M. Kojima, S. Mizuno and A. Yoshise. An O( nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331-342, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....like a path following algorithm than a potentialreduction algorithm. The rst pure potential reduction algorithm with O( nL) iteration complexity bound was due to Ye [39] see also Freund [5] This algorithm was not a symmetric primal dual method. In the same year, Kojima, Mizuno and Yoshise [17] gave a symmetric primal dual potential reduction algorithm with O( nL) iteration complexity bound. We would like to refer to some good references here. Gonzaga [8] gave an excellent review on path following methods. Later, Todd [30] gave an excellent survey paper on potential function. Also, ....

M. Kojima, S. Mizuno, and A. Yoshise. An O( nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331-342, 1991.

....analytic centers that allows the unified analysis of several important families of interior point methods which share a method of search direction generation, but which differ in their strategies for the sequen 3 tial adjustment of the step size and various other parameters. See for instance [30, 23, 26, 19, 20, 40]. The contributions of this thesis fit into this context as follows: Chapter 2 provides a simplification of unification number 1 in the framework of unification number 2. Chapters 3 and 4 deal with a generalization of the tools that allowed unification number 3. This generalization is done in the ....

....analytic centers of v. The properties of the target map and weighted analytic centers opened up the possibility for the construction of target following algorithms, a family of algorithms that unifies several well established classes of interior point methods for linear programming. See, e.g. [30, 23, 26, 19, 20, 40]. This is what we referred to as unification number 3 at the outset of this section. 1.1.5 Synopsis of Chapter 2 Self scaled barrier functionals are characterized via a rather lengthy list of axioms (see Sections 1.2.1 and 1.2.2) and one may ask the question whether the set of functions ....

[Article contains additional citation context not shown here]

M. Kojima, S. Mizuno and A. Yoshise. An O( p nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331--342, 1991.

....possible le d evelopment d une classe d algorithmes appell es algorithmes poursuiveurs de cibles ( target following algorithms ) et ce cadre se prete a 8 l analyse unifi ee de plusieures classes importantes d algorithmes aux points int erieurs pour la programmation lin eaire. Voir par exemple [7, 13, 10, 4, 5, 20]. 3 Description des r esultats obtenus Le sujet central de ma th ese s adresse a la g en eralisation du cadre unifi e d efini dans la Section 2.3 dans le contexte de l optimisation lin eaire sur les cones symm etriques introduite dans la Section 2.2 ci dessus. Je dirai quelques mots sur cette ....

M. Kojima, S. Mizuno and A. Yoshise. An O( p nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331--342, 1991.

....613 304 200 for the project High Performance Methods for Mathematical Optimization. 1 2 T. ILL ES, J. PENG, C. ROOS AND T. TERLAKY Kojima, Mizuno and Yoshise [15] presented a polynomial time algorithm that produces an exact solution for LCPs where M is positive semide nite. The same authors [16] established an O( p nL) 1 iteration bound for a potential reduction algorithm. Ji, Potra and Huang [11] developed a polynomial, O( p nL) predictor corrector method for positive semide nite LCPs under the assumption that the sequence of iterates generated by their interior point algorithm ....

.... In this section we assume that we have given a point x (0) 2 0 close to the central path (i.e. c (x (0) for some 1) We de ne 0 by n 0 = x (0) T s (0) Starting at x 0 interior point methods for solving (LCP ) need O( p n log(n 0 = iterations (see, e.g. [12, 16, 18, 34]) or O(n log(n 0 = see, e.g. 8] iterations to generate a point x such that c (x) and x T s(x) The rst bound holds for methods with small updates of the barrier parameter whereas the second bound is typical for methods using large updates, and also for methods using a ....

M. Kojima, S. Mizuno and A. Yoshise. An O( p nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331-342, 1991.

....30.9 Table 3: Solving Model 2 Problems with the Termination Test 6 2.3. Progress in Convex Nonlinear Optimization Several interior point algorithms currently exist for convex nonlinear optimization (e.g. 10] 14] 18] 20] 24] We augmented those with a potential reduction algorithm (e.g. [16][22] Y5] Y10] Y14] which has exhibited promising practical efficiency. Han, Pardalos and myself implemented and tested the algorithm using vectorization on an IBM 3090 600S computer with Vector Facilities using the VS Fortran compiler. All numerical results were obtained in double precision. ESSL ....

M. Kojima, S. Mizuno and A. Yoshise. An O( p nL) iteration potential reduction algorithm for linear complementarity problems, Research Report, Department of Information Science, Tokyo Institute of Technology (Tokyo, Japan, 1988).

....extensions from the interior 249 250 HIGH PERFORMANCE OPTIMIZATION TECHNIQUES point methods for linear optimization. A survey on recent results is written by Yoshise [17] Kojima, Mizuno and Yoshise [7] presented a polynomial time algorithm for positive semidefinite LCPs. The same authors [8] established an O( p nL) iteration bound 1 for a potential reduction algorithm. Ji, Potra and Huang [6] developed a polynomial, O( p nL) predictor corrector method for positive semidefinite LCPs under the assumption that the sequence of iterates generated by their IPM converges to a strictly ....

Kojima, M., S. Mizuno and A. Yoshise. An O( p nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331-342, 1991.

....reduces at least with a certain constant. Using this result, we proof that the number of iterations required by the algorithm to converge to an ffl optimal solution is bounded by a polynomial in ffl, the dimension of the problem and the Lipschitz constant. We note that Kojima, Mizuno and Yoshise [7] already proposed a primal dual potential reduction algorithm for for linear complementarity problems. To our knowledge our algorithm is the first large step algorithm for (a class of) smooth convex programming. Our algorithm can also be viewed as a natural implementation of the classical method ....

Kojima, M., Mizuno, S., Yoshise, A. (1988), An O( p n) Iteration Potential Reduction Algorithm for Linear Complementarity Problems, Research Report, Department of Information Sciences, Tokyo Institute of Technology, Tokyo, Japan. 16

....613 304 200 for the project High Performance Methods for Mathematical Optimization. 1 2 T. ILL ES, J. PENG, C. ROOS AND T. TERLAKY Kojima, Mizuno and Yoshise [15] presented a polynomial time algorithm that produces an exact solution for LCPs where M is positive semidefinite. The same authors [16] established an O( p nL) 1 iteration bound for a potential reduction algorithm. Ji, Potra and Huang [11] developed a polynomial, O( p nL) predictor corrector method for positive semidefinite LCPs under the assumption that the sequence of iterates generated by their interior point algorithm ....

.... assume that we have given a point x (0) 2 Gamma 0 close to the central path (i.e. ffi c (x (0) for some 1) We define 0 by n 0 = Gamma x (0) Delta T s (0) Starting at x 0 interior point methods for solving (LCP ) need O( p n log(n 0 =ffl) iterations (see, e.g. [12, 16, 18, 33]) or O(n log(n 0 =ffl) see, e.g. 8] iterations to generate a point x such that ffi c (x) and x T s(x) ffl. The first bound holds for methods with small updates of the barrier parameter whereas the second bound is typical for methods using large updates, and also for methods using a ....

M. Kojima, S. Mizuno and A. Yoshise. An O( p nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331-342, 1991.

....barrier approach leads to polynomial time algorithms [6, 7, 16, 15] The most popular interior point algorithms nowadays are the so called primal dual methods. They were first introduced as path following methods, 9, 12] but later they have been extended to a potential reduction approach [10]. As their name indicates, primal dual methods operate simultaneously on the primal and on the dual problem, an appealing feature due to the intimate relationship between the primal and the dual problem. The search direction in these methods is obtained by applying Newton s method to the system ....

M. Kojima, S. Mizuno, and A. Yoshise. An O( p nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, Series A, 50:331--342, 1991.

....exactly coincides with the long step method. See also [16] The most popular interior point algorithms nowadays are the so called primal dual methods. They were first introduced as path following methods, 11] and [20] but later they have been extended to a potential reduction approach [12] that is more practical but still has a bound of O( p nL) iterations. In fact, an impressive amount of papers on primal dual methods has been published in the last years. It is not our aim to give an extensive survey at this place. Instead we refer to the bibliography of Kranich [13] for more ....

Kojima, M., Mizuno, S. and Yoshise, A. (1991), An O( p nL) iteration potential reduction algorithm for linear complementarity problems, Mathematical Programming 50, 331--342.

....[7, 8] have introduced the class of sucient matrices properly containing the two classes, for which the decision problem is co NP. For solving such good classes of LCP s, there are two major approaches, the classical pivoting approach and the recent approach using interior point algorithms [16, 26]. While polynomial time algorithms for LCP s have been obtained only by the latter approach, a satisfactory foundation of linear complementarity theory depends much on the combinatorial nature of the former approach. Todd [23, 24] was the rst to explain the essence of combinatorics used in the ....

Kojima, M., Mizuno, S., and Yoshise, A., (1988), An O( p nL) iteration potential reduction algorithm for linear complementarity problems, Research Report, Department of Information Sciences, Tokyo Institute of Technology, Tokyo, Japan. 20

....required by the algorithm to converge to an ffl optimal solution is polynomial in ffl, the dimension of the problem and the Lipschitz constant. We note that Liu and Goldfarb [8] already proposed a potential reduction algorithm for convex quadratic programming and Kojima, Mizuno and Yoshise [7] for linear complementarity problems. To our knowledge our algorithm is the first potential reduction algorithm for (a class of) smooth convex programming. To extend the method to all smooth convex programs (without the Relative Lipschitz Condition) is currently under research. This paper is ....

Kojima, M., Mizuno, S., Yoshise, A. (1988), An O( p n) Iteration Potential Reduction Algorithm for Linear Complementarity Problems, Research Report, Department of Information Sciences, Tokyo Institute of Technology, Tokyo, Japan. 17

....(k) Z (k) Gamma 0:24: Once again, the theorem holds for an algorithm that uses an appropriate approximate plane search. In the case of an LP, with F = diag(Ax b) and Z = diag(z) this symmetric scaling coincides with the primal dual symmetric scaling used in Kojima, Mizuno and Yoshise [KMY91], for example, where search directions are computed from FZ Gamma1 A A T 0 # ffiz sym ffix sym # = GammaaeF e Z Gamma1 e 0 # : 73) The three algorithms we discussed so far differ only in the scaling matrices S used in (62) In linear programming, the equivalent of ....

M. Kojima, S. Mizuno, and A. Yoshise. An O( p nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331-- 342, 1991.

....zero. Primal dual methods are most conveniently studied in a framework that exhibits the symmetry between the two optimization problems jointly solved by the algorithm. In the linear programming literature such a framework known as the 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 ....

M. Kojima, S. Mizuno and A. Yoshise. An O( p nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331-- 342, 1991.

.... and was first investigated by Nesterov Todd [21, 22] In [7] we began the generalization of the so called V space approach, a theory that allows the unified analysis of various important classes 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 ....

....direction fields, and they all lend themselves to the construction of so called predictor corrector methods. Moreover, target maps are closely connected to the notion of weighted analytic centers. Weighted centers were introduced to the linear programming literature by Kojima, Mizuno and Yoshise [12]. There they play an important role as a unifying framework for the analysis of interior point 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 ....

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M. Kojima, S. Mizuno and A. Yoshise. An O( p nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331-- 342, 1991. 25

....analytic centers to the framework of self scaled conic programming. In the linear programming literature target directions and weighted analytic centers have led to a unified theory for the 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 ....

....a point on the central path (see below) Since target directions are 2 Newton directions this implies that N T directions are genuine Newton directions. This is a direct extension of the corresponding result for the special case of linear programming, developed by Kojima Mizuno Yoshise [18]. However, since practically all of the difficulties that come up in the theory of square root fields collapse in the linear programming case (see [10] Section 5) a completely different set of analytic tools had to be developed in this paper to tackle with the general case. For the special case ....

M. Kojima, S. Mizuno and A. Yoshise. An O( p nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331-- 342, 1991.

.... and was first investigated by Nesterov Todd [20, 21] In [5] we began the generalization of the so called V space approach, a theory that allows the unified analysis of various important classes of primal dual interior point methods for linear programming (see e.g. Kojima Mizuno Yoshise [11], Monteiro Adler [17] Mizuno [13] Jansen Roos Terlaky Vial [8, 9] Todd [27] 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 ....

....[20, 21] and Todd Toh Tutuncu [28] We also showed that a certain class of target maps 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 ....

M. Kojima, S. Mizuno and A. Yoshise. An O( p nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331-- 342, 1991.

.... : T 2 (s) x; T 2 ( GammaF 0 (x) GammaF 0 (s) 1 j k uk 2 2 F 00 (s) T 2 j k uk 2 2 h F 00 (x) i Gamma1 9 = for j 1. We illustrate the analysis on a generalization of the primal dual interior point algorithm of Kojima, Mizuno and Yoshise [3] for LCP and the joint scaling primal dual interior point algorithm of Nesterov and Todd [8] also see the results on a potential reduction approach by Guler and the author in [12] which is concerned with the special case when K is symmetric) We define u : Gamma # p # # v w; ....

M. Kojima, S. Mizuno and A. Yoshise, An O( p nL) iteration potential reduction algorithm for linear complementarity problems, Math. Prog. 50 (1991) 331--342.

No context found.

M. Kojima, S. Mizuno and A. Yoshise. An O( p nL) iteration potential reduction algorithm for linear complementarity problems. Math. Programming, to appear.

....be essentially equivalent to self concordance. Zhu et al. 49, 29, 44] used the scaled Lipschitz condition to analyze path following methods. The fundamental work of McLinden [31] brought us a lot of ideas to develop interior point algorithms for linear complementarity problems (LCP) and NCPs ([6, 8, 7, 13, 24, 26, 21, 22, 23, 20, 18, 27, 33, 38, 42, 46, 48, 47] etc. The global convergence of these algorithms has been shown by using the existence of the central path, which can be seen as a minimum requirement for interior point methods to be applicable. Similarly to the case of nonlinear convex problems, the study of the convergence rate has also ....

....p 1 2) 2 kp v k 2 (1 fl ) 1 2)kp v k 2 : This completes the proof of the lemma. 8 The above lemmas give us some tools for analyzing primal dual algorithms applying to NCP. In the linear case, Lemma 3. 1 is important to provide the polynomiality of many primal dual algorithms ( [25, 24, 34, 35, 26, 32, 33, 20, 19, 18], etc. On the other hand, Lemma 3.4 suggests us that these analyses may be extended to nonlinear cases. Before proceeding we mention that for the monotone NCP the bounds in Lemma 3.4 can be improved by using Deltax T Deltas( 1 2 ( Deltax) T (f(x Deltax) Gamma f(x) 0: 4 ....

M. Kojima, S. Mizuno, and A. Yoshise. An O( p nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331--342, 1991.

....(IPMs) as well. IPMs for solving (LCP ) are widely studied in the last decade. A survey on recent results is written by Yoshise [33] Kojima, Mizuno and Yoshise [14] presented a polynomial time algorithm that produces an exact solution for LCPs where M is positive semidefinite. The same authors [15] established an O( p nL) 1 iteration bound for a potential reduction algorithm. Ji, Potra and Huang [10] developed a polynomial, O( p nL) predictor corrector method for positive semidefinite LCPs under the assumption that the sequence of iterates generated by their interior point algorithm ....

.... assume that we have given a point x (0) 2 Gamma 0 close to the central path (i.e. ffi c (x (0) for some 1) We define 0 by n 0 = Gamma x (0) Delta T s (0) Starting at x 0 interior point methods for solving (LCP ) need O( p n log(n 0 =ffl) iterations (see, e.g. [11, 15, 17, 31]) or O(n log(n 0 =ffl) see, e.g. 8] iterations to generate a point x such that ffi c (x) and x T s(x) ffl. The first bound holds for methods with small updates of the barrier parameter whereas the second bound is typical for methods using large updates, and also for methods using a ....

M. Kojima, S. Mizuno and A. Yoshise. An O( p nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331-342, 1991.

No context found.

M. Kojima, S. Mizuno and A. Yoshise. An O( nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331-342, 1991.

No context found.

M. Kojima, S. Mizuno, and A. Yoshise. An O( nL) iteration potential reduction algorithm for linear complementarity problems. Mathematical Programming, 50:331-- 342, 1991.

No context found.

M. KOJIMA, S. MIZUNO, and A. YOSHISE. An O( # nL) iteration potential reduction algorithm for linear complementarity problems. Math. Programming, 50:331--342, 1991.

No context found.

M. Kojima, S. Mizuno and A. Yoshise, An O( p nL) iteration potential reduction algorithm for linear complementarity problems. Math. Prog. 50 (1991) 331--342.

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