Fierce competition in today's global market forces industrial companies to better design and manage their supply chain networks. In particular, making the right decisions regarding one of the core supply chain processes, goods production, directly affects the productivity and hence the competitiveness of a company. Industrial production management involves, among others, deciding about which products should be made, when and in which quantity. Despite its rather simple definition, production planning is most often a complex task for industrial managers who can be overwhelmed by the complexity of the problem. This is particularly the case when production planning involves lot-sizing and scheduling decisions. This arises whenever start-up operations such as tool changes are required between production runs of different products on a machine. In this situation, finding the right quantity to produce after a start-up, i.e. the lot size, requires reaching a good trade-off between start-up costs (indicating large lot sizes) and inventory holding costs (indicating small lot sizes) . Lot-sizing and scheduling leads to the formulation of difficult combinatorial optimization problems. A wide variety of solution techniques from the Operations Research field have been proposed to solve them. The scientific challenge here is to develop optimization methods in which the production system is modelled with the required accuracy and which are capable of providing guaranteed optimal or near-optimal production plans within reasonable computation times. In this context, project LotRelax focuses on: (i) improving the production system representation in the optimization method by taking into account a complicating feature frequently encountered in practice: the presence of sequence-dependent start-up costs and times, (ii) computing guaranteed optimal or near-optimal production plans by exploiting recent theoretical advances in the mathematical programming field, in particular the latest developments in semidefinite programming. Semidefinite programming can be broadly described as the extension of linear programming from the space of real vectors to the space of symmetric matrices. This rather new area of mathematical programming has witnessed important developments during the last twenty years and has proved successful at solving prominent difficult combinatorial optimization problems. However, as can be seen from recent reviews in the academic literature, there seems to be no previous attempt at using semidefinite relaxations to solve lot-sizing and scheduling problem. The main objective of project LotRelax is thus to develop solution approaches based on semidefinite relaxations to solve several variants of lot-sizing problems involving sequence-dependent start-up costs and times.