Computing safe dike heights at minimal costs

This is a technical report that belongs to the RIZA project Analyse complex besliskundig model Optimale Veiligheid Overstromen (RI-4772). It is an appendix of reference [3], which is the report of phase I of the mentioned project. Below we briefly sketch the contents of this technical report. This report starts in the next section (Section 2) with the notations that are used in the athematical model of a dike ring. We discuss the investment costs and expected damage costs during the planning period. Since our aim is to minimize (the sum of) these costs we pay much attention to the question whether or not these costs are convex in the decision variables. The reason for this is that convex optimization problems have the nice property that every local minimum is also a global minimum. When a problem is not convex it is usually much harder to solve, and a solver may generate a solution that is local optimal, but not global optimal; it is then hard to decide if the given solution is global or not. In Section 3 we present several optimization models for solving the dike ring problem. Section 4 is devoted to the case where a dike ring is not homogeneous, but consists of several segments, each with their own characteristics. Computational results are reported only for the case of a homogeneous dike ring, in Appendix B. These results were obtained by using the dynamic programming approach, as discussed in Section 3.4. They are compared with solutions generated by the package OptimaliseRing (version 1.1), currently in use by RIZA. There are two other Appendices. Appendix A contains some (non)convexity results for the exponential investment cost function. In Appendix C we deal with the case of infinite time horizon, and it is investigated if there exists a periodical solution which is optimal. As such, this Appendix tries to answer an intriguing question that was one of the tasks in phase I of the current project, namely if a solution proposed by C.J.J. Eijgenraam is a global optimal solution or not. Unfortunately, this question has yet been answered only partially.