1. IIASA Yuri Ermoliev International Institute for Applied Systems Analysis Mathematical methods for robust solutions

2. IIASA Facets of robustness  Variability of goals  Explicit risk measures  Concept of flexible solutions

3. IIASA 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?

4. IIASA 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?

5. IIASA 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 InputsMODELOutputs

6. IIASA 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 InputsMODELDecisions

7. IIASA Possible definitions of robustness Risk aversion, proneness, neutrality; mean – variance efficiency Other goals (liabilities, targets, thresholds)? Underestimation of low probability scenarios Partially know distributions?

8. IIASA 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

9. IIASA Mean–variance efficiency Returns (costs, benefits, etc.) and additional risk measure: the variance of returns Symmetric risk measure Normal distribution

10. IIASA 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., …

11. IIASA 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