STOCHASTIC PROGRAMMING & RISK MANAGEMENT: FUNDAMENTALS

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 Abstract. In the field of mathematical optimization, Stochastic Programming (SP) is a framework for modeling optimization problems that involve uncertainty. Below is presented the SP based on scenarios, also known as non-anticipative stochastic programming. SP considers multiple scenarios, associating each of them with a probability of occurrence, and the model determines the "best" decision before the occurrence of the scenario (non-anticipative decisions). This alternative requires a process to generate the scenarios, they may be exogenous (calculated outside of optimization model using a special stochastic model whose results are read as optimization parameters), or endogenous (calculated within the model optimization). It should be note, that in stochastic optimization there is no the "best" solution, in contrast to the deterministic optimization, already a fundamental part of the analysis is the management of risk associated with decisions taken; optimize just the expected value of the objective function, may lead to decision highly risks (risk-prone) with greater risk (volatility) than other decisions that do not seek this goal.

 INDEX

 1.     Stochastic Optimization

1.1. Stochastic Decision-Making

1.2. Decision Process Modeling

1.3. Multi-Stage Decision Process

1.3.1.       Static Decisions – Two Stage Stochastic Programming

1.3.2.       Dynamic Decisions – Multi Stage Stochastic Programming

1.4. Scenario Generation

1.5. Performance Indexes & Bounding

1.5.1.       The Stochastic Optimization Problem

1.5.2.       The Perfect Information Problem

1.5.3.       The Deterministic Expected Value Problem

1.5.4.       EVPI and VSS

1.5.5.       EVPI and VSS Bounds

1.5.6.       Jensen's Lower Bounds

1.6. Probabilistic Constraints

1.7. Risk Management

1.7.1.       Risk Modeling

1.7.2.       Decision-Maker Criterium

 2.     Large Scale Methodologies

2.1. Benders Decomposition

2.2. Lagrangean Relaxation

1.     Introduction

 1.1. Stochastic Decision-Making

 According to Powell (2017) the problem of making decisions uncertainty has motivated applications in several disciplines: spanning business, science, engineering, economics and finance, health and transportation, supply chain design and operations, … ; the decisions may be binary, discrete, continuous. Even richer are the different ways that uncertainty arises that creates a virtually unlimited range of problems. This diversity has generated the evolution of different mathematical modeling styles and solution approaches.

 Powell brings to the stochastic optimization communities a canonical modeling framework that covers all the different perspectives that have evolved over time. From this foundation, Powell reduce the rich array of solution approaches into a relatively small number of fundamental strategies, he is not “replacing these fields but rather building on them, somewhat like standing on the shoulders of giants”.

 The type of stochastic optimization models described by Powell are decision trees, stochastic search, optimal stopping, optimal control, Markov decision processes, approximate/adaptive/neuro dynamic programming, reinforcement learning, online computation, model predictive control, stochastic programming, robust optimization, ranking and selection, simulation optimization and multiarmed bandit.

In this chapter is presented the Stochastic Programming (SP) methodology to solve stochastic optimization models. For this it is necessary to define a process that considers:

1.     Decision Tree: It defines how to simulate how the decisions under uncertainty environment

2.     Stochastic Process, the uncertainty dimensions must be defined by the users, considering which is the more convenient stochastic process.

3.     Risk Management: The biggest advantage of stochastic models is the inclusion of risk measures in the stochastic model. Nowadays, the risk measure most used is the CVaR (Conditional-Value-at-Risk)

4.     Solution Process: the solution of the stochastic model can be accomplished through direct solution of equivalent deterministic problem (the random variables are fixed during the optimization process) or a “real” stochastic model (the random variables change during the optimization process).

https://www.doanalytics.net/Documents/DOA-Stochastic-Programming-Fundamentals.pdf   

Other Chapters:

• JACQUES F. BENDERS Theory:

https://www.dhirubhai.net/pulse/j-f-benders-theory-applications-past-present-future-jesus-velasquez/

• Stochastic & Dynamic Benders Theory – Electricity Sector Applications

https://www.dhirubhai.net/pulse/stochastic-dynamic-benders-theory-jesus-velasquez/

• Stochastic Lagrangean Relaxation & Decomposition
• The Future: Mathematical Programming 4.0

https://www.dhirubhai.net/pulse/future-mathematical-programming-jesus-velasquez/