Quadratic programming and combinatorial minimum weight product problems

We present a fully polynomial time approximation scheme (FPTAS) for minimizing an objective (a T x + ¿)(b T x + d) under linear constraints A x = d. Examples of such problems are combinatorial minimum weight product problems such as the following: given a graph G = (V,E) and two edge weights find an s - t path P that minimizes a(P)b(P), the product of its edge weights relative to a and b.