Yuri Ermoliev International Institute for Applied Systems Analysis

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Yuri ErmolievInternational Institute for Applied
Systems Analysis
Mathematical methods for robust solutions
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Facets of robustness

• Variability of goals
• Explicit risk measures
• Concept of flexible solutions

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A simple standard example

• The problem How to spend 10 units of money?
• Invest now with 100 return (A) or
• Keep money under mattress (B)
• The deterministic model Maximize the return
  function
• Optimal solution (10, 0). Return 20
• Is this a desirable solution ? Uncertainty is
  50/50 with return of 40 or 0.
• How to deal with such uncertainty?

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Option 1 Scenario analysis

• Scenario 1 Real returns 40. Solution (10, 0)
  is still optimal
• Scenario 2 Insolvency. Optimal solution (0, 10)
• Solution (0,10) is not optimal for the
  deterministic model, but it is robust against
  all possible scenarios
• Is there a better solution?
• In which sense?
• What about mixed solutions?
• How can we find them?

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Option 2 Straightforward sensitivity and
uncertainty analysis

• Keep changing scenarios of input (uncertainties,
  decision variables, )
• Evaluations can easily take 100s of years CPU
  time
• Provides only frequency distributions of output,
  no direct information for decision making
• How can we find a desirable solution without
  evaluating all feasible alternatives? Need for
  optimization methods

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Option 3 Decisionoriented methods for
sensitivity and uncertainty analyses

• Preference structure is more stable than
  outputs
• What-if scenario analysis
• Stochastic models
• Expected utility theory
• Meanvariance efficiency
• Stochastic optimization

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Possible definitions of robustness

• Risk aversion, proneness, neutrality mean
  variance efficiency
• Other goals (liabilities, targets, thresholds)?
• Underestimation of low probability scenarios
• Partially know distributions?

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Expected utility theory

• Summarizes all outcomes and attitudes to risks
  into one preference index
• Quadratic, logarithmic, exponential, linear,
  convex, concave, utility function?
• Shape of utility function reflects attitudes to
  risk risk aversion, proneness, neutrality

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Meanvariance efficiency

• Returns (costs, benefits, etc.) and additional
  risk measure the variance of returns
• Symmetric risk measure
• Normal distribution

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Stochastic optimization

• Explicitly deals with different outputs and
  interactions among decisions x and uncertainties
  ?
• Different goal functions (costs, benefits,
  balances)
• Concepts of robust solutions involve goals,
  different risk measures and concepts of
  solutions, feasibility, and, in particular,
  their flexibility, which cant be formalized
  within deterministic models.

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Conclusions

• All practical problems are solved somehow,
  how is the most important question
• New problems often require new methods
• Robustness is characterized by different goals,
  risk measures, and concepts of feasible solutions
• Formalized in terms of STO models
• Different methods either exist or can be
  developed, e.g., adaptive Monte Carlo
  optimization procedures