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

....programming , semidefinite programming , problems with convex quadratic constraints and various combinations of such problems ) with the help of a simple and unifying Jordan algebraic technique. In the present paper we analyze a primal dual potential reduction algorithm which has been developed in [K M] for the case of the linear programming problem. This algorithm has been generalized with the help of a rather sophisticated technique of the convex analysis to the Research supported in part by NSF grant DMS98 03191. 1 class of problems involving symmetric cones in [N T] see also [T] Our ....

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

....recent development of interior point methods is the path of centers of the feasible region. This path was first studied by McLinden [31] and later by several authors [2, 3, 32, 47] The idea of approximately tracing the central path has led to the development of many interior point methods; see [12, 17, 20, 22, 23, 24, 33, 34, 35, 43, 44, 49, 55, 57, 60], to name a few references. One common approach to keep the iterates close to the central path is to use a perturbed Newton direction to search the next iterate. Interestingly, such a direction is often a descent direction for some potential function associated with the problem. Our proposed ....

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

.... Sciences, The University of Iowa, Iowa City, Iowa 52242 Since Karmarkar proposed his projective algorithm [5] various primal dual potential reduction algorithms for linear programming have been developed by Anstreicher and Bosch [1] Freund [2] Gonzaga and Todd [4] Kojima, Mizuno and Yoshise [6], Liu and Goldfarb [7] McShane, Monma and Shanno [8] and Ye [10] 11] among others. All of these algorithms are based on reducing a primal dual potential function that is first appeared in Todd and Ye [9] They showed that a Newton type step can reduce the function by a constant from an interior ....

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

....a primal dual potential function Phi ae (x; s) that has been the main tool of recent potential reduction algorithms. A succession of articles showed that this potential function can be reduced by at least a constant amount starting from any feasible interior point, for large enough values of ae [15, 19, 5]. However, these methods are feasible methods, i.e. they have to start at a strictly feasible initial solution and obtaining such a solution is often hard. Even worse, feasible or strictly feasible solutions may not exist for particular instances of the LP problem. One alternative is to consider ....

....functions defined on original infeasible variables can be reduced at the same rate as the TTY potential function on the feasible variables of this artificial problem and analyze a particular member of this family. The significance of this result lies in the fact that it mimics the result in [5] for infeasible point potential functions, i.e. it provides a recipe to guarantee reductions from any infeasible point and ensures global polynomial convergence. This guarantee also gives the method the flexibility to incorporate heuristic tricks to make it more efficient. In particular, once the ....

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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 (1991) pp. 331-342.

....of piecewiselinear homotopy methods, see, e.g. Kojima (1974, 1979) Kojima, Nishino and Sekine (1976) Saigal (1971, 1976) Todd (1976b) Complementarity problems can also be considered from an interior point algorithm viewpoint, see Section 4. 9, hence by following a smooth path, see, e.g. Kojima, Mizuno and Noma (1990b) Kojima, Mizuno and Yoshise (1991d) Kojima, Megiddo and Noma (1991b) Kojima, Megiddo and Mizuno (1990a) Mizuno (1992) We present the Lemke algorithm as an example of a piecewise linear algorithm since it played a crucial role in the development of subsequent piecewise linear algorithms. Let ....

....Saigal (1971, 1976) Todd (1976b) Complementarity problems can also be considered from an interior point algorithm viewpoint, see Section 4. 9, hence by following a smooth path, see, e.g. Kojima, Mizuno and Noma (1990b) Kojima, Mizuno and Yoshise (1991d) Kojima, Megiddo and Noma (1991b) Kojima, Megiddo and Mizuno (1990a) Mizuno (1992) We present the Lemke algorithm as an example of a piecewise linear algorithm since it played a crucial role in the development of subsequent piecewise linear algorithms. Let us consider the following linear complementarity problem: Given an affine map g : R n R n , find ....

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

....programming , semidefinite programming , problems with convex quadratic constraints and various combinations of such problems ) with the help of a simple and unifying Jordan algebraic technique. In the present paper we analyze a primal dual potential reduction algorithm which has been developed in [6] for the case of the linear programming problem. This algorithm has been generalized with the help of a rather sophisticated technique of the convex analysis to Department of Mathematics, University of Notre Dame, Room 370, CCMB, Notre Dame,IN, 46556 5683, USA the class of problems involving ....

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

....Nagatsuta cho, Midori ku, Yokohama 227, Japan 1. Introduction. There are numerous variations and extensions of primal dual interior point algorithms for linear programs, convex quadratic programs, linear complementarity problems, convex programs and nonlinear complementarity problems ([8, 9, 10, 11, 13, 14, 17, 18, 19, 20, 24, 26, 29], etc. A common basic idea behind the algorithms in this class is moving in a Newton direction for approximating a point on the central trajectory at each iteration. Among others, primal dual infeasible interior point algorithms are known to solve large scale practical linear programs very ....

.... the central trajectory which has been playing an essential role in the former class can be characterized as the set of minimizers of the primal dual logarithmic barrier function = a special case of self concordant barrier functions (see [9, 17] and primal dual potential reduction algorithms ([8, 11, 18], etc. utilize the logarithmic potential function = a special case of self concordant potential functions. Such close relationships support the issue (a) in the class of primal dual interior point algorithms. A substantial difference, however, lies in their search directions. Roughly speaking, we ....

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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 (1991) 331--342.

.... given by Kojima Mizuno Yoshise [14] for the monotone LCP (linear complementarity problem) and Monteiro Adler [25] for the LP (linear program) to the monotone SDLCP (2) b) A potential reduction feasible interior point algorithm which is an extension of the algorithm given by Kojima Mizuno Yoshise [15] for the monotone LCP to the monotone SDLCP (2) c) A potential reduction infeasible interior point algorithm which is an extension of the constrained potential reduction algorithm (Algorithm I) given by Mizuno Kojima Todd [22] for the LP to the monotone SDLCP (2) Quite recently, much progress ....

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

....potential function in the neighborhood N Gamma 1 (fi) The primal dual potential function introduced by Todd and Ye [19] is (x; s) ae log(x T s) Gamma n X j=1 log(x j s j ) 1) where ae n. Using this, Ye [21] Freund [3] Anstreicher and Bosch [1] and Kojima, Mizuno and Yoshise [6] have developed O(n 0:5 L) iteration potential reduction algorithms with the choice of ae = n (n 0:5 ) However, practical experiments indicate that a big ae is much better (McShane et al. 10] and Lustig et al. 9] We show that if we set ae = n (n 2 ) the potential reduction algorithm ....

M. Kojima, S. Mizuno, and A. Yoshise, "An O( p nL) iteration potential reduction algorithm for linear complementarity problems," Research Reports on Information Sciences B-217, Dept. of Information Sciences, Tokyo Institute of Technology (Meguroku, Tokyo, Japan, 1988).

....a neighborhood plays a key role in gaining sufficient reduction in the duality gap at each iteration to ensure the polynomial time convergence. There has been another development in PD algorithms, namely, an O( p nL) iteration PD potential reduction algorithm given by Kojima, Mizuno and Yoshise [13]. They have taken a search direction parameter fi = n= n p n) and a step length ff such that in each iteration there is at least a constant reduction in the primal dual potential function of Todd and Ye [29] rather than the duality gap. Kojima, Megiddo, Noma and Yoshise [9] generalized the ....

....method using Rule P. In [4; 9; 12; 13; 22] PD algorithms were presented for the complementarity problem, rather than pairs of primal and dual linear programs. All the results obtained there can be easily adapted to the primal dual pair of linear programs (P) and (D) See the concluding remarks of [13]. Many interior point algorithms have been proposed which work on the primal dual pair of problems (P) and (D) but are not covered by the GPD method. Among others, we refer to the following: i) An O(n 3 L) algorithm using a sequence [20] ii) An O( p nL) iteration potential reduction ....

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

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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 (1991) 331--342.

....Kojima Mizuno Yoshise [8] for the monotone LCP (linear complementarity problem) and Monteiro Adler [14] for the linear program to the monotone SDLCP (3) in symmetric matrices. b) A potential reduction feasible interior point algorithm which is based on the algorithm given by Kojima Mizuno Yoshise [9] for the monotone LCP to the monotone SDLCP (3) c) A potential reduction infeasible interior point algorithm based on the constrained potential reduction algorithm (Algorithm I) given by Mizuno Kojima Todd [12] for the linear program to the monotone SDLCP (3) In Section 2, we present (d) a ....

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

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