Towards new solution techniques in mathematical programming with scenarios

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Operational research now occupies an important place in research and industrial world. Along with the development of computational capacities, researchers and practitioners are more concerned about the optimization of operational processes, or the estimation of predictive models. Such models however suffer from many underlying simplifications, a major one being the assumption that all parameters are perfectly known. This is unfortunately rarely true, and a growing interest exists to incorporate uncertainty in mathematical problems. This allows to produce more flexible and robust solutions, as they better take changing conditions into account. The operational decisions can greatly differ when such an uncertainty is taken into account. However, the modeling task is then much more complex, and numerical efforts are more costly. The purpose of this proposal is to explore developments for algorithms in stochastic and dynamic programming, and to apply them to various fields. We focus on uncertainty modeled by scenarios, that represents a sequence of possible outcomes. We first study new developments in multistage stochastic programming, in order to propose faster while accurate techniques, capitalizing on the integration of ideas originally proposed in nonlinear programming. As a main application, we consider the hydroelectric generation, that is dependent of uncertain water intputs. A better management can lead to cost savings, a larger production capacity, and also a reduced impact of environment. Secondly, we consider dynamic discrete choice models, describing people choices among a finite set of alternative, but instead of studying standard punctual models, we integrate how people plan decisions over a time horizon. We apply the ideas to the description of consumers behavior, and we also consider the sequence of decisions made in a traffic network when selecting a road between two points. All these problems require efficient estimation techniques, and we develop new strategies for the underlying optimization algorithms.