tackles decision-making under uncertainty. It splits choices into "here-and-now" decisions before uncertainty unfolds and "wait-and-see" decisions after. This approach helps balance immediate actions with future flexibility.
The method uses scenarios to represent possible outcomes and aims to minimize overall costs. It's a powerful tool for optimizing complex systems, from energy production to , where future conditions are uncertain but critical to success.
Two-Stage Stochastic Programming Formulations
Fundamental Concepts and Structure
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Two-stage stochastic programming models decision-making under uncertainty with decisions made in two stages: before and after uncertain parameter realization
First stage involves "here-and-now" decisions before uncertainty revelation, second stage involves "wait-and-see" decisions after uncertainty resolution
minimizes sum of first-stage costs and of second-stage costs
Scenario generation creates finite set of possible uncertain parameter realizations (weather patterns, market demands)
Formulation includes first-stage (production capacity), second-stage decision variables (actual production), and scenario-dependent linking stages
ensure second-stage decisions depend only on available information at that stage
Mathematical Representation and Solving Approaches
(DEP) represents all scenarios simultaneously as large-scale linear or mixed-integer program
DEP solved using standard optimization techniques (simplex method, )
General form of two-stage stochastic program:
min[x](https://www.fiveableKeyTerm:x)cTx+Eξ[Q(x,ξ)]s.t. Ax=b,x≥0
where Q(x,ξ)=min[y](https://www.fiveableKeyTerm:y)qTys.t. Tx+Wy=h(ξ),y≥0
x represents first-stage decisions, y represents second-stage decisions, ξ represents random vector
T called , W called
Advanced Modeling Techniques
Incorporate () into objective function balancing expected performance and risk exposure
Multi-stage extensions model sequential decision-making processes (energy production planning)
handle probabilistic constraints (meeting demand with certain probability)
address ambiguity in
Integration with machine learning methods for improved scenario generation and solution approaches
Solving Two-Stage Stochastic Programs
Decomposition Methods
() solves two-stage stochastic linear programs by iteratively generating cutting planes
Algorithm decomposes problem into master problem (first stage) and subproblems (second stage)
uses scenario decomposition, effective for problems with integer first-stage variables
Method relaxes non-anticipativity constraints and uses augmented Lagrangian approach
(SDDP) algorithm approximates multi-stage problems as series of two-stage problems
SDDP builds piecewise linear approximation of future cost function
Simulation-Based Approaches
uses Monte Carlo simulation for problems with large scenario sets
SAA generates sample of scenarios, solves resulting approximation, repeats process to obtain statistical estimates
(stochastic gradient descent) update solution based on sampled scenarios