On the dual coordinate ascent approach for nonlinear networks

Curet, Norman D.

doi:10.1016/0305-0548(93)90069-u

Cited by 3 publications

(4 citation statements)

References 9 publications

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“…The only hitherto published primal feasibility schemes are of this type. The scheme of Ventura [Ven91] is based on a minimum cost network flow problem; the scheme of Curet [Cur93] is a pure feasibility problem. We then provide two new feasibility schemes; the first is a combination of Ventura's and Curet's schemes, while the second is an improvement over Ventura's scheme.…”

Section: It Follows By the Continuity Of The Projection Operation Thamentioning

confidence: 99%

“…In Curet's [Cur93] scheme, the size of any element of δ is bounded by a constant ε > 0 which is also used in his dual algorithm, acting there as a relaxation parameter much like in ε-relaxation methods. His scheme ignores the objective function, and simply asks the vector δ to be such that…”

Section: Curet's Schemementioning

confidence: 99%

“…Curet uses an a priori given value of ε > 0, and therefore provides a primal feasible solution finitely. (This solution also has a guaranteed quality in terms of the value of φ, measured in terms of the value of ε; see [Cur93,Proposition 3.3]. ) Curet's overall algorithm however has the drawback that there is a trade-off between getting fast convergence of the dual algorithm and primal feasible solutions quickly (from using large values of ε), and getting good primal solutions (which requires a small value of ε).…”

Section: Curet's Schemementioning

confidence: 99%

“…Previously, only two Lagrangian heuristics for the problem SSCNFP have been presented, namely those of Ventura [Ven91] and Curet [Cur93]. Both are based on the solution of linear SSCNFPs, and they are not convergent in the sense that if they are applied from a sequence of Lagrangian dual vectors that converge to an optimal one, then the resulting primal vectors generally will not converge to an optimal primal flow vector; we, however, construct a convergent heuristic based on a combination of the two.…”

Section: Introductionmentioning

confidence: 99%

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Convergent Lagrangian heuristics for nonlinear minimum cost network flows

Larsson

Marklund

Olsson

et al. 2008

European Journal of Operational Research

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We consider the separable nonlinear and strictly convex single-commodity network flow problem (SSCNFP). We develop a computational scheme for generating a primal feasible solution from any Lagrangian dual vector; this is referred to as "early primal recovery". It is motivated by the desire to obtain a primal feasible vector before convergence of a Lagrangian scheme; such a vector is not available from a Lagrangian dual vector unless it is optimal. The scheme is constructed such that if we apply it from a sequence of Lagrangian dual vectors that converge to an optimal one, then the resulting primal (feasible) vectors converge to the unique optimal primal flow vector. It is therefore also a convergent Lagrangian heuristic, akin to those primarily devised within the field of combinatorial optimization but with the contrasting and striking advantage that it is guaranteed to yield a primal optimal solution in the limit. Thereby we also gain access to a new stopping criterion for any Lagrangian dual algorithm for the problem, which is of interest in particular if the SSCNFP arises as a subproblem in a more complex model. We construct instances of convergent Lagrangian heuristics that are based on graph searches within the residual graph, and therefore are efficiently implementable; in particular we consider two shortest path based heuristics that are based on the optimality conditions of the original problem. Numerical experiments report on the relative efficiency and accuracy of the various schemes.

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Section: It Follows By the Continuity Of The Projection Operation Thamentioning

confidence: 99%

Section: Curet's Schemementioning

confidence: 99%

Section: Curet's Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Convergent Lagrangian heuristics for nonlinear minimum cost network flows

Larsson

Marklund

Olsson

et al. 2008

European Journal of Operational Research

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A survey on the continuous nonlinear resource allocation problem

Patriksson

2008

European Journal of Operational Research

184

137

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Our problem of interest consists of minimizing a separable, convex and differentiable function over a convex set, defined by bounds on the variables and an explicit constraint described by a separable convex function. Applications are abundant, and vary from equilibrium problems in the engineering and economic sciences, through resource allocation and balancing problems in manufacturing, statistics, military operations research and production and financial economics, to subproblems in algorithms for a variety of more complex optimization models. This paper surveys the history and applications of the problem, as well as algorithmic approaches to its solution. The most common techniques are based on finding the optimal value of the Lagrange multiplier for the explicit constraint, most often through the use of a type of line search procedure. We analyze the most relevant references, especially regarding their originality and numerical findings, summarizing with remarks on possible extensions and future research.

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A partial linearization method for the traffic assignment problem

1993

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This paper presents a new solution technique for the traffic assignment problem. The approach is based on an iteratively improved nonlinear and separable approximation of the originally nonseparable objective function, and resembles the Frank-Wolfe algorithm in the sense that the subproblem separates with respect to commodities. Since the singlecommodity subproblems are strictly convex, the new algorithm will not suffer from the poor convergence behaviour of the Frank-Wolfe algorithm, which is a consequence of the extreme solutions of its linear subproblems. The solution method is outlined along with convergence results, and a dual approach to the solution of the strictly convex subproblems is described. The performance of the algorithm is illustrated with two numerical examples.

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On the dual coordinate ascent approach for nonlinear networks

Cited by 3 publications

References 9 publications

Convergent Lagrangian heuristics for nonlinear minimum cost network flows

Convergent Lagrangian heuristics for nonlinear minimum cost network flows

A survey on the continuous nonlinear resource allocation problem

A partial linearization method for the traffic assignment problem

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