T. Tsuchiya and M. Muramatsu. Global convergence of a long-step affine scaling algorithm for degenerate linear programming problems. SIAM Journal on Optimization, 5(3):525--551, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is essential for obtaining a strictly complementary optimal solution pair in polynomial time. Without using the centering component, i.e. in case of a pure (primal) AFS method, no polynomial time convergence proof exists. Surprisingly enough, very recent results of Tsuchiya and Muramatsu [29], 1992, make clear that there is some centering effect in the (primal) AFS direction. Tsuchiya was the first who showed the convergence of some (primal) AFS method without any nondegeneracy assumption [28] The strongest result [29] is that the AFS method with step size 2 3 of the maximal step ....

....Surprisingly enough, very recent results of Tsuchiya and Muramatsu [29] 1992, make clear that there is some centering effect in the (primal) AFS direction. Tsuchiya was the first who showed the convergence of some (primal) AFS method without any nondegeneracy assumption [28] The strongest result [29] is that the AFS method with step size 2 3 of the maximal step size (up to the boundary of the feasible region) converges to an optimal solution of (P ) In fact, Tsuchiya s work has made clear that the dual solution generated by his AFS method is the analytic center of the dual optimal set. On ....

Tsuchiya, T. and Muramatsu, M. (1992), Global Convergence of a Long-- Step Affine Scaling Algorithm for Degenerate Linear Programming Problems, Research Memorandum No. 423, The Institute of Statistical Mathematics, Tokyo, Japan.

....= i ADA T E j Gamma1 i ADA T E j : The second identity follows by multiplying the right hand side by ADA T E and then employing the first identity. 2. Affine Scaling Directions. The affine scaling algorithm is the simplest interior point method and has been studied extensively [5, 2, 21, 1, 3, 20, 12, 13, 15, 14, 8]. The affine scaling step directions depend on the specific formulation of the linear program. If the linear program is formulated with equality constraints and nonnegative variables, then the resulting expressions for the step directions are called the primal affine scaling directions. If, on the ....

....a primal dual algorithm) Traditional affine scaling, on the other hand, is an algorithm that updates only the primal variables. It computes dual estimates as a side calculation. There is no analog of this side calculation in the primaldual affine scaling method. Recently, Muramatsu and Tsuchiya [15] proved that for step lengths less than or equal to 2=3 of the distance to the nearest face, the primal variables and the associated dual estimates converge to optimal solutions to their respective problems. They also showed that the value 2=3 is sharp (in a certain weak sense) Subsequently, Hall ....

T. Tsuchiya and M. Muramatsu. Global convergence of a long-step affine scaling algorithm for degenerate linear programming problems. Technical Report 423, The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo, 106, Japan, January 1992. Revised September 1992.

....[2] remained unnoticed for a long time. After Karmarkar [6] initiated the dynamically developing field of interior point methods (IPMs) affine scaling became one of the basic concept in IPMs. Primal or dual affine scaling methods were studied by e.g. Barnes [1] Vanderbei et al. 13] Tsuchiya [12] and Saigal [11] A primal dual affine scaling algorithm for linear programming (LP) was analyzed by Monteiro, Adler and Resende [10] For a general framework of IPMs for LCP see [7] Recently, the authors proposed a new primal dual affine scaling method for LP [5] Given a nearly centered ....

T. Tsuchiya and M. Muramatsu. Global convergence of the long--step affine scaling algorithm for degenerate linear programming problems. Research Memorandum 423, The Institute of Statistical Mathematics, 4--6--7 Minami--Azabu, Minato--ku, Tokyo 106, Japan, January 1992. Revised September 1992.

....Then, in 1991, Dikin [5] considered the case where b = 0, which is called the homogeneous case, and assuming that c 0 showed convergence of the dual iterates when taking = 1=2. Later, Dikin [4] got the same result for the general case. Inspired by this work of Dikin, Tsuchiya and Muramatsu [16] could extend and improve these results. They showed that any step size corresponding to 2=3 ensures global convergence of the primal iterates to an interior point of the optimal face as well as convergence of the dual iterates to the so called analytic center of the dual optimal face. In ....

T. Tsuchiya and M. Muramatsu. Global convergence of the long--step affine scaling algorithm for degenerate linear programming problems. Research Memorandum 423, The Institute of Statistical Mathematics, 4--6--7 Minami--Azabu, Minato--ku, Tokyo 106, Japan, January 1992. Revised September 1992.

....was first introduced by Dikin [6] in 1967 but remained unknown to the western community until the late 80 s. The method was later rediscovered independently by Barnes [3] and Vanderbei et al. 44] Since then, there have appeared a number of papers which study its global and local convergence [7, 8, 12, 21, 37, 38, 39, 41, 42, 43], the behavior of its associated continuous trajectories [2, 4, 22, 24, 45] and its computational efficiency [1, 23] In 1980, Dikin [9] proposed the second order affine scaling algorithm for convex quadratic programming (QP) problem, where the next iterate minimizes the objective function over ....

T. Tsuchiya and M. Muramatsu, Global convergence of a long--step affine scaling algorithm for degenerate linear programming problems, SIAM Journal on Optimization, 5 (1995), pp. 525-- 551.

.... starts with an arbitrary interior feasible point x 0 and generates a sequence of primal feasible solutions by repeated application of T : x k 1 = T (x k ) Associated with each iterate x k of the primal solution is a corresponding dual estimate y k : y(x k ) In a recent paper [2], Tsuchiya and Muramatsu study this algorithm under very weak assumptions. Namely, they assume that the primal and the dual are both feasible, that there is a strictly interior primal feasible solution, and that the constraint matrix A has full row rank. They then show that if fl 2=3, x k and ....

....= 2=3 and he conjectured that fl = 2=3 is sharp. We prove this conjecture by exhibiting a linear program for which the sequence of dual estimates fails to converge for every fl 2=3. After a first version of our paper was written and submitted for publication, we received a revised version of [2] which contained the proof of convergence for fl 2=3 and also an example where the duals fail to converge to the analytic center of the dual optimal face. However, in contrast to our example, their sequence of duals is convergent. It is interesting to contrast this result with the analogous ....

T. Tsuchiya and M. Muramatsu. Global convergence of a long-step affine scaling algorithm for degenerate linear programming problems. Technical Report 423, The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo, 106, Japan, January 1992.

....after the publication of Karmarkar s interior point algorithm, was among the first interior point algorithms to be shown to be competitive with the Simplex Method. Affine scaling algorithms for linear, convex quadratic, and network programming have been extensively studied and implemented (e.g. [1, 22, 29, 31]) We now consider an affine scaling algorithm for general nonconvex quadratic programming (1 2) Given an arbitrary feasible point x k 2 P = fx 2 n j Ax = b; x 0g, using the scaling technique, we form the suboptimization problem min q(x) 1 2 x T Qx c T x (42) subject to ....

T. Tsuchiya and M. Muramatsu. Global convergence of a long-step affine scaling algorithm for degenerate linear programmingproblems. Technical Report 423, The Institute of Statistical Mathematics, Tokyo, 1992. To appear in SIAM J. Opt.

....just affine scaling steps for the linearized barrier function OE lin and the convex quadratic objective function kF 2 (x) Ask. By (46) the step lengths converge to 1 2 when Delta 0. The corrector steps reduce the value of the barrier function, but they may not change F 2 very much. In [24], global convergence of the affine scaling method with step lengths up to 2 3 is proved for linear programs. In our case, the objective function of the predictor step (kF 2 k) is nonlinear, and typically (when problem (9) is not ill posed) there is a solution such that kF 2 k = 0 in the ....

....to 2 3 is proved for linear programs. In our case, the objective function of the predictor step (kF 2 k) is nonlinear, and typically (when problem (9) is not ill posed) there is a solution such that kF 2 k = 0 in the interior of the domain of OE. Thus, the situation is different form the one in [24]. If the result of [24] generalizes to the concept used here, then it follows that x minimizes the linearization of kF 2 k over the linearized domain of problem (9) which is the statement made in Proposition 3. The articles [14, 23] show that even for linear programs the affine scaling ....

[Article contains additional citation context not shown here]

T. Tsuchiya and M. Muramatsu, "Global Convergence of a Long-Step Affine Scaling Algorithm for Degenerate Linear Programming Problems", SIAM Journal on Optimization, Vol. 5, No. 3 (1995) 525--551.

....0. At each iteration, a tentative primal solution [21] is given by x k = D 2 k A (AD 2 k A ) Gamma1 b: 1. 3) Though polynomial time convergence has not been established for the dual affine scaling algorithm, several authors (e.g. Dikin [5] Tsuchiya [22, 23] Tsuchiya Muramatsu [25], Monteiro, Tsuchiya Wang [15] Tsuchiya Monteiro [24] Saigal [20] and Hall Vanderbei [8] have shown the global convergence of the primal and dual iterates of the dual affine scaling algorithm. Most of the computational effort in the dual affine scaling algorithm is related to building and ....

T. Tsuchiya and M. Muramatsu, Global convergence of the long-step affine scaling algorithm for degenerate linear programming problems, tech. report, The Institute of Statistical Mathematics, Tokyo, January 1992.

.... converges to a strictly complementary solution, there is no theoretical guarantee that its primal iterates converge to a point in the center of the optimal face (the dual iterates do, but the primal iterates can only be guaranteed to converge to a point in the relative interior of the optimal face [11, 40, 54]) Most primal dual algorithms enjoy this guarantee [21] This property has been used to develop the robust stopping criteria for the network interior point method described in this paper. Before concluding this introduction, we present some notation and outline the remainder of the paper. We ....

T. Tsuchiya and M. Muramatsu, Global convergence of the long-step affine scaling algorithm for degenerate linear programming problems, SIAM Journal on Optimization, 5 (1995), pp. 525--551.

....[5] Vanderbei, Meketon and Freedman [39] and others, after Karmarkar [16] proposed the first polynomial time interior point method. Though polynomial timecomplexity has not been proved yet for this algorithm, global convergence using so called long steps was proved by Tsuchiya and Muramatsu [37]. This algorithm is often called the primal (or dual) affine scaling algorithm because the algorithm is based on the primal (or dual) problem only. There is also a notion of primal dual affine scaling algorithm. In fact, for LP, there are two different types of primal dual affine scaling algorithm ....

T. Tsuchiya and M. Muramatsu. Global convergence of a long-step affine scaling algorithm for degenerate linear programming problems. SIAM Journal on Optimization, 5(3):525--551, 1995.

....[5] Vanderbei, Meketon and Freedman [39] and others, after Karmarkar [16] proposed the first polynomial time interior point method. Though polynomial time complexity has not been proved yet for this algorithm, global convergence using so called long steps was proved by Tsuchiya and Muramatsu [37]. This algorithm is often called the primal (or dual) affine scaling algorithm because the algorithm is based on the primal (or dual) problem only. There is also a notion of primal dual affine scaling algorithm. In fact, for LP, there are two different types of primal dual affine scaling algorithm ....

T. Tsuchiya and M. Muramatsu. Global convergence of a long-step affine scaling algorithm for degenerate linear programming problems. SIAM Journal on Optimization, 5(3):525--551, 1995.

....[5] Vanderbei, Meketon and Freedman [38] and others, after Karmarkar [16] proposed the first polynomial time interior point method. Though polynomial time complexity has not been proved yet for this algorithm, global convergence using so called long steps was proved by Tsuchiya and Muramatsu [36]. This algorithm is often called the primal (or dual) affine scaling algorithm because the algorithm is based on the primal (or dual) problem only. There is also a notion of primal dual affine scaling algorithm. In fact, for LP, there are two different types of primal dual affine scaling ....

T. Tsuchiya and M. Muramatsu. Global convergence of a long-step affine scaling algorithm for degenerate linear programming problems. SIAM Journal on Optimization, 5(3):525--551, 1995.

....[5] Vanderbei, Meketon and Freedman [42] and others, after Karmarkar [16] proposed the first polynomial time interior point method. Though polynomial time complexity has not been proved yet for this algorithm, global convergence using so called long steps was proved by Tsuchiya and Muramatsu [40]. This algorithm is often called the primal (or dual) affine scaling algorithm because the algorithm is based on the primal (or dual) problem only. There is also a notion of primal dual affine scaling algorithm. In fact, for LP, there are two different types of primal dual affine scaling algorithm ....

T. Tsuchiya and M. Muramatsu. Global convergence of a long-step affine scaling algorithm for degenerate linear programming problems. SIAM Journal on Optimization, 5(3):525--551, 1995.

....scaling algorithm has been widely implemented and extensively studied, but unlike many other interior point methods, the question of whether the affine scaling algorithm is polynomially convergent is still an open problem. The strongest convergence result so far is due to Tsuchiya and Muramatsu [30], which establishes global convergence of the affine scaling algorithm where the step is taken as a fixed fraction less than or equal to 2=3 of the whole step to the boundary of the feasible region. Simpler proofs of the same global convergence results can also be found in Monteiro, Tsuchiya and ....

T. Tsuchiya and M. Muramatsu, "Global convergence of a long-step affine scaling algorithm for degenerate linear programming problems", SIAM Journal on Optimization 5, No.3(1995)525-551.

....for the primal or dual variants of the algorithm. The exception is the primal dual affine scaling algorithm, for which a polynomial time proof exists [9] but that uses very short steps, rendering it impractical. Recently, several authors (e.g. Dikin [4] Tsuchiya [16, 17] Tsuchiya Muramatsu [19], Monteiro, Tsuchiya Wang [10] Tsuchiya Monteiro [18] Saigal [14] and Hall Vanderbei [6] have presented exciting convergence results for the affine scaling algorithm. In this paper, we use some results in [4, 10, 19] to derive indicators that identify the optimal primal and dual faces for ....

....several authors (e.g. Dikin [4] Tsuchiya [16, 17] Tsuchiya Muramatsu [19] Monteiro, Tsuchiya Wang [10] Tsuchiya Monteiro [18] Saigal [14] and Hall Vanderbei [6] have presented exciting convergence results for the affine scaling algorithm. In this paper, we use some results in [4, 10, 19] to derive indicators that identify the optimal primal and dual faces for a linear program, and present an approach for using one such indicator in the context of minimum cost network flow problems solved via the dual affine scaling algorithm. At the same time, we study some practical ....

[Article contains additional citation context not shown here]

T. Tsuchiya and M. Muramatsu, Global convergence of the long-step affine scaling algorithm for degenerate linear programming problems, tech. report, The Institute of Statistical Mathematics, Tokyo, January 1992.

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