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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

Inputs MODEL Outputs

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Option 3: Decision–oriented methods for sensitivity and uncertainty analyses

Preference structure is more “stable” than outputs

• “What-if” scenario analysis

• Stochastic models:

– Expected utility theory

– Mean–variance efficiency

– Stochastic optimization

Inputs MODEL Decisions

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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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Mean–variance 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 ω

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 can’t be formalized within deterministic models.

),( xf1 ),( xf2 ),( xf3 , …

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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