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Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization

Thomas L. Magnanti(magnanti***at***mit.edu)
Dan Stratila(dstrat***at***rci.rutgers.edu)

Abstract: We study the problem of minimizing a nonnegative separable concave function over a compact feasible set. We approximate this problem to within a factor of 1+epsilon by a piecewise-linear minimization problem over the same feasible set. Our main result is that when the feasible set is a polyhedron, the number of resulting pieces is polynomial in the input size of the polyhedron and linear in 1/epsilon. For many practical concave cost problems, the resulting piecewise-linear cost problem can be formulated as a well-studied discrete optimization problem. As a result, a variety of polynomial-time exact algorithms, approximation algorithms, and polynomial-time heuristics for discrete optimization problems immediately yield fully polynomial-time approximation schemes, approximation algorithms, and polynomial-time heuristics for the corresponding concave cost problems. We illustrate our approach on two problems. For the concave cost multicommodity flow problem, we devise a new heuristic and study its performance using computational experiments. We are able to approximately solve significantly larger test instances than previously possible, and obtain solutions on average within 4.27% of optimality. For the concave cost facility location problem, we obtain a new 1.4991+epsilon approximation algorithm.


Category 1: Combinatorial Optimization

Category 2: Network Optimization


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Entry Submitted: 01/15/2012
Entry Accepted: 01/15/2012
Entry Last Modified: 01/15/2012

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