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

Abstract

We present a fully polynomial time approximation scheme (FPTAS) for optimizing a very general class of non-linear 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.

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Correspondence to Shashi Mittal.

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Mittal, S., Schulz, A.S. An FPTAS for optimizing a class of low-rank functions over a polytope. Math. Program. 141, 103–120 (2013). https://doi.org/10.1007/s10107-011-0511-x

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Keywords

  • Non-convex optimization
  • Combinatorial optimization
  • Approximation schemes

Mathematics Subject Classification (2000)

  • 90C20
  • 90C26
  • 90C27
  • 90C29