Optimising in many-objective search spaces, i.e. search spaces with more than three objectives, is a challenging task. Scalarization functions transform the multi-objective search space into a single objective search space. In order to scale-up optimisation in many-objective search spaces, we use Cartesian product of scalarization functions or simpler product functions to reduce the number of objectives of the search space. Stochastic product local search (SprLS) uses product functions to evaluate solutions within a local search run with the goal of generating the entire Pareto front. To improve the performance of SprLS algorithms: 1) we pursuit a fixed set the product function that improves the most the performance of the algorithm, or 2) we adapt the direction of the component scalarization functions using solutions in the current (possible suboptimal) Pareto front. We compare the performance of local search algorithms on many-objective quadratic assignment instances with correlated flow matrices.