Bertsekas, D. P., and Eckstein, J., "Dual Coordinate Step Methods for Linear Network Flow Problems," Mathematical Programming, Vol. 42, 1988, pp. 203-243.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... of research that has been invested in developing e#cient solution algorithms for (MCF) problems [1, 3] either by specializing LP algorithms such as the Simplex method [1, 16, 17] or the Interior Point method [20] to the network case, or by developing ad hoc approaches such as those of [5, 6, 15]. It is therefore extremely interesting, both for practitioners and for algorithms developers, to evaluate which algorithm is the most e#cient in practice to solve (MCF) This is usually done as follows: a large set of usually randomly generated instances of di#erent classes is collected, ....

D.P. Bertsekas, J. Eckstein, Dual Coordinate Step Methods for Linear Network Flow Problems, Mathematical Programming, Series B, Vol. 42, pp. 203-243, 1988.

....iterations which do not lead to an increase of the number of items assigned to customers, and switches to the Hungarian method as soon as the counter exceeds an appropriately defined parameter. It can be shown that the worst case complexity of this combination is O(n 3 ) Bertsekas and Eckstein [35] modified the above described auction algorithm to obtain a polynomial time algorithm with worst case complexity O(n 3 log(nR) where R is defined as above. The algorithm works with a pair of a primal and a dual solution which fulfills the ffl complementarity conditions x ij (c ij Gamma u i ....

....ffl relaxed problem for a large ffl, and then to decrease ffl gradually until it reaches 1=n. When ffl is decreased to ffl 0 the optimal solution of the previous ffl relaxed problem is updated with few efforts to obtain the optimal solution of the ffl 0 relaxed problem. Bertsekas et al. [35] implement this ffl scaling in terms of a scaling of the costs. All costs c ij are multiplied by n 1 and a 1 relaxed problem (ffl = 1) is solved. If x ij and u i , v j are optimal primal and dual solutions of the 1 relaxed problem with costs c 0 ij = n 1)c ij , then x ij and u i n 1 , v ....

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D. P. Bertsekas and J. Eckstein, Dual coordinate step methods for linear network flow problems, Mathematical Programming 42, 1988, 203--243.

....good initial prices for the next. The practical performance of auction algorithms has shown to be very efficient. Computational experience in [3] shows that an auction algorithm for the LAP is considerably faster than, for example, the sequential shortest path algorithm. Bertsekas and Eckstein [2] show that the worst case complexity of the forward auction algorithm for the LAP using e scaling is O(nmlognC) where m is the number of possible links between persons and objects and C is the maximum absolute benefit. The computational analysis performed by Schwartz [17] indicates an expected ....

D.P. Bertsekas and J. Eckstein, "Dual Coordinate Step Methods for Linear Network Flow Problems," Mathematical Programming 42, 203-243 (1988).

....on these delays is imposed; the latter seems to be the case especially for gradient type algorithms (see [11, Sec. 6.3. 2] Examples of totally asynchronous algorithms include dynamic programming methods, fixed point methods, and coordinate search methods for linear and convex network flows (e.g. [5, 6, 10, 9, 11, 7]) Partially asynchronous versions of instances of the CA algorithm have been studied only to a limited extent; variable metric (or, deflected gradient) methods in unconstrained optimization [11, Sec. 7.5] gradient projection methods [70, 11, 67] and coordinate ascent methods for dual ....

D. P. Bertsekas and J. Eckstein, Dual coordinate step methods for linear network flow problems, Math. Programming, 42 (1988), pp. 203--243.

....pushes for general networks. They show that a specific implementation of the generic implementation, called the first active method algorithm, performs O(n 3 ) nonsaturating pushes, as does a related method, the wave method. The wave method was developed independently by Bertsekas and Eckstein[5]. We shall show that an adaptation of the first active method for bipartite networks performs O(n 3 1 ) nonsaturating bipushes. The first active method uses the acyclicity of the admissible network. As is well known, the vertices of an acyclic network can be ordered so that for each edge (v; w) ....

D. Bertsekas and J. Eckstein, Dual coordinate step methods for linear network flow problems, Mathematical Programming, 42 (1988), pp. 202--243.

....algorithm solves the fractional assignment problem in O(v(v log v e) log(vCD) time. In [19] Orlin and Ahuja proposed an O( p ve log(vW ) time algorithm for the assignment problem and it is said that their algorithm is competitive with the existing strongly polynomial algorithm (see [2] also) In their algorithm, it is assumed that the edge weights are integer and W denotes the maximum of the absolute value of edge weights. To incorporate their algorithm into Dinkelbach s algorithm, we need to replace the objective function of the subproblem Q( solved at the rth iteration by ....

D. P. Bertsekas and J. Eckstein. Dual coordinate step methods for linear network flow problems. Mathematical Programming, 42 pp.203--243, 1988.

....the next. 12 R. FRELING et al. The practical performance of auction algorithms has shown to be very efficient. Computational experience in [3] shows that an auction algorithm for the LAP is considerably faster than, for example, the sequential shortest path algorithm. Bertsekas and Eckstein [2] show that the worst case complexity of the forward auction algorithm for the LAP using e scaling is O(nm lognC) where m is the number of possible links between persons and objects and C is the maximum absolute benefit. The computational analysis performed by Schwartz [17] indicates an expected ....

D.P. Bertsekas and J. Eckstein, "Dual Coordinate Step Methods for Linear Network Flow Problems," Mathematical Programming 42, 203-243, 1988.

.... of different processors; examples of algorithms DECOMPOSITION METHODS FOR DIFFERENTIABLE OPTIMIZATION OVER PRODUCT SETS 25 that have been analyzed using this model are dynamic programming methods, fixed point methods, and coordinate search methods for linear and convex network flows (e.g. [5, 6, 10, 9, 11, 7]) A partially asynchronous algorithm model assumes the existence of an upper bound on the communication delays; this has been the main computing model by which asynchronous gradienttype algorithms have been analyzed (see [11, Sec. 6.3.2] Partially asynchronous versions of instances of the CA ....

D. P. Bertsekas and J. Eckstein, Dual coordinate step methods for linear network flow problems, Math. Programming, 42 (1988), pp. 203--243.

.... may not improve the primal or the dual cost at any iteration, and they are based on a relaxed version of the CS conditions, called # complementary slackness (# CS for short) They have an excellent worst case computational complexity, when properly implemented, as shown in [Gol87] see also [BeE88], BeT89] GoT90] Their practical performance is also very good and they are well suited for parallel implementation (see [BCE95] LiZ91] NiZ93] We will extend two such methods, the # relaxation method and the auction sequential shortest path method, to the general convex cost case. One ....

....##, where 0 # 1. Based on these porperties, we can proceed to the second step of our analysis and prove a bound for the total number of price increases that the naive method can perform on any node. The proof is patterned after that for the # relaxation method for linear cost case [Ber86a] [BeE88]. A variant of the proof specialized to the convex cost # relaxation methods (and with # = 0.5) was presented in [BPT96] We present here the proof for the naive method for completeness. for the naive method satisfies K# CS together with some feasible flow vector x . Then, the naive method ....

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Bertsekas, D. P., and Eckstein, J., "Dual Coordinate Step Methods for Linear Network Flow Problems," Mathematical Programming, Vol. 42, 1988, pp. 203-243.

....log(NC) where C is the cost range C = max a ij , can be proved for the resulting method. The unscaled version of the method, where # is kept fixed at 1 (N 1) is pseudopolynomial. These complexity bounds can be derived using well known lines of analysis [Ber86b] BeE87] Gol87] [BeE88], BeT89] GoT90] and will not be proved here. 3. VARIATIONS AND EFFICIENT IMPLEMENTATION In this section we describe a number of variations of the algorithms of the preceding section, which we have empirically found to improve performance. Some of these variations are similar to corresponding ....

Bertsekas, D. P., and Eckstein, J., "Dual Coordinate Step Methods for Linear Network Flow Problems," Math. Programming, Series B, Vol. 42, 1988, pp. 203-243.

.... primal simplex and primal dual (or sequential shortest path) methods [Ber91] KeH80] PaS82] Roc84] In this paper we will focus on auction algorithms, first proposed in [Ber79] for both symmetric and asymmetric problems, and subsequently developed in several other papers [Ber85] Ber88] [BeE88], BCT91] The textbook [Ber91] contains an extensive discussion of these methods and their extensions to other network flow problems. Recent experimental evidence suggests that auction algorithms outperform their competitors by a substantial margin, particularly for sparse assignment problems ....

....[BCT91] The textbook [Ber91] contains an extensive discussion of these methods and their extensions to other network flow problems. Recent experimental evidence suggests that auction algorithms outperform their competitors by a substantial margin, particularly for sparse assignment problems [BeE88], Ber90] BCT91] and are also well suited for parallel computation [BeC89] KKZ90] PhZ88] WeZ90] WeZ91] Zak91] In the original proposal of the auction algorithm there is a price for each object, and at each iteration, one or more unassigned persons bid simultaneously for their best ....

Bertsekas, D. P., and Eckstein, J., "Dual Coordinate Step Methods for Linear Network Flow Problems," Math. Progr., Series B, Vol. 42, 1988, pp. 203-243.

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Bertsekas, D. P. and Eckstein, J. Dual coordinate step methods for linear network flow problems. Math. Programming, 42(2):203--244, 1988.

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