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 Many efficient interior-point methods (IPMs) are based on the use of a self-concordant barrier function for the domain of the problem that has to be solved. Recently, a wide class of new barrier functions has been introduced in which the functions are not self-concordant, but despite this fact give rise to efficient IPMs. Here, we introduce the notion of locally self-concordant barrier functions and we prove that the new barrier functions are locally self-concordant. In many cases, the (local) complexity numbers of the new barrier functions along the central path are better than the complexity number of the logarithmic barrier function by a factor between 0.5 and 1.

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Many regions in the world are protected against flooding by a dike, which may be either natural or artificial. We deal with a model for finding the optimal heights of such a dike in the future. It minimizes the sum of the investments costs for upgrading the dike in the future and the expected costs due to flooding. The model is highly nonlinear, nonconvex, and infinite-dimensional. Despite this, the model can be solved analytically if there is no backlog in maintenance. If there is a backlog in maintenance, then the optimal solution can be found by minimizing a convex function over a finite interval. However, if the backlog becomes extremely large we show that the model breaks down. Our model has been used in The Netherlands to define legal safety standards for the coming decades.

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We deal with a recently proposed method of Chubanov [1], for solving linear homogeneous systems with positive variables. We use Nesterov's excessive gap method in the basic procedure. As a result, the iteration bound for the basic procedure is reduced by the factor $nsqrt{n}$. The price for this improvement is that the iterations are more costly, namely $O(n^2 )$ instead of $O(n)$. The overall gain in the complexity hence becomes a factor of $sqrt{n}$.