In this paper we deal with the problem of finding the first K shortest paths from a single origin node to all other nodes of a directed graph. In particular, we define the necessary and sufficient conditions for a set of distance label vectors, on the basis of which we propose a class of methods which can be viewed as an extension of the generic label-correcting method for solving the classical single-origin all-destinations shortest path problem. The data structure used is characterized by a set of K lists of candidate nodes, and the proposed methods differ in the strategy used to select the node to be extracted at each iteration. The computational results show that: 1. some label-correcting methods are generally much faster than the double sweep method of Shier (1979); 2. the most efficient node selection strategies, used for solving the classical single-origin all-destinations shortest path problem, have proved to be effective also in the case of the K shortest paths.
We propose a model to select the optimal portfolio which underlies insurance policies with a guarantee. The objective function is defined in order to minimise the conditional value-at-risk (CVaR) of the distribution of the losses with respect to a target return. We add operational and regulatory constraints to make the model as flexible as possible when used for real applications. We show that the integration of the asset and liability side yields superior performances with respect to naive fixed-mix portfolios and asset based strategies. We validate the model on out-of-sample scenarios and provide insights on policy design.
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