual algorithms for linear programming based on the logarithmic barrier method. J. Optim. Theory Appl., 83(1):1--26, 1994.

....behavior. This is especially true for so called primal dual large update methods, which are the most efficient methods in practice (see, e.g. Andersen et al. 1] The aim of this paper is to present a novel analysis of Newton s method based on a proximity measure introduced by Jansen et al. [7]. To be more concrete we need to go into more detail at this stage. We consider the following linear optimization problem: P ) minfc T x : Ax = b; x 0g; where A 2 m Thetan ; b 2 m ; c 2 n , and its dual problem (D) maxfb T y : A T y s = c; s 0g: We assume that both (P) and ....

.... introduced by Kojima et al. 10] and used in many other papers (cf. McShane, Monma and Shanno [13] Mehrotra [15] Mizuno [17, 18] Monteiro and Adler [20] Todd [24] Zhang and Tapia [29] and also in the books of Wright [26] and Ye [28] The third measure, ffi, was introduced by Jansen et al. [7], and thoroughly used in [22] and Zhao [30] It was also used in the analysis of interior point methods for semidefinite optimization by Jiang [8] de Klerk [4] and by de Klerk et al. 3] Full Newton step methods have the best theoretical performance, with an iteration bound O Gamma p n log n ....

[Article contains additional citation context not shown here]

B. Jansen, C. Roos, T. Terlaky, and J.-Ph. Vial. Primal-dual algorithms for linear programming based on the logarithmic barrier method. Journal of Optimization Theory and Applications, 83:1--26, 1994.

....behavior. This is especially true for so called primal dual large update methods, which are the most efficient methods in practice (see, e.g. Andersen et al. 1] The aim of this paper is to present a novel analysis of Newton s method based on a proximity measure introduced by Jansen et al. [7]. To be more concrete we need to go into more detail at this stage. We consider the following linear optimization problem: P ) minfc T x : Ax = b; x 0g; where A 2 m Thetan ; b 2 m ; c 2 n , and its dual problem (D) maxfb T y : A T y s = c; s 0g: We assume that both (P) and ....

.... introduced by Kojima et al. 10] and used in many other papers (cf. McShane, Monma and Shanno [13] Mehrotra [15] Mizuno [17,18] Monteiro and Adler [20] Todd [24] Zhang and Tapia [29] and also in the books of Wright [26] and Ye [28] The third measure, ffi, was introduced by Jansen et al. [7], and thoroughly used in [22] and Zhao [30] It was also used in the analysis of interior point methods for semidefinite optimization by Jiang [8] de Klerk [4] and by de Klerk et al. 3] Full Newton step methods have the best theoretical performance, with an iteration bound O Gamma p n log n ....

[Article contains additional citation context not shown here]

B. Jansen, C. Roos, T. Terlaky, and J.-Ph. Vial. Primal-dual algorithms for linear programming based on the logarithmic barrier method. Journal of Optimization Theory and Applications, 83:1--26, 1994.

.... by Kojima et al. 9] these methods have proven efficient in computational studies [11] The worst case complexity of long step algorithms with O(1) step sizes is O(n ln 1=ffl) iterations, and for medium step sizes of O(1= p n) one has a worst case bound of O( p n ln 1=ffl) iterations [3, 2, 8]. Although the long step methods have a worse complexity bound than the short and medium step variants, the number of iterations performed in practice are often lower as becomes clear from the cited references. Jansen et al. 6, 7, 4] provided a unifying framework of analysis for these important ....

B. Jansen, C. Roos, T. Terlaky, and J.-Ph. Vial. Primal--dual algorithms for linear programming based on the logarithmic barrier method. Journal of Optimization Theory and Applications, 83:1--26, 1994.

....) fl fl fl fl xs Gamma e fl fl fl fl ; 10.4) ffi(xs; 1 2 fl fl fl fl r xs Gamma r xs fl fl fl fl ; 10.5) where xs ; q xs and q xs denote the vectors whose i th component is x i s i ; q x i s i and q x i s i respectively. The first measure Phi [5] has turned out to be appropriate for the analysis of large update methods, while the second measure has been used by many authors to analyze the behavior of the path following methods based on full Newton step, which are the simplest small update methods. Some variants of Phi are also applied in ....

....small update methods. Some variants of Phi are also applied in the analysis of the so called potential reduction methods [13] One reason for this is that Phi has some barrier properties while ffi K does not. In [12] we have used the third measure ffi which is introduced by Jansen et al. [5] to analyze the complexity of the primal dual newton method for linear programming and presented a unified proof for both small and large update methods. We notice that a variant of the proximity ffi(xs; had been used by Kojima et al. in [9] In [11] Mizuno and Nagasawa also used the ....

Jansen, B., C. Roos, T. Terlaky and J.-Ph. Vial. Primal-dual algorithms for linear programming based on the logarithmic barrier method. Journal of Optimization Theory and Applications, 83:1-26, 1994.

....the target point. We will introduce a quantity that measures the closeness of a point to the target, and prove some nice properties of this quantity. We point out that this proximity measure is completely in the spirit of the Roos Vial measure [33] and the primal dual measures as discussed in [19]. After the first version of this paper was written, we found out that it also appears in Mizuno [25, 27] 3.1 The Newton step in the (x; s) space Let (x; s) be a pair of primal dual interior feasible solutions, and let v be the corresponding point in the v space. Furthermore, let v be the ....

.... 2 Gamma v 2 jfl fl fl = 1 2 min(v) fl fl fl v i u Gamma u Gamma1 jfl fl fl : 8) Let us indicate that if v 2 = e for some positive then this amounts to ffi(v; v) 1 2 fl fl flu Gamma u Gamma1 fl fl fl ; which, up the factor 1 2 , is equal to the proximity measure used in [19]. The next lemma relates the proximity measure to the ratio v i =v i and shows that componentwise the elements of v cannot differ too much from the elements of v. Lemma 3.2 Let ffi : ffi (v; v) and u as defined in (7) Then one has 1 ae(ffi) u i ae(ffi) i = 1; n; with ae(ffi) ....

B. Jansen, C. Roos, T. Terlaky, and J.-Ph. Vial. Primal--dual algorithms for linear programming based on the logarithmic barrier method. Technical Report 92--104, Faculty of Technical Mathematics and Informatics, TU Delft, NL--2600 GA Delft, The Netherlands, 1992. (To appear in JOTA, Vol. 83).

....behavior. This is especially true for so called primal dual large update methods, which are the most efficient methods in practice (see, e.g. Andersen et al. 1] The aim of this paper is to present a novel analysis of Newton s method based on a proximity measure introduced by Jansen et al. [5]. To be more concrete we need to go into more detail at this stage. We consider the following linear optimization problem: P ) minfc T x : Ax = b; x 0g; where A 2 m Thetan ; b 2 m ; c 2 n , and its dual problem (D) maxfb T y : A T y s = c; s 0g: We assume that both (P) and ....

.... by Kojima et al. 8] and used in many other papers (cf. McShane, Monma and Shanno [10] Mehrotra [12] Mizuno [14, 15] Monteiro and Adler [16] Todd [20] Zhang and Tapia [25] Zhao [26] and also in the books of Wright [22] and Ye [24] The third measure, ffi, was introduced by Jansen et al. [5], and thoroughly used in [18] Full Newton step methods have the best theoretical performance, with an iteration bound O Gamma p n log n Delta , but practically they are outperformed by so called damped Newton step methods, or large update methods, that use larger value of and several ....

[Article contains additional citation context not shown here]

B. Jansen, C. Roos, T. Terlaky, and J.-Ph. Vial. Primal-dual algorithms for linear programming based on the logarithmic barrier method. Journal of Optimization Theory and Applications, 83:1--26, 1994.

....v min 1, which is exactly the same as we will state in Lemma 3.8 of this work. However, we failed to prove a similar inequality for the case v min 1 and hence could not improve the complexity of the large update IPM in [16] The measure , up to a factor 1 2 , was introduced by Jansen et al. [5], and thoroughly used in [17] Zhao [26] more recently [16] Its SDO analogue was also used in the analysis of interior point methods for semide nite optimization [3] We notice that variants of the proximity (xs; had been used by Kojima et al. in [9] and Mizuno et al. in [13] The paper is ....

B. Jansen, C. Roos, T. Terlaky, and J.-Ph. Vial. Primal-dual algorithms for linear programming based on the logarithmic barrier method. Journal of Optimization Theory and Applications, 83:1-26, 1994.

....barrier function, with respect to the barrier parameter ; its usefulness is known already for a long time (cf. Frisch [5] Lootsma [12] and Fiacco and McCormick [4] and is due to the fact that it has the barrier property. The measure, up to a factor 1 2 , was introduced by Jansen et al. [7], and thoroughly used in [20, 21, 22, 32] Its SDO analogue was also used in the analysis of interior point methods for semide nite optimization [3] We notice that variants of the proximity (xs; had been used by Kojima et al. in [11] and Mizuno et al. in [16] There are also many others ....

B. Jansen, C. Roos, T. Terlaky, and J.-Ph. Vial. Primal-dual algorithms for linear programming based on the logarithmic barrier method. Journal of Optimization Theory and Applications, 83:1-26, 1994.

....min 1, which is exactly the same as we will state in Lemma 3.8 of this work. However, we failed to prove a similar inequality for the case v min 1 and hence could not improve the complexity of the large update IPM in [16] The measure ffi, up to a factor 1 2 , was introduced by Jansen et al. [5], and thoroughly used in [17] Zhao [26] more recently [16] Its SDO analogue was also used in the analysis of interior point methods for semidefinite optimization [3] We notice that variants of the proximity ffi(xs; had been used by Kojima et al. in [9] and Mizuno et al. in [13] The paper is ....

B. Jansen, C. Roos, T. Terlaky, and J.-Ph. Vial. Primal-dual algorithms for linear programming based on the logarithmic barrier method. Journal of Optimization Theory and Applications, 83:1--26, 1994.

....It is very much in the spirit of the work of [7] It uses the concept of logarithmic barrier function. The convergence analysis is quite simple: it is based on the observation that each iteration decreases the barrier function by a sizable amount. Our presentation is a simplified version of [16]. In Section 5 we discuss some of the implementation issues that make the theoretical algorithm efficient in solving practical problems. We also present some computational experience. Section 5 relies on the large body of literature in the field, but more specifically on [27, 44, 3] In Section 6 ....

....number of steps that are required until the proximity measure falls below the specified level. We state below the fundamental lemma that defines the default value for the step length and gives the estimate for the barrier function decrease. We omit its proof. For a simple proof of it, we refer to [16]. Lemma 12 Let ffi : ku Gamma u Gamma1 k 0 and let ff : ffi 2 ( ffi 2 ) with : p kX Gamma1 d s k 2 kS Gamma1 d x k 2 . Then 0 and x ffd x 0 and s ffd s 0. Moreover Deltaf (ff) f(x ffd x ; s ffd s ; Gamma f (x; s) Gamma ln(1 ....

[Article contains additional citation context not shown here]

B. Jansen, C. Roos, T. Terlaky and J.--P. Vial (1992), Primal--Dual Algorithms for Linear Programming Based on the Logarithmic Barrier Method, Technical Report 92--104, Faculty of Technical Mathematics and Informatics, Technical University Delft, Delft, The Netherlands.

....methods. They were first introduced as path following methods [15, 26] but they have been extended later to a potential reduction and logarithmic barrier approaches [16] that are more practical but, depending on the step size, still have a bound of O( p nL) and O(nL) iterations (see also [12, 27, 23]) As their name indicates, the PD algorithms operate simultaneously on the primal (2) and the dual (3) problems. Having an initial primal dual interior point pair, the search direction in these methods is given as the Newton direction associated to the following asymmetric system of equations ....

B. Jansen, C. Roos, T. Terlaky, and J. Vial, Primal-dual algorithms for linear programming based on the logarithmic barrier method, Tech. Report 92-104, Faculty of Technical Mathematics and Informatics, Delft University of Technology, P.O.Box 5031, 2600 GA Delft, The Netherlands, 1992.

....algorithms in [8] by pushing the target far away. Consequently, one Newton step will not suffice to reach the new target, but more (damped) Newton steps are needed. Such so called long or medium step algorithms are in the spirit of the algorithms of Den Hertog [4] Gonzaga [3] and Jansen et al. [7]. The long step 1 As far as notation is concerned, if x; s 2 IR n then x T s denotes the dot product of the two vectors, whereas xs, p x and x ff for ff 2 IR denote the vectors obtained from componentwise operations. methods use a step size which is O(1) and converge in O(nL) ....

....et al. 5] in which the search direction has the property of both centering and improving complementarity simultaneously. Secondly, we can easily deduce, using the analysis of the long step Dikin method, the complexity bounds for medium and long step logarithmic barrier methods, as given in [7]. We extend the analysis to the well known weighted path following methods. One remarkable and unfortunate outcome of this is that the complexity bound for the medium step algorithm gets a factor O( p n) worse by using targets that are not on the central path. Finally, we use the present ....

[Article contains additional citation context not shown here]

B. Jansen, C. Roos, T. Terlaky, and J.-Ph. Vial. Primal--dual algorithms for linear programming based on the logarithmic barrier method. Technical Report 92--104, Faculty of Technical Mathematics and Informatics, TU Delft, NL--2600 GA Delft, The Netherlands, 1992. (To appear in JOTA, Vol. 83).

....methods for semidefinite programming have recently been studied intensively, due to their polynomial complexity and practical efficiency. Most of these methods are extensions of linear programming algorithms. The primaldual central path following method for linear programming by Jansen et al. [8] has recently been extended to semidefinite programming by Jiang [9] utilizing the Nesterov Todd direction and introducing a new distance measure. In this note we refine and extend this analysis: A weaker condition for a feasible full Newton step is established, and quadratic convergence to ....

.... to be a special case of the primal dual directions for monotone semidefinite complementarity problems introduced in [11] A long step primal dual path following method using the NT direction was recently presented by Jiang [9] This paper is an extension of the LP analysis by Jansen et al. [8]. This method targets a specific point on the primal dual central path, which is then updated if the current iterates are close enough to it. The novelty of this analysis lies in the use of a new centrality measure, which is analogous to the LP measure in [8] In this paper we extend and refine ....

[Article contains additional citation context not shown here]

B. Jansen, C. Roos, T. Terlaky, and J.-Ph. Vial. Primal--dual algorithms for linear programming based on the logarithmic barrier method. Journal of Optimization Theory and Applications, 83:1--26, 1994.

.... s Deltax) ff 2 Deltax Deltas = xs(1 Gamma ff) ff(e ff Deltax Deltas) The property (x(ff) s(ff) 0 holds for all 0 ff 1, since k Deltax Deltask 1 k Deltax Deltask 1. Remark 4. 4 The primal dual direction was introduced in [16, 18] A different proximity measure is used in [14]. 5 Improving the complexity The generic path following algorithm of Section 3 does not enjoy the optimal complexity of O( p n) iterations. We can only guarantee an estimate with O(n) and the reason for this is quite simple. There are two key points in the demonstration of convergence. ....

....can be replaced by the relevant bound in the two sided inequality z 0 Gamma b T y 0 z 0 Gamma z c T x 0 z 0 : 5.3 Newton directions The derivation of Newton directions in the weighted case has been performed by many authors. Examples and proofs can be found in e.g. [10, 14]. In the statement of the results, we do not treat the distinguished row explicitly. Alternatively, b T is supposed to be the first row of A. 5.3.1 Primal direction The Newton direction for the weighted potential is defined by the solution Deltax of minf 1 2 Deltax T WX Gamma2 Deltax ....

Jansen, B., Roos, C., Terlaky, T., and Vial, J.-P. (1994),"Primal-dual algorithms for linear programming based on the logarithmic barrier method", Journal of Optimization Theory and Applications 83, 1--26.

....sequences, in which the target can be pushed far away . Consequently, one Newton step will not suffice to reach the new target, but more (damped) Newton steps are needed. Such long or medium step algorithms are in the spirit of the algorithms of Den Hertog [6] Gonzaga [4] and Jansen et al. [10]. The long step methods use a step size which is O(1) and typically converge in O(n ln 1=ffl) iterations to an ffl approximate solution, whereas the medium step methods converge in O( p n ln 1=ffl) iterations with a step size O(1= p n) To analyze algorithms of this type, we introduce a ....

.... Defining u : v Gamma1 v; 6) this can be rewritten as ffi(v; v) 1 2min(v) fl fl flv Gamma1 i v 2 Gamma v 2 jfl fl fl = 1 2min(v) fl fl flv i u Gamma u Gamma1 jfl fl fl : 7) We mention that if v 2 = e for some positive then this measure reduces to the measure used in [10]. The next lemma from Jansen et al. 10, 9] relates the proximity measure to the ratio v i =v i and shows that componentwise the elements of v cannot differ too much from the elements of v. Lemma 2.1 Let ffi : ffi(v; v) and u as defined in (6) Then one has 1 ae(ffi) u i ae(ffi) i = 1; ....

[Article contains additional citation context not shown here]

B. Jansen, C. Roos, T. Terlaky, and J.-Ph. Vial. Primal--dual algorithms for linear programming based on the logarithmic barrier method. Journal of Optimization Theory and Applications, 83:1--26, 1994.

No context found.

B. Jansen, C. Roos, T. Terlaky, and J.P. Vial, Primal-dual algorithms for linear programming based on the logarithmic barrier method, J. Optim. Theory Appl. 83(1994) 1-26.

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