Efficient minimization over products of simplices and its application to nonlinear multicommodity network problems

Karakitsiou, Athanasia; Mavrommati, Athanasia; Migdalas, Athanasios

doi:10.1007/bf02943605

Cited by 6 publications

(3 citation statements)

References 14 publications

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“…[This subproblem is equivalent to that of the regularized Frank-Wolfe method ( [63,42]), and a special case of that of the cost approximation method ( [77,74,76,75]). ] With the appropriate choices of the function ϕ, multi-dimensional search versions of gradient projection and Newton-type methods, for example, are possible to construct.…”

Section: Nonlinear Simplicial Decompositionmentioning

confidence: 99%

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: Nonlinear Simplicial Decompositionmentioning

confidence: 99%

A generic column generation principle: derivation and convergence analysis

2015

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“…However, the solution of the restricted master problems does require the application of, in general, infinitely converging algorithms, several of which have been proposed in the literature, including projected second-order methods such as Newton methods (Bertsekas 1982b;Bertsekas and Gafni 1983;Bertsekas et al (1984)), reduced gradient methods (Larsson and Patriksson 1992;Patriksson 2001), and modified Frank-Wolfe algorithms. In Damberg and Migdalas (1997) and Karakitsiou et al (2005) the restricted master problems are solved by applying a regularized Frank-Wolfe approach of Subsect. 3.2.1.…”

Section: Simplicial Decompositionmentioning

confidence: 99%

“…3.2.1. Computational results in Karakitsiou et al (2005) show that the overall approach is about 50 % more efficient when regularized Frank-Wolfe is used to solve the master problem in DSD instead of Wolfe's reduced gradient algorithm [eg. Bazaraa and Shetty (1979)] employed by Larsson and Patriksson (1992).…”

Section: Simplicial Decompositionmentioning

confidence: 99%

Convex optimization problems in supply chain planning and their solution by a column generation method based on the Frank Wolfe method

Karakitsiou

Migdalas

2015

Oper Res Int J

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Many problems in supply chain optimization concern the minimization of a differentiable convex objective function subject to a set of linear constraints. The aim of this work is to present a number of such problems and to propose an efficient method for their solution. The proposed method is based on improvements of the well known Frank-Wolfe algorithm. The computational results of the proposed algorithm demonstrate its effectiveness and efficiency.

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Nonlinear Programming in Telecommunications

Migdalas

Handbook of Optimization in Telecommunications

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Efficient minimization over products of simplices and its application to nonlinear multicommodity network problems

Cited by 6 publications

References 14 publications

A generic column generation principle: derivation and convergence analysis

A generic column generation principle: derivation and convergence analysis

Convex optimization problems in supply chain planning and their solution by a column generation method based on the Frank Wolfe method

Nonlinear Programming in Telecommunications

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