Dike height optimization is of major importance to the Netherlands as a large part of the country lies below sea level and high water levels in rivers can cause floods. A cost-benefit analysis is discussed in Eijgenraam et al. (2010), which is an improvement of the model Van Dantzig (1956) introduced after a devastating flood in the Netherlands in 1953. We consider the extension of this model to nonhomogeneous dike rings, which may also be applicable to other deltas in the world. A nonhomogeneous dike ring consists of different segments with different characteristics with respect to flooding and investment costs. The individual segments can be heightened independently at different moments in time and by different amounts, making the problem considerably more complex than the homogeneous case. We show how the problem can be modeled as a MINLP problem and present an iterative algorithm that can be used to solve the problem. Moreover, we consider a robust optimization approach to deal with uncertainty in the model parameters. The method has been implemented and integrated in software, which is used by the government to determine how the safety standards in the Dutch Water Act should be changed.

In the Netherlands, flood protection is a matter of national survival. In 2008, the Second Delta Committee recommended increasing legal flood protection standards at least tenfold to compensate for population and economic growth since 1953; this recommendation would have required dike improvement investments estimated at 11.5 billion euro. Our research group was charged with developing efficient flood protection standards in a more objective way. We used cost-benefit analysis and mixed-integer nonlinear programming to demonstrate the efficiency of increasing the legal standards in three critical regions only. Monte Carlo analysis confirms the robustness of this recommendation. In 2012, the state secretary of the Ministry of Infrastructure and the Environment accepted our results as a basis for legislation. Compared to the earlier recommendation, this successful application of operations research yields both a highly significant increase in protection for these regions (in which two-thirds of the benefits of the proposed improvements accrue) and approximately 7.8 billion euro in cost savings. Our methods can also be used in decision making for other flood-prone areas worldwide.

Flood prevention policy is of major importance to the Netherlands since a large part of the country is below sea level and high water levels in rivers may also cause floods. In this paper we propose a dike height optimization model to determine economically efficient flood protection standards. We improve the model proposed by David van Dantzig [van Dantzig D (1956) Economic decision problems for flood prevention. Econometrica 24(3):276–287] after a devastating flood in the Netherlands in 1953. Our model is nonconvex, but we derive an explicit simple expression for the global optimal solution, which is periodic. We also discuss how to use this optimal investment policy to derive an efficient flood protection standard. The rather simple expression for this standard gives us much insight into how it depends on several relevant economic and climate model parameters. This approach has been applied to all dike rings in the Netherlands, and the resulting standards have been stated in the new Delta Programme 2015, which has been accepted by the government. This paper was accepted by Yinyu Ye, optimization.

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Robust optimization is a methodology that can be applied to problems that are affected by uncertainty in the problem's parameters. The classical robust counterpart (RC) of the problem requires the solution to be feasible for all uncertain parameter values in a so-called uncertainty set, and offers no guarantees for parameter values outside this uncertainty set. The globalized robust counterpart (GRC) extends this idea by allowing controlled constraint violations in a larger uncertainty set. The constraint violations are controlled by the distance of the parameter to the original uncertainty set. We derive tractable GRCs that extend the initial GRCs in the literature: our GRC is applicable to nonlinear constraints instead of only linear or conic constraints, and the GRC is more flexible with respect to both the uncertainty set and distance measure function, which are used to control the constraint violations. In addition, we present a GRC approach that can be used to provide an extended trade-off overview between the objective value and several robustness measures.

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This paper presents a new sequential method for constrained non-linear optimization problems. The principal characteristics of these problems are very time consuming function evaluations and the absence of derivative information. Such problems are common in design optimization, where time consuming function evaluations are carried out by simulation tools (e.g., FEM, CFD). Classical optimization methods, based on derivatives, are not applicable because often derivative information is not available and is too expensive to approximate through finite differencing. The algorithm first creates an experimental design. In the design points the underlying functions are evaluated. Local linear approximations of the real model are obtained with help of weighted regression techniques. The approximating model is then optimized within a trust region to find the best feasible objective improving point. This trust region moves along the most promising direction, which is determined on the basis of the evaluated objective values and constraint violations combined in a filter criterion. If the geometry of the points that determine the local approximations becomes bad, i.e. the points are located in such a way that they result in a bad approximation of the actual model, then we evaluate a geometry improving instead of an objective improving point. In each iteration a new local linear approximation is built, and either a new point is evaluated (objective or geometry improving) or the trust region is decreased. Convergence of the algorithm is guided by the size of this trust region. The focus of the approach is on getting good solutions with a limited number of function evaluations (not necessarily on reaching high accuracy).