Stability and Robustness: Reliability in the World of Uncertainty

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Main contributors•Ben-Tal and Nemirovski - ellipsoidal uncertainty for continiousoptimization problems•Kouvelis and Yu, as well as Averbakh - minmax regret forscenario-based uncertainty, worst case optimization•Yaman and Karasan, Mintemanni and Gambardella –absolute and relative robustness for the minimum spanning treeand shortest path problems•Bertsimas and Sim - robust optimization with control ofconservatism of a solution16/04/2009 Yury Nikulin, Sensitivity analysis 5
Classical approachKouvelis P. and Yu G. (1997). Robust discrete optimization and itsapplications. Kluwer Academic Publishers, Norwell, M.A.A comprehensive treatment of the state of the art (up to 1997) inrobust discrete optimization and extensive references arepresented in this work. However, there still are more open problemsthan solved ones. Most of the known results correspond toscenario-represented models of uncertainty, i.e. where there existsa finite number of possible scenarios each of which is givenexplicitly by listing the corresponding values of parameters. It isshown that most classical polynomially solvable combinatorialoptimization problems loose this nice property and become NPhardin a robust version with scenario-represented uncertainty.16/04/2009 Yury Nikulin, Sensitivity analysis 6
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OutlineThe World of UncertaintyRobu
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OutlineOutlineThe World of Uncertai
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Page 11 and 12: OutlineThe World of UncertaintyRobu
Page 13 and 14: Interval UncertaintyOutlineThe Worl
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Page 51 and 52: ExampleOutlineThe World of Uncertai
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Page 59 and 60: Introduction: Sensitivity analysis
Page 61: Dealing with uncertainty16/04/2009
Page 65 and 66: Telecommunication and MST16/04/2009
Page 67 and 68: Multicommodity reformulation formin
Page 69 and 70: Dual problem16/04/2009 Yury Nikulin
Page 71 and 72: Interval data and scenariosG = ( E,
Page 73 and 74: RST problem reformulation as mixedi
Page 75 and 76: Robust Spanning Tree Problem within
Page 77 and 78: Example: weak edgesweaknon-weak12-5
Page 79 and 80: Example: Strong edgesstrong12-553-4
Page 81 and 82: CPLEX results on complete graphs16/
Page 83 and 84: Branching strategyNode d with the s
Page 85 and 86: Lower bound16/04/2009 Yury Nikulin,
Page 87 and 88: SA: preprocessing, initial solution
Page 89 and 90: SA: neighborhood search moveCase2.1
Page 91 and 92: Running time16/04/2009 Yury Nikulin
Page 93 and 94: SA summary• In spite of SA is gen
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Page 97 and 98: Heuristics for Central Tree Problem
Page 99 and 100: Distance MeasureThe Central TreePro
Page 101 and 102: Distance MeasureThe Central TreePro
Page 103 and 104: The Central Tree ProblemThe Central
Page 105 and 106: The CTP vs. MDTPThe Central TreePro
Page 107 and 108: The CTP vs. MDTPThe Central TreePro
Page 109 and 110: Short ExampleThe Central TreeProble
Page 111 and 112: Complexity of the CTPThe Central Tr
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Complexity of the CTPThe Central Tr
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Equivalence between 0 − 1 RSTP an
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One more evidence of NP-hardnessThe
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MILP FormulationCorollary from the
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Approximation for the CTPThe Centra
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The Central TreeProblemHeuristics f
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Greedy Construction HeuristicThe Ce
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ExampleThe Central TreeProblemHeuri
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Computational ExperimentsThe Centra
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ConclusionThe Central TreeProblemHe
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Sensitivity Analysis in Game Theory
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Game descriptionProblemFormulationS
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Forming a coalition, bansProblemFor
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EquilibriaProblemFormulationStabili
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Original problemProblemFormulationS
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ExampleProblemFormulationStability
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Stability FunctionProblemFormulatio
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SufficiencyProblemFormulationStabil
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Formula for Stability FunctionProbl
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Sketch of the proofProblemFormulati
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Formula for Stability RadiusProblem
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Independent players, no bansProblem
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Accuracy Function and RadiusProblem
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ExampleProblemFormulationStability
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Concluding remarksProblemFormulatio
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Fuzziness: one more way of dealingw
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Crisp and Fuzzy Numbers28/04/2009 Y
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Geometrical interpretation: Union28
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Geometrical interpretation:Intersec
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Fuzzy sets28/04/2009 Yury Nikulin,
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Comparison of fuzzy numbers:strong
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Maximal and Minimalfuzzy numbersμ(
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Fuzzy maximum and minimum28/04/2009
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Statistical data about flight delay
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Possibility distribution28/04/2009
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Fuzzy Number representing delays28/
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Stability and Robustness: Reliability in the World of Uncertainty

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