Direct Search Techniques for Mixed Stochastic Nonlinear Programming Model

Abstract

Stochastic programming is a methodology utilized for the purpose of achieving optimal planning and decision-making outcomes when faced with uncertain data. The subject of investigation pertains to a stochastic optimization problem wherein the results of stochastic data are not disclosed during runtime, and the optimization of the decision does not necessitate foresight into forthcoming outcomes. This establishes a strong correlation with the imperative need for immediate optimization in uncertain data settings, enabling effective decision-making in the present moment. The present study introduces a novel methodology for achieving global optimization of the model for nonlinear mixed-stochastic programming problem. The present study centers on stochastic problems that are two-staged and entail non-linearities in both the objective function and constraints. The first stage variables are discrete in nature, whereas the second stage variables are a combination of continuous and mixed types. Scenario-based representations are utilized for formulating problems. The fundamental approach to address the non-linear mixed-stochastic programming problem involves converting the model into a deterministic non-linear mixed-count program that is equivalent in form. The feasibility of this proposition stems from the discrete distribution assumption of uncertainty, which can be represented by a limited set of scenarios. The magnitude of the model size will increase significantly due to the quantity of scenarios and time horizons involved. The utilization of filtered probability space in conjunction with data mining techniques will be employed for the purpose of scenario generation. The methodology employed for addressing nonlinear mixed-integer programming problems of significant scale involves elevating the value of a non-basic variable beyond its boundaries in order to compel a basis variable to attain a cumulative value. Subsequently, the problem is simplified by maintaining a constant count variable and modifying it incrementally in discrete intervals to achieve an optimal solution at a global level.