The aim of this paper is two-fold. First, the so-called ‘optimal level solutions’ method is described in a new unifying framework with the aim to provide an algorithmic scheme able to approach various different classes of problems. Then, the ‘optimal level solutions’ method is used to solve a class of low-rank programmes involving linear and quadratic functions and having a polyhedral feasible region. In particular, the considered class of programmes covers, among all, rank-three d.c., multiplicative and fractional programmes. Some optimality conditions are used to improve the performance of the proposed algorithm.