Convergence Properties of Projection and Contraction Methods for Variational Inequality Problems

Xiu, Naihua; Wang, C.; Zhang, J.

doi:10.1007/s002450010023

Cited by 29 publications

(6 citation statements)

References 34 publications

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“…We have chosen to solve the saddle point formulation of the robust PCP using a projection contraction method because it is relatively easy to implement, uses little storage, and therefore it is an attractive alternative for solving large‐scale problems in general. (See for example and the references therein for an extensive discussion on the properties and advantages of projection type algorithms for solving variational inequalities.) Most of the recent papers in robust and distributionally robust optimization solve formulations with min‐max objective functions by first taking the dual of the inner problem as we have also discussed in formulations (3.9), (3.17) and (3.24).…”

Section: Saddle Point Methods For Project Crashingmentioning

confidence: 99%

Distributionally robust project crashing with partial or no correlation information

Natarajan

Shi

2019

Networks

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Crashing is shortening the project makespan by reducing activity times in a project network by allocating resources to them. Activity durations are often uncertain and an exact probability distribution itself might be ambiguous. We study a class of distributionally robust project crashing problems where the objective is to optimize the first two marginal moments (means and SDs) of the activity durations to minimize the worst‐case expected makespan. Under partial correlation information and no correlation information, the problem is solvable in polynomial time as a semidefinite program and a second‐order cone program, respectively. However, solving semidefinite programs is challenging for large project networks. We exploit the structure of the distributionally robust formulation to reformulate a convex‐concave saddle point problem over the first two marginal moment variables and the arc criticality index variables. We then use a projection and contraction algorithm for monotone variational inequalities in conjunction with a gradient method to solve the saddle point problem enabling us to tackle large instances. Numerical results indicate that a manager who is faced with ambiguity in the distribution of activity durations has a greater incentive to invest resources in decreasing the variations rather than the means of the activity durations.

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Section: Saddle Point Methods For Project Crashingmentioning

confidence: 99%

Distributionally robust project crashing with partial or no correlation information

Natarajan

Shi

2019

Networks

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“…Among numerous effective numerical algorithms for solving VI, especially LVI, one famous one is the projectioncontraction (PC) method which was originally proposed by Uzawa [46]. The attractive characteristics of the PC method, for example, simpleness and effectiveness, have motivated further development on VI especially in computational aspects; see, for example, [39,[47][48][49]. In this section, we will summarize some concepts and results about linear variational inequalities and then adopt the projection-contraction method in [48] for solving LVI (21)- (22).…”

Section: A Projection-contraction Methods For LVI (21)-(22)mentioning

confidence: 99%

A Heuristic Algorithm for Constrained Multi-Source Location Problem with Closest Distance under Gauge: The Variational Inequality Approach

Jiang

Assani

Cheng

et al. 2013

Abstract and Applied Analysis

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This paper considers the locations of multiple facilities in the space , with the aim of minimizing the sum of weighted distances between facilities and regional customers, where the proximity between a facility and a regional customer is evaluated by the closest distance. Due to the fact that facilities are usually allowed to be sited in certain restricted areas, some locational constraints are imposed to the facilities of our problem. In addition, since the symmetry of distances is sometimes violated in practical situations, the gauge is employed in this paper instead of the frequently used norms for measuring both the symmetric and asymmetric distances. In the spirit of the Cooper algorithm (Cooper, 1964), a new location-allocation heuristic algorithm is proposed to solve this problem. In the location phase, the single-source subproblem with regional demands is reformulated into an equivalent linear variational inequality (LVI), and then, a projection-contraction (PC) method is adopted to find the optimal locations of facilities, whereas in the allocation phase, the regional customers are allocated to facilities according to the nearest center reclassification (NCR). The convergence of the proposed algorithm is proved under mild assumptions. Some preliminary numerical results are reported to show the effectiveness of the new algorithm.

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“…Although the amount of computation per iteration for Algorithm PC is less than for Algorithms FM and IFM, the results in Table 6 show that Algorithm PC is not suitable for solving large dense linear inequality systems. The main reason of its slow convergence is that the iteration sequence {(x k , z k )} generated by Algorithm PC may not be contained in the feasible set H in (1.13), so it is called an infeasible PC method in [28]. As shown in Tables 3,4,5, almost identical numbers of iteration are required by Algorithms FM and IFM in most cases, but less CPU time is consumed by Algorithm IFM because of its lower computation amount per iteration, and the facts are also confirmed by the figures in Table 6.…”

Section: Examplementioning

confidence: 99%

The inexact fixed matrix iteration for solving large linear inequalities in a least squares sense

Yuan

2014

Numer Algor

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A fixed matrix iteration algorithm was proposed by A. Dax (Numer. Algor. 50, 97-114 2009) for solving linear inequalities in a least squares sense. However, a great deal of computation for this algorithm is required, especially for large-scale problems, because a least squares subproblem should be solved accurately at each iteration. We present a modified method, the inexact fixed iteration method, which is a generalization of the fixed matrix iteration method. In this inexact iteration process, the classical LSQR method is implemented to determine an approximate solution of each least squares subproblem with less computational effort. The convergence of this algorithm is analyzed and several numerical examples are presented to illustrate the efficiency of the inexact fixed matrix iteration algorithm for solving large linear inequalities.

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Convergence Properties of Projection and Contraction Methods for Variational Inequality Problems

Cited by 29 publications

References 34 publications

Distributionally robust project crashing with partial or no correlation information

Distributionally robust project crashing with partial or no correlation information

A Heuristic Algorithm for Constrained Multi-Source Location Problem with Closest Distance under Gauge: The Variational Inequality Approach

The inexact fixed matrix iteration for solving large linear inequalities in a least squares sense

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