Iterative methods for variational and complementarity problems

Pang, Jong Shi; Chan, Dorian

doi:10.1007/bf01585112

Cited by 265 publications

(83 citation statements)

References 28 publications

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“…summarizes the traffic in the network as generated by the users and the parameters k i of the user objective. The user traffic rates are coupled through the constraint of the form N i=1 A li x i ≤ C l for all l ∈ L , where C l is the maximum aggregate traffic through link l. The constraint can be compactly written as Ax ≤ C, where C is the link capacity vector and is given by C = (10,15,20,10,15,20,20,15,25).…”

Section: A Regularized Dualmentioning

confidence: 99%

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Multiuser Optimization: Distributed Algorithms and Error Analysis

2011

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Abstract. Traditionally, a multiuser problem is a constrained optimization problem characterized by a set of users, an objective given by a sum of user-specific utility functions, and a collection of linear constraints that couple the user decisions. The users do not share the information about their utilities, but do communicate values of their decision variables. The multiuser problem is to maximize the sum of the users-specific utility functions subject to the coupling constraints, while abiding by the informational requirements of each user. In this paper, we focus on generalizations of convex multiuser optimization problems where the objective and constraints are not separable by user and instead consider instances where user decisions are coupled, both in the objective and through nonlinear coupling constraints. To solve this problem, we consider the application of gradient-based distributed algorithms on an approximation of the multiuser problem. Such an approximation is obtained through a Tikhonov regularization and is equipped with estimates of the difference between the optimal function values of the original problem and its regularized counterpart. In the algorithmic development, we consider constant steplength primal-dual and dual schemes in which the iterate computations are distributed naturally across the users, i.e., each user updates its own decision only. Convergence in the primal-dual space is provided in limited coordination settings, which allows for differing steplengths across users as well as across the primal and dual space. We observe that a generalization of this result is also available when users choose their regularization parameters independently from a prescribed range. An alternative to primal-dual schemes can be found in dual schemes which are analyzed in regimes where approximate primal solutions are obtained through a fixed number of gradient steps. Per-iteration error bounds are provided in such regimes and extensions are provided to regimes where users independently choose their regularization parameters. Our results are supported by a case-study in which the proposed algorithms are applied to a multi-user problem arising in a congested traffic network.

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Section: A Regularized Dualmentioning

confidence: 99%

“…Related is also the literature on centralized projection-based methods for optimization (see for example books [3,8,21]) and variational inequalities [8,[10][11][12]20,24]. Recently, efficient projection-based algorithms have been developed in [1,2,18,26] for optimization, and in [17,19] for variational inequalities.…”

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confidence: 99%

Multiuser Optimization: Distributed Algorithms and Error Analysis

2011

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“…We leave the issue of computing x(t) and z(t) open (see [2,5,20,32] Hence, by the definition of c and 0, :ll(t)ll 2 > I$I(t+l)1 2 -c(t) 2 -11Ax(t)11 2 + 2c(t)cFllx(t)11 2 + c(t) 2 11BA(t)ll 2 > lI(t+1)112 + c(t)(2a-c(t)p(ATA))Il2(t)112 + c(t) 2 11B2(t)11 2 .…”

Section: Application To Variational Inequalitiesmentioning

confidence: 99%

Applications of a Splitting Algorithm to Decomposition in Convex Programming and Variational Inequalities

Tseng¹

1991

SIAM J. Control Optim.

410

337

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Recently Han and Lou [18] proposed a highly parallelizable decomposition algorithm for convex programming involving strongly convex costs. We show in this paper that their algorithm, as well as the method of multipliers [17,19,34] and the dual gradient method [8,40], are special cases of a certain multiplier method for separable convex programming. This multiplier method is similar to the alternating direction method of multipliers [10,15] but uses both Lagrangian and augmented Lagrangian functions. We also apply this method to symmetric linear complementarity problems to obtain a new class of matrix splitting algorithms. Finally, we show that this method is itself a dual application of an algorithm of Gabay [12] for finding a zero of the sum of two maximal monotone operators. We give an extension of Gabay's algorithm that allows dynamic stepsizes and show that, under certain conditions, it has a linear rate of convergence. We also apply this algorithm to variational inequalities.

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“…where M is the minimum cost of travel between o and n. Alternately, a diagonalisation algorithm referred to as the nonlinear Jacobi method has been used by many authors including Abduaal and LeBlanc (1979), Ahn (1979), Florian and Spiess (1982), Dafermos (1982), and Pang and Chan (1982) among others. In a diagonalisation algorithm, at each iteration n, diagonalisation is performed such that all the flows on links other than the flow on current link, v are fixed.…”

Section: Satisfaction Of Conditions For Good Behaviourmentioning

confidence: 99%

Mathematical programming algorithms for equilibrium road traffic assignment

1997

Transportation Research Part A: Policy and Practice

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The equilibrium approach to representing interactions between the supply and demand sides of traffic assignment has been used widely in the estimation of traffic flows on road networks. Although this approach is quite reasonable, there is a considerable gap between the observed and modelled values of cost and flow. This gap can be reduced by relaxing some of the restrictive assumptions behind the models used in order to enhance their realism.This study investigates the solutions of various advanced road traffic assignment models. Priority and signal controlled junctions are modelled in traffic assignment in order to enhance the realism of junction analysis. A multiclass assignment is modelled to represent different groups of users. These problems are known to be non-separable because traffic cannot be segmented in such a way that the costs incurred by any one segment vary only with the flow within that segment. Existence, uniqueness and stability properties of solutions to these problems are investigated. These analyses are important to know the reliability and repeatability of any solutions that are calculated. Analyses of these properties lead to some guidelines for using these detailed models. A number of new solution algorithms are developed to solve the resulting traffic assignment problems. These algorithms belong to the general category of simplicial decomposition which solves the problem by dividing it into two subproblems: a linear and a master subproblem which are solved alternately. One of the advantages of these algorithms is that they operate in a lower dimensional space than that of original feasible region and hence allow large-scale problems to be solved with improved accuracy and speed of convergence. These improved algorithms give many choices to the traffic management studies.Two substantial networks have been used to compare the performance of new algorithms on the various models developed. They have performed favourably by comparison with existing algorithms. A small example network has been used to investigate existence, uniqueness and stability properties using the models. In a priority controlled model, a unique stable solution has been obtained using the model whilst in a signal controlled model, multiple and unstable solutions have been obtained. In a multiclass model, a unique solution has been obtained in terms of the total class flow whilst multiple solutions have been obtained in terms of each class flow. These results correspond well to the theoretical analyses of these models, which has shown to have indeterminate behaviour and by the nature of these models assumed, the degree of non-separability is ordered according to priority controlled, multiclass and signal controlled models. 3 ACKNOWLEDGEMENTS

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Iterative methods for variational and complementarity problems

Cited by 265 publications

References 28 publications

Multiuser Optimization: Distributed Algorithms and Error Analysis

Multiuser Optimization: Distributed Algorithms and Error Analysis

Applications of a Splitting Algorithm to Decomposition in Convex Programming and Variational Inequalities

Mathematical programming algorithms for equilibrium road traffic assignment

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