A parallelizable augmented Lagrangian method applied to large-scale non-convex-constrained optimization problems

Boland, Natashia; Christiansen, Jeffrey H.; Dandurand, Brian; Eberhard, Andrew; Oliveira, Fabrício

doi:10.1007/s10107-018-1253-9

Cited by 25 publications

(17 citation statements)

References 43 publications

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“…After uncovering a collection of theoretical properties on the asymptotic behavior of the inner PCG procedure for MCCFBN problems, we provide extensive computational experiments with linear and quadratic instances of up to one billion arcs and 200 and five million nodes in each subset of the node partition. This is in line with the the growing effort to address the design of scalable algorithmic solutions for large-scale optimization [5,9,12,13]. In fact, the presented results provide both a theoretical and computational support of the scalable property of the the specialized IPM for MCCFBN problems.…”

Section: Mathematics Subject Classificationsupporting

confidence: 79%

“…For the computational executions we considered the following algorithms implemented in CPLEX 12. 5 While the barrier algorithm can be applied both to linear and (quadratic) convex instances, the remaining solvers are only applied to linear instances. Thus, the barrier algorithm can be regarded as the current benchmark for general MCCFBN problems.…”

Section: Computational Experimentsmentioning

confidence: 99%

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A specialized interior-point algorithm for huge minimum convex cost flows in bipartite networks

Castro

Nasini

2021

European Journal of Operational Research

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The computation of the Newton direction is the most time consuming step of interior-point methods. This direction was efficiently computed by a combination of Cholesky factorizations and conjugate gradients in a specialized interior-point method for block-angular structured problems. In this work we apply this algorithmic approach to solve very large instances of minimum cost flows problems in bipartite networks, for convex objective functions with diagonal Hessians (i.e., either linear, quadratic or separable nonlinear objectives). After analyzing the theoretical properties of the interior-point method for this kind of problems, we provide extensive computational experiments with linear and quadratic instances of up to one billion arcs and 200 and five million nodes in each subset of the node partition. For linear and quadratic instances our approach is compared with the barriers algorithms of CPLEX (both standard path-following and homogeneous-self-dual); for linear instances it is also compared with the different algorithms of the state-of-the-art network flow solver LEMON (namely: network simplex, capacity scaling, cost scaling and cycle canceling). The specialized interior-point approach significantly outperformed the other approaches in most of the linear and quadratic transportation instances tested. In particular, it always provided a solution within the time limit and it never exhausted the 192 Gigabytes of memory of the server used for the runs. For assignment problems the network algorithms in LEMON were the most efficient option.

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Section: Mathematics Subject Classificationsupporting

confidence: 79%

Section: Computational Experimentsmentioning

confidence: 99%

A specialized interior-point algorithm for huge minimum convex cost flows in bipartite networks

Castro

Nasini

2021

European Journal of Operational Research

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“…For example, the Hestenes-Powell-Rockafellar augmented Lagrangian [1][2][3], the cubic augmented Lagrangian [4], Mangasarian's augmented Lagrangian [5,6], the exponential penalty function [7,8], the log-sigmoid Lagrangian [9], modified barrier functions [8,10], the p-th power augmented Lagrangian [11], and nonlinear augmented Lagrangian functions [12][13][14][15]. The other related discussion on augmented Lagrangians regarding special constrained optimization includes second-order cone programming [16,17], semidefinite programming [18][19][20], cone programming [21][22][23], semi-infinite programming [24,25], min-max programming [26], distributed optimization [27], mixed integer programming [28], stochastic mixed-integer programs [29], generalized Nash equilibrium problems [30], quasi-variational inequalities [31], composite convex programming [32], and sparse discrete problems [33]. The duality theory is closely related to the perturbation of primal problem.…”

Section: Introductionmentioning

confidence: 99%

“…Augmented Lagrange multipliers are an important concept in duality theory. Their existence is important for the global convergence analysis of primal-dual type algorithms based on the use of augmented Lagrangians [7,19,29,32,33]. In addition, augmented Lagrange multipliers are closely related to saddle points, the zero duality gap property, and exact penalty representation.…”

Section: Introductionmentioning

confidence: 99%

Existence of Generalized Augmented Lagrange Multipliers for Constrained Optimization Problems

Wang

Zhou

Tang

2020

MCA

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The augmented Lagrange multiplier as an important concept in duality theory for optimization problems is extended in this paper to generalized augmented Lagrange multipliers by allowing a nonlinear support for the augmented perturbation function. The existence of generalized augmented Lagrange multipliers is established by perturbation analysis. Meanwhile, the relations among generalized augmented Lagrange multipliers, saddle points, and zero duality gap property are developed.

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“…Although the relaxed problem is convex, the resulting Lagrangian relaxation subproblem is usually nonconvex and as difficult to solve as the original problem. To obtain a tighter relaxation with convex subproblems, augmented Lagrangian relaxation [9,36] can be used, at the expense of introducing nonlinear terms. Moreover, both traditional Lagrangian relaxation and the augmented version usually result in nonsmooth problems.…”

Section: Introductionmentioning

confidence: 99%

Enhancing the normalized multiparametric disaggregation technique for mixed-integer quadratic programming

et al. 2018

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We propose methods for improving the relaxations obtained by the normalized multiparametric disaggregation technique (NMDT). These relaxations constitute a key component for some methods for solving nonconvex mixed-integer quadratically constrained quadratic programming (MIQCQP) problems. It is shown that these relaxations can be more efficiently formulated by significantly reducing the number of auxiliary variables (in particular, binary variables) and constraints. Moreover, a novel algorithm for solving MIQCQP problems is proposed. It can be applied using either its original NMDT or the proposed reformulation. Computational experiments are performed using both benchmark instances from the literature and randomly generated instances. The numerical results suggest that the proposed techniques can improve the quality of the relaxations.

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A parallelizable augmented Lagrangian method applied to large-scale non-convex-constrained optimization problems

Cited by 25 publications

References 43 publications

A specialized interior-point algorithm for huge minimum convex cost flows in bipartite networks

A specialized interior-point algorithm for huge minimum convex cost flows in bipartite networks

Existence of Generalized Augmented Lagrange Multipliers for Constrained Optimization Problems

Enhancing the normalized multiparametric disaggregation technique for mixed-integer quadratic programming

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