Optimizing the Sum of Linear Fractional Functions

Falk, James E.; Palocsay, Susan W.

doi:10.1515/9781400862528.221

Cited by 51 publications

(41 citation statements)

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“…In order to find a 1-link shortest path one needs to solve a number of optimization problems of the form Thus, the functions we try to optimize are 1-dimensional SOFs with generic term r i (x) = √ 1 + x 2 (a i /(b i x + c i )) rather than 1-dimensional SOLFs, where the generic term would have the form r i (x) = a i /(x + c i ). While in general one may not be able to apply the d-dimensional SOLF algorithm in [8] for a SOF problem, we will show below that this is possible for our objective function. Our choice of method is based on the results in [4], which show that the one-dimensional SOLF algorithm is very fast in practice.…”

Section: Optimal 1-links: a Sum Of Fractionals Approachmentioning

confidence: 97%

“…To compute 1-link shortest paths, we adapt an algorithm for minimizing a sum of linear fractional functions (SOLF) [8], to the sum of fractional functions (SOF) problem that describes an optimal 1-link path. In order to find a 1-link shortest path one needs to solve a number of optimization problems of the form Thus, the functions we try to optimize are 1-dimensional SOFs with generic term r i (x) = √ 1 + x 2 (a i /(b i x + c i )) rather than 1-dimensional SOLFs, where the generic term would have the form r i (x) = a i /(x + c i ).…”

Section: Optimal 1-links: a Sum Of Fractionals Approachmentioning

confidence: 99%

“…The second algorithm relies only on computation of (1 + /6)-approximate 1-link shortest paths and produces a 14k-link path whose weight is within factor (1 + ) of optimal. (3) We give a new (in this context) algorithm for computing 1-link shortest paths, based on solving a variant of the sum-of-linear-fractionals (SOLF) problem [8]. (4) We have implemented a system, based on the CORE library [5], for computing 1-link shortest paths.…”

Section: Introductionmentioning

confidence: 99%

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k-Link Shortest Paths in Weighted Subdivisions

Daescu

Mitchell

Ntafos

et al. 2005

Lecture Notes in Computer Science

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Abstract. We study the shortest path problem in weighted polygonal subdivisions of the plane, with the additional constraint of an upper bound, k, on the number of links (segments) in the path. We prove structural properties of optimal paths and utilize these results to obtain approximation algorithms that yield a path having O(k) links and weighted length at most (1 + ) times the weighted length of an optimal k-link path, for any fixed > 0. Some of our results make use of a new solution for the 1-link case, based on computing optimal solutions for a special sum-of-fractionals (SOF) problem. We have implemented a system, based on the CORE library, for computing optimal 1-link paths; we experimentally compare our new solution with a previous method for 1-link optimal paths based on a prune-and-search scheme.

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Section: Optimal 1-links: a Sum Of Fractionals Approachmentioning

confidence: 97%

Section: Optimal 1-links: a Sum Of Fractionals Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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k-Link Shortest Paths in Weighted Subdivisions

Daescu

Mitchell

Ntafos

et al. 2005

Lecture Notes in Computer Science

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“…Further, Matsui (1996) has shown that optimizing functions of this form over a polytope is an NP-hard problem. Problems of this form arise, for example, in multi-stage stochastic shipping problems where the objective is to maximize the profit earned per unit time (Falk and Palocsay 1992). For more applications, see the survey paper by Schaible and Shi (2003) and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…A common approach for solving these problems is to linearize the objective function by introducing a parameter for each ratio in the objective (see e.g. Falk and Palocsay (1992)). In contrast, our algorithm does not need to parametrize the objective function.…”

Section: Introductionmentioning

confidence: 99%

An FPTAS for optimizing a class of low-rank functions over a polytope

Mittal

Schulz

2012

Math. Program.

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We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of nonlinear functions of low rank over a polytope. Our approximation scheme relies on constructing an approximate Pareto-optimal front of the linear functions which constitute the given low-rank function. In contrast to existing results in the literature, our approximation scheme does not require the assumption of quasi-concavity on the objective function. For the special case of quasi-concave function minimization, we give an alternative FPTAS, which always returns a solution which is an extreme point of the polytope. Our technique can also be used to obtain an FPTAS for combinatorial optimization problems with non-linear objective functions, for example when the objective is a product of a fixed number of linear functions. We also show that it is not possible to approximate the minimum of a general concave function over the unit hypercube to within any factor, unless P = NP. We prove this by showing a similar hardness of approximation result for supermodular function minimization, a result that may be of independent interest. † Last updated on September 7, 2011.

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Optimal service and arrival rates in Jackson queueing networks

Bretthauer

2000

Naval Research Logistics

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In this paper we present an algorithm for solving a class of queueing network design problems. Specifically, we focus on determining both service and arrival rates in an open Jackson network of queueing stations. This class of problems has been widely studied and used in a variety of applications, but not well solved due to the difficulty of the resulting optimization problems. As an example, consider the classic application in computer network design which involves determining the minimum cost line capacities and flow assignments while satisfying a queueing performance measure such as an upper limit on transmission delay. Other application areas requiring the selection of both service and arrival rates in a network of queues include the design of communication, manufacturing, and health care systems. These applications yield optimization problems that are difficult to solve because typically they are nonconvex, which means they may have many locally optimal solutions that are not necessarily globally optimal. Therefore, to obtain a globally optimal solution, we develop an efficient branch and bound algorithm that takes advantage of the problem structure. Computational testing on randomly generated problems and actual problems from a health care organization indicate that the algorithm is able to solve realistic sized problems in reasonable computing time on a laptop computer. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 1–17, 2000

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Optimizing the Sum of Linear Fractional Functions

Cited by 51 publications

References 0 publications

k-Link Shortest Paths in Weighted Subdivisions

k-Link Shortest Paths in Weighted Subdivisions

An FPTAS for optimizing a class of low-rank functions over a polytope

Optimal service and arrival rates in Jackson queueing networks

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