Scenario Cluster Lagrangian Decomposition in two stage stochastic mixed 0-1 optimization

In this paper we introduce four scenario Cluster based Lagrangian Decomposition (CLD) procedures for obtaining strong lower bounds to the (optimal) solution value of two-stage stochastic mixed 0-1 problems. At each iteration of the Lagrangian based procedures, the traditional aim consists of obtaining the solution value of the corresponding Lagrangian dual via solving scenario submodels once the nonanticipativity constraints have been dualized. Instead of considering a splitting variable representation over the set of scenarios, we propose to decompose the model into a set of scenario clusters. We compare the computational performance of the four Lagrange multiplier updating procedures, namely the Subgradient Method, the Volume Algorithm, the Progressive Hedging Algorithm and the Dynamic Constrained Cutting Plane scheme for different numbers of scenario clusters and different dimensions of the original problem. Our computational experience shows that the CLD bound and its computational effort depend on the number of scenario clusters to consider. In any case, our results show that the CLD procedures outperform the traditional LD scheme for single scenarios both in the quality of the bounds and computational effort. All the procedures have been implemented in a C++ experimental code. A broad computational experience is reported on a test of randomly generated instances by using the MIP solvers COIN-OR and CPLEX for the auxiliary mixed 0-1 cluster submodels, this last solver within the open source engine COIN-OR. We also give computational evidence of the model tightening effect that the preprocessing techniques, cut generation and appending and parallel computing tools have in stochastic integer optimization. Finally, we have observed that the plain use of both solvers does not provide the optimal solution of the instances included in the testbed with which we have experimented but for two toy instances in affordable elapsed time. On the other hand the proposed procedures provide strong lower bounds (or the same solution value) in a considerably shorter elapsed time for the quasi-optimal solution obtained by other means for the original stochastic problem.

New climate scenario framework implementation in the GCAM integrated assessment model

48 p.

The outset of mental torture. Through the lens of the Ticking Time Bomb Scenario

No doubt shall be placed when qualifying torture as one of the cruellest crime offences against human beings. It is widely known that the first torture practices go back to the Middle Ages, where torture mechanisms and devices were used as a legitimate means of punishment, extraction of confessions or executions. Brutal techniques such as ‘Judas Cradle’, ‘The Rack’ or the ‘Rat Torture’ were indeed, the ones commonly used. Moreover, some centuries onwards, torture warrants were permitted and authorised by Privy Councils in legislations such as the English one. However, examples like that were the only ones which public accountability was given to, whereas off-the-book practices remained in silence in other countries for long lasting years. Nowadays, in the 21st century, there are innumerable enforced laws and provisions that prohibit the act of torture, to be precise, physical and psychological torture. Nonetheless, not only are these legislations necessary for fighting torture, but also ad hoc courts and specialised committees continuously report the existence of this crime offence.