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

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

A simple proof of a primal affine scaling method

In this paper, we present a simpler proof of the result of Tsuchiya and Muramatsu on the convergence of the primal affine scaling method. We show that the primal sequence generated by the method converges to the interior of the optimum face and the dual sequence to the analytic center of the optimal dual face, when the step size implemented in the procedure is bounded by 2/3. We also prove the optimality of the limit of the primal sequence for a slightly larger step size of 2 q /(3 q −1), where q is the number of zero variables in the limit. We show this by proving the dual feasibility of a cluster point of the dual sequence.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44263/1/10479_2005_Article_BF02206821.pd