Complexity and algorithms for nonlinear optimization problems

Hochbaum, Dorit S.

doi:10.1007/s10479-007-0172-6

Cited by 86 publications

(64 citation statements)

References 61 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Possible future work is to investigate more on different cost models to account for economies of scale or other effects. There has been some work on complexity and algorithms for network flow problem with nonlinear cost functions (,e.g., [6], [2] ) but we think that this is worth extending. While the resulting problems are mostly hard, it would still be interesting to devise tailored algorithms for practical use.…”

Section: Discussionmentioning

confidence: 99%

Network Flow Optimization with Minimum Quantities

Seedig

2011

Operations Research Proceedings

View full text Add to dashboard Cite

Economies of scale often appear as decreasing marginal costs, but may also occur as a constraint that at least a minimum quantity should be produced or nothing at all. Taking the latter into account can support decisions about keeping a production site, setting up a new one, establishing new routes in logistics network, and many more. In our work, we define the corresponding problem Minimum Cost Network Flow with Minimum Quantities and prove its computational hardness. Following that, we devise a tailored Branch-&-Bound algorithm with an efficient update step that usually affects only small parts of the network. Experiments are conducted on real problems and artificial networks. In our results, the update step made the inevitable but well-studied subproblem of the initial computation of a minimum cost network flow problem taking most of the actual total running time. As our method does also compare favorably with a heuristic algorithm that has been implemented before, we recommend it for practical use.

show abstract

Section: Discussionmentioning

confidence: 99%

Network Flow Optimization with Minimum Quantities

Seedig

2011

Operations Research Proceedings

View full text Add to dashboard Cite

show abstract

“…Given these γ jj ′ values, theγ jj ′ values can be computed by the Floyd-Warshall algorithm and the optimal ∆ obtained by (17) as for the linear case. It can be verified that the algorithm is strongly polynomial the same way as for the linear case.…”

Section: Fisher's Market With Spending Constraint Utilitiesmentioning

confidence: 99%

“…in [16,4,17,12,34]). The survey paper [17] gives an overview of special cases solvable in strongly polynomial time. These include a fixed number of suppliers (Cosares and Hochbaum,[4]), and series-parallel graphs (Tamir [34]).…”

Section: Introductionmentioning

confidence: 99%

A Strongly Polynomial Algorithm for a Class of Minimum-Cost Flow Problems with Separable Convex Objectives

Végh¹

2016

SIAM J. Comput.

View full text Add to dashboard Cite

A well-studied nonlinear extension of the minimum-cost flow problem is to minimize the objective ij∈E C ij (f ij ) over feasible flows f , where on every arc ij of the network, C ij is a convex function. We give a strongly polynomial algorithm for the case when all C ij 's are convex quadratic functions, settling an open problem raised e.g. by Hochbaum [16]. We also give strongly polynomial algorithms for computing market equilibria in Fisher markets with linear utilities and with spending constraint utilities, that can be formulated in this framework (see Shmyrev [33], Devanur et al. [2]). For the latter class this resolves an open question raised by Vazirani [37]. The running time is O(m 4 log m) for quadratic costs, O(n 4 + n 2 (m+ n log n) log n) for Fisher's markets with linear utilities and O(mn 3 + m 2 (m + n log n) log m) for spending constraint utilities.All these algorithms are presented in a common framework that addresses the general problem setting. Whereas it is impossible to give a strongly polynomial algorithm for the general problem even in an approximate sense (see Hochbaum [16]), we show that assuming the existence of certain black-box oracles, one can give an algorithm using a strongly polynomial number of arithmetic operations and oracle calls only. The particular algorithms can be derived by implementing these oracles in the respective settings.

show abstract

“…, n. The resulting Problem PosHPKr, belongs to a general area of Quadratic Programming in which it is required to optimize a quadratic function subject to linear constraints. See Hochbaum (2005Hochbaum ( , 2008 for detailed reviews of quadratic optimization with integer and continuous variables.…”

Section: Continuous Relaxation and Constant-ratio Approximationmentioning

confidence: 99%