arXiv:2411.03520v3 Announce Type: replace-cross 
Abstract: We study a class of two-stage stochastic programs, namely, those with fixed recourse matrix and fixed costs, and linear second stage. We show that, under mild assumptions, the problem can be solved with just one scenario, which we call an ``optimal scenario.'' Such a scenario does not have to be unique and may fall outside the support of the underlying distribution. Although finding an optimal scenario in general might be hard, we show that the result can be particularly useful in the case of stochastic optimization problems with contextual information, where the goal is to optimize the expected value of a certain function given some contextual information (e.g., previous demand, customer type, etc.) that accompany the main data of interest. The contextual information allows for a better estimation of the quantity of interest via machine learning methods. We focus on a class of learning methods -- sometimes called in the literature decision-focused learning -- that integrate the learning and optimization procedures by means of a bilevel optimization formulation, which determines the parameters for pointwise forecasts. By using the optimal scenario result, we prove that when such models are applied to the class of contextual two-stage problems considered in this paper, the pointwise forecasts computed from the bilevel optimization formulation actually yield asymptotically the best approximation of an optimal scenario within the modeler's pre-specified set of parameterized forecast functions. Numerical results conducted with inventory problems from the literature (with synthetic data) as well as a bike-sharing problem with real data demonstrate that the proposed approach performs well when compared to benchmark methods from the literature.

تظهر دراسة حديثة حول البرامج العشوائية ذات المرحلتين أنه من الممكن حل مشكلات التحسين المعقدة باستخدام سيناريو 'مثالي' واحد فقط. هذه النتيجة مهمة لأنها تبسط عملية التنبؤ، مما يجعلها أكثر كفاءة ويسرًا لمجموعة متنوعة من التطبيقات. من خلال إثبات أن هذا السيناريو المثالي يمكن أن يوجد خارج نطاق التوزيع المعتاد، تفتح الأبحاث آفاقًا جديدة للتطبيقات العملية في مجالات مثل المالية واللوجستيات.

Un estudio reciente sobre programas estocásticos de dos etapas revela que es posible resolver problemas complejos de optimización utilizando solo un 'escenario óptimo.' Este hallazgo es significativo porque simplifica el proceso de pronóstico, haciéndolo más eficiente y accesible para diversas aplicaciones. Al demostrar que este escenario óptimo puede existir fuera del soporte de distribución típico, la investigación abre nuevas avenidas para implementaciones prácticas en campos como las finanzas y la logística.

Une étude récente sur les programmes stochastiques en deux étapes révèle qu'il est possible de résoudre des problèmes d'optimisation complexes en utilisant un seul 'scénario optimal.' Cette découverte est importante car elle simplifie le processus de prévision, le rendant plus efficace et accessible pour diverses applications. En démontrant que ce scénario optimal peut exister en dehors du support de distribution typique, la recherche ouvre de nouvelles voies pour des applications pratiques dans des domaines comme la finance et la logistique.

A recent study on two-stage stochastic programs reveals that it's possible to solve complex optimization problems using just one 'optimal scenario.' This finding is significant because it simplifies the forecasting process, making it more efficient and accessible for various applications. By demonstrating that this optimal scenario can exist outside the typical distribution support, the research opens new avenues for practical implementations in fields like finance and logistics.

Forecasting Outside the Box: Application-Driven Optimal Pointwise Forecasts for Stochastic Optimization

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