Applications of Simplicial Decomposition with Nonlinear Column Generation to Nonlinear Network Flows

Larsson, Torbjörn; Patriksson, Michael; Rydergren, Clas

doi:10.1007/978-3-642-59179-2_17

Cited by 19 publications

(49 citation statements)

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“…As for the Frank-Wolfe method, there might also be a potential for enhancements of simplicial decomposition methods through the introduction of a nonlinear direction-finding subproblem. This was the original motivation for the work in [49] and the present one.…”

Section: Background and Motivationmentioning

confidence: 70%

“…As has been remarked upon already, the motivation for the work in [49] and the present one was the potential for improving the convergence of simplicial decomposition schemes through the use of nonlinear column generation subproblems. In [49] convergence is established for an exact version of the nonlinear simplicial decomposition method for the solution of CP , in which the column generation problem is assumed to be constructed through the approximation of f with a function Π of the form Π(x, y) = ∇f (x)…”

Section: Nonlinear Simplicial Decompositionmentioning

confidence: 88%

“…The origin of the work which led to the development of the generic column generation principle, to be presented in this paper, was the idea to generalize simplicial decomposition type methods to include nonlinear subproblems, the reason being that the use of subproblems arising from more accurate approximations of the original problem might enhance the overall performance of such methods. (Some numerical results for this generalization are presented in Larsson et al [49] as well as in [30].) A further motivation for this work was also the interest to extend the previous analysis to cover also non-differentiable convex objective functions.…”

Section: Background and Motivationmentioning

confidence: 99%

“…The choice t k = max{ t | x k + t(y k+1 − x k ) ∈ X }, which gives the largest augmentation, has in [49] been observed to yield a substantial improvement over the choice t k = 1 for some cases of nonlinear column generation problems CG(x).…”

Section: Remarksmentioning

confidence: 99%

“…In [49] convergence is established for an exact version of the nonlinear simplicial decomposition method for the solution of CP , in which the column generation problem is assumed to be constructed through the approximation of f with a function Π of the form Π(x, y) = ∇f (x)…”

Section: Nonlinear Simplicial Decompositionmentioning

confidence: 99%

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A generic column generation principle: derivation and convergence analysis

2015

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Given a non-empty, compact and convex set, and an a priori defined condition which each element either satisfies or not, we want to find an element belonging to the former category. This is a fundamental problem of mathematical programming which encompasses nonlinear programs, variational inequalities, and saddle-point problems.We present a conceptual column generation scheme, which alternates between solving a restriction of the original problem and a column generation phase which is used to augment the restricted problems. We establish the general applicability of the conceptual method, as well as to the three problem classes mentioned. We also establish a version of the conceptual method in which the restricted and column generation problems are allowed to be solved approximately, and of a version allowing for the dropping of columns.We show that some solution methods (e.g., Dantzig-Wolfe decomposition and simplicial decomposition) are special instances, and present new convergent column generation methods in nonlinear programming, such as a sequential linear programming (SLP) type method. Along the way, we also relate our quite general scheme in nonlinear programming presented in this paper with several other classic, and more recent, iterative methods in nonlinear optimization.

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Section: Background and Motivationmentioning

confidence: 70%

Section: Nonlinear Simplicial Decompositionmentioning

confidence: 88%

Section: Background and Motivationmentioning

confidence: 99%

Section: Remarksmentioning

confidence: 99%

Section: Nonlinear Simplicial Decompositionmentioning

confidence: 99%

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