The objective of the course consists of the presentation of the basic concepts, modeling technologies and algorithmic developments for treating in the model the risk inherent to the decisión making in any stochastic envrironment. The paraneters` uncertainty is represented via scenario trees. The model`s solution, so-called robust solution, is intended to minimize the regret of a wrong decision making. Two types of merit functions are considered, namely the minimization of the expected deviation of the model`s solution from the scenario optimal solution and the minimization of the expected objetive funcion over all escenrios. Three types of recourse philosophies are considered, namely, simple recourse, parcial recourse and full recourse. Several mathematical representations of the deterministc equivalent model are discussed, namely, compact represenation, splitting variables representation via scenario group and splitting variables representation via scenario. Three types of problem`s decomposition are studied, namely, Benders decomposition, (augmented) Lagrangian decomposition and a decomposition based on stochastic dynamic programming.
Theory subject (Contents)
Topic 1. Optimization under uncertainty. Motivation
1.1 Introduction. Illustrative cases
1.2 General modeling
1.3 Risk aversion modeling and probabilistic conditions
1.4 Basic concepts for the analysis of the solution goodness
Topic 2. Stochastic modeling via scenario analysis
2.1 Introduction
2.2 Scenario immunization modeling
2.2.1 General presentation
2.2.2 Mathematical representation
2.2.3 Scenario infeasibility function
2.3 Two-steps hierarchical procedure
2.4 Robust optimization
2.4.1 Simple, partial and full recourse
2.4.2 Non-anticipativity principle
2.5 Equivalent mathematical representations
2.6 Scenario tree generation
2.6.1 Artificial neural networks
2.6.2 Clustering analysis
2.6.3 Mont Carlo simulation
Topic 3. Compact representation of stochastic models
3.1 Multi-stage linking constraints
3.2 Mathematical representation
3.3 Methodology for obtaining the initial solution
3.4 Benders decomposition
Topic 5. Two-stage stochastic programming
5.1 Mathematical representation
5.2 Geometric interpretation of the Benders decomposition
5.3 Utilization of the Benders decomposition. Compact representation
5.3.1 Algorithm DBC
5.3.2 Parallel computing methodology
5.4 Utilization of the Benders decomposition. Splitting variables representation
5.4.1 Algorithm DBVD
5.4.2 Parallel computing methodology
5.5 Utilization of the Augmented Lagrangian decomposition. Splitting variables representation via scenario group
5.5.1 Algorithm DLAG
5.5.2 Parallel computing methodology
5.6 Utilization of the Augmented Lagrangian decomposition. Splitting variables representation via scenario
Topic 6. Stochastic dynamic programming
6.1 General presentation
6.2 Algorithmic framework
6.2.1 Basic modules
6.2.2 Proposed aapproach
6.3 Utilization of the expected future cost curve
6.4 Obtaining the expected future cost curve
6.4.1 Mathematical representation
6.4.2 Expected future cost in function of the stage linking variables
6.5 Estimation of the reference levels for the stage linking variables
Evaluation system
The assignment of a computer work by implementing the appropriate model and algorithm using any optmization engine for a given case will be the 50% of the global evaluation. Note 1: The assignment can be carried jointly by a team of two students, at most, and it will be auditable. Note 2: The same case will be proposed for the consecutive evaluations 1 and 2, 3 and 4, etc. The other 50% of the global evaluation will consist of a written exam including a set of questions. Note 3. No help will books, notes and any other written material will be allowed in the written exam.
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