Given an optimization problem, combining knowledge from both (i) structural or algorithmic known results and (ii) new solving techniques, helps gain insight and knowledge on the aforementioned problem by tightening the gap between lower and upper bounds on the sought optimal value. Additionally, this gain may be further improved by iterating (i) and (ii) until a fixed point is reached. In this paper, we illustrate the above through the classical Cyclic Bandwidth problem, an optimization problem which takes as input an undirected graph G = (V, E) with |V | = n, and asks for a labeling φ of V in which every vertex v takes a unique value φ(v) ∈ [1; n], in such a way that Bc (G, φ) = ax{minuv∈E(G) {|φ(u)−φ(v)|, n−|φ(u)−φ(v)|}}, called the cyclic bandwidth of G, is minimized. Using the classic benchmark from the Harwell-Boeing sparse matrix collection, we show how to combine (i) previous results from the Cyclic Bandwidth literature, and (ii) new solving techniques, which we first present, and then implement, starting from the best results obtained in step (i). We show that this process allows us to determine the optimal cyclic bandwidth value for half of the instances of our benchmark, and improves the best known bounds for a large number of the remaining instances.