A neural network-based distributional constraint learning methodology for mixed-integer stochastic optimization Articles uri icon

publication date

  • June 2023


  • 232

International Standard Serial Number (ISSN)

  • 0957-4174

Electronic International Standard Serial Number (EISSN)

  • 1873-6793


  • The use of machine-learning methods helps to improve decision-making in different fields. In particular, the
    idea of bridging predictions (predictive models) and prescriptions (optimization problems) is gaining attention
    within the scientific community. One of the main ideas to address this trade-off is the Constraint Learning (CL)
    methodology, where the structure of the machine learning model can be treated as a set of constraints to be
    embedded within the optimization problem, establishing the relationship between a direct decision variable
    𝑥 and a response variable 𝑦. However, most CL approaches have focused on making point predictions, not
    considering the statistical and external uncertainty faced in the modeling process. In this paper, we extend
    the CL methodology to deal with uncertainty in the response variable 𝑦. The novel Distributional Constraint
    Learning (DCL) methodology makes use of a piece-wise linearizable neural network-based model to estimate
    the parameters of the conditional distribution of 𝑦 (dependent on decisions 𝑥 and contextual information),
    which can be embedded within mixed-integer optimization problems. In particular, we formulate a stochastic
    optimization problem by sampling random values from the estimated distribution by using a linear set of
    constraints. In this sense, DCL combines both the predictive performance of the neural network method and
    the possibility of generating scenarios to account for uncertainty within a tractable optimization model. The
    behavior of the proposed methodology is tested in the context of electricity systems, where a Virtual Power
    Plant seeks to optimize its operation, subject to different forms of uncertainty, and with price-responsive


  • Statistics


  • stochastic optimization; constraint learning; distribution estimation; neural networks; mixed-integer optimization