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To solve real life problems under uncertainty in Economics, Finance, Energy, Transportation and Logistics, the use of stochastic optimization is widely accepted and appreciated. However, the nature of stochastic programming leads to a conflict between adaptability to reality and tractability. To formulate a multistage stochastic model, two types of formulations are typically adopted: the so-called stage-scenario formulation named also formulation with explicit non-anticipativity constraints and the so-called nodal formulation named also formulation with implicit non-anticipativity constraints. Both of them have advantages and disadvantages. This work aims at helping the scholars and practitioners to understand the two types of notation and, in particular, to reformulate with the nodal formulation a model that was originally defined with the stage-scenario formulation presenting this implementation in the algebraic language GAMS. In addition, this work presents an empirical analysis applying the two formulations both without any further decomposition to perform a fair comparison. In this way, we show that the difficulties to implement the model with the nodal formulation are somehow reworded making the problem tractable without any decomposition algorithm. Still, we remark that in some other applications the stage-scenario formulation could be more helpful to understand the structure of the problem since it allows to relax the non-anticipativity constraints.

We analyse an optimal goal-based households’ asset-liability management problem characterised by a real estate target and a retirement goal over a long-term planning horizon. The problem is formulated as a multistage stochastic program and we evaluate the impact of second order stochastic dominance (SSD) constraints on different specifications of a family objective function and with respect to three alternative benchmark policies. We define a stochastic linear program in which the SSD constraints are based on a double stochastic matrix, whose effectiveness in determining the decision maker strategies is studied in a case study developed in the second part of the article. We show that depending on the adopted benchmark policy, SSD constraints even if binding far on the planning horizon, may influence the root node investment decision and affect both the investment and the liability optimal policies. Based on an extended computational study we analyse under which conditions and problem formulation, an SSD condition may also imply first order stochastic dominance (FSD). Finally we analyse the relationship between the specification of a minimum shortfall objective with respect to the goals and the introduced SSD constraints at the terminal horizon.

Background Bacterial infection is a major cause of morbidity and mortality for persons who inject drugs (PWID). Injection cessation may help abrogate such infections, but maintaining complete cessation is challenging. Limited data exists on the role of reduced injection intensity on invasive bacterial infection risk. We sought to evaluate decreased risk for bacterial infections following cessation and substantive reduction in the injection intensity. Methods Participants were persons in the AIDS Linked to the Intravenous Experience (ALIVE) cohort with initial high-frequency injection drug use (> 1 daily). Pooled logistic regression with generalized estimating equations was used to estimate risk for invasive bacterial infection (pneumonia, endocarditis, or sepsis) among participants achieving complete injection cessation or reduced injection intensity relative to those with sustained high-frequency use. Results Of 2247 study participants with 12,469 paired study visits, complete injection cessation was achieved at 13.5% and reduced injection intensity at 25.5% of study visits. Adjusting for sociodemographics and HIV status, injection cessation was associated with a 54% reduction of bacterial infection at 3 months (odds ratio [OR] 0.46, 95% CI 0.25–0.84) and a 46% reduction at 6 months (OR 0.54, 95% CI 0.36–0.81). Reduced injection intensity was associated with a 36% reduction of infection at 3 months (OR 0.64, 95% CI 0.43–0.96) and a 26% reduction at 6 months (OR 0.74, 95% CI 0.56–0.98). Conclusions Both complete cessation and reduced injection frequency demonstrate substantial benefit in reducing invasive bacterial infection risk among PWID. With high rates of relapse into injection use, targeting sustained reductions in drug use intensity may be a key harm reduction modality for improving clinical outcomes in this population. Electronic supplementary material The online version of this article (10.1186/s12954-019-0312-8) contains supplementary material, which is available to authorized users.

To solve a decision problem under uncertainty via stochastic programming means to choose or to build a suitable stochastic programming model taking into account the nature of the real-life problem, character of input data, availability of software and computer technology. In applications of multistage stochastic programs additional rather complicated modeling issues come to the fore. They concern the choice of the horizon, stages, methods for generating scenario trees, etc. We shall discuss briefly the ways of selecting horizon and stages in financial applications. In our numerical studies, we focus on alternative choices of stages and their impact on optimal first-stage solutions of bond portfolio optimization problems.

The bond portfolio management problem is formulated as a stochastic program based on interest rate scenarios. The coefficients of the resulting program are subject to errors of various kind. In this paper, we complement the theoretical stability results of by simulation experiments. Adapting the approach of to problems based on perturbed yield curves, we then provide bounds for the optimality gap for various candidate first-stage solutions.