Stability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionStability and Robustness:Reliability in the World of UncertaintyYury NikulinDepartment of MathematicsUniversity of TurkuApril 9, 2009Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineOutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusion1 The World of UncertaintyReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling Uncertainty2 Robust Optimization3 Stability Analysis4 ConclusionYury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineOutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling Uncertainty1 The World of UncertaintyReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling Uncertainty2 Robust Optimization3 Stability Analysis4 ConclusionYury NikulinStability and Robustness: Reliability in the World of Uncertainty

EpigraphOutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling UncertaintyButterfly effect... Superfluous one tenth degree in somepoint – and a cyclone bursts out here insteadof there, it storms over countries which wouldbe spared if it were not this tenth...... We find here again the same discrepancybetween a smallest reason which is notsensed by an observer, and a significanteffect causing sometimes terribleconsequences...Jules Henri Poincaré. The science and themethod. 1908. Paris.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionReasons for UncertaintyReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling UncertaintyUncertainty in Optimizationinaccuracy of initial datanon-adequacy of models to real processeserrors of numerical methodserrors of rounding offabsence or lack of precise and reliable information etc.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionReasons for UncertaintyReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling UncertaintyUncertainty in Scheduling and Time-tablingequipment (machine) breakdowns or activity (job) disruptionearliness or tardiness, changes of job processingchanges of release dates, due dates, deadlines or resourceavailability etc.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling UncertaintyExtreme Consequences of UncertaintySelf-Propagated Chain Reactionsnuclear chain reactiondomino effectbutterfly effectthermal runawaycascading failureelectron avalancheYury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling UncertaintyWorld Economic Crisis: Domino Effect1. PessimisticAccountReportProfit10%downTransfer100billiondollarsattemporarydisposalPurchasesharesoftheGasProducingCompany100WorldEconomicCrisisasanexampleoftheDominoEffectsGasProducingCompanyBankDeposit100GlobalFinancialService1004. DepositWithdrawGetting75b.d.tobuysharesPensionFund3. Sellingsharesonlowerprices.Getting75b.d.Deficit15b.d.CommercialBankBuyingfixedcostshares5. Selling2. Withdrawmoney90billiondollars.Lost10b.d.CommercialEnterprise5050shares.Getting50b.dDeficit25b.d.LongtermcreditIndustrialEnterpriseGovernment25+15=40b.d.RescuebailoutAverytemporarymeasuretosecurethefinancialmarketplayersandovercomeliquiditydeficit.Otherwise,badlongtermconsequences:financialsectorcollapse,industrialsectorslowsdown,unemploymentgrows,economicfallsintorecession“Hidinganelephantbehindamouse!”Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling UncertaintyIf only being predicted and controlleduncertainty may profit!Complicated system + artificially caused uncertainty ↦→ systemnon-stability ↦→ controlled to some extent chain reactions ↦→ systembeing changed ↦→ diversification, selection or separation ↦→measuring new data and getting the result.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling UncertaintyFrom Microwave Heating to Electron CollidingThermal runaway used in microwave heatingMicrowaves + materials ↦→ energyabsorption ↦→ materials heating ↦→ controlledthermal runaway ↦→ selective localoverheating ↦→ material separation based ontemperature.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling UncertaintyFrom Microwave Heating to Electron CollidingProton avalanche used in Large Hadron ColliderThe Large Hadron Collider (LHC) is the world’slargest and highest-energy particle acceleratorcomplex, intended to collide opposing beams ofprotons with very high kinetic energy (8 billiondollars, 14 years, 40 countries)GoalsTo confirm the Big Bang theoryTo unveil the secrets of dark matterTo get more knowledge about mass andgravityYury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionModeling UncertaintyReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling UncertaintyUncertainty can be modeled by means of:possible scenarios (parameter realizations)fuzzy numbers and fuzzy objective functionsvarious probabilistic measuresinterval dataYury NikulinStability and Robustness: Reliability in the World of Uncertainty

Interval UncertaintyOutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling UncertaintyAdvantagesLower and upper interval bounds are easy to specifyNo statistical observations needed for the modelEasy implementation and practical useDisadvantagesInfinite number of problem input parameter realization -scenarios (can be overcome for some models!)Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

Interval UncertaintyOutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling UncertaintyAdvantagesLower and upper interval bounds are easy to specifyNo statistical observations needed for the modelEasy implementation and practical useDisadvantagesInfinite number of problem input parameter realization -scenarios (can be overcome for some models!)Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineOutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusion1 The World of UncertaintyReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling Uncertainty2 Robust Optimization3 Stability Analysis4 ConclusionYury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionWhat is Robust Optimization?Playing against the Worst-CaseInstead of producing an optimal solution for a normal situation, whichis described by deterministic models but rarely occurs in practice, andwhere recovery to optimality can be complicated, the aim of robustoptimization is to produce solutions that optimize additionallyconstructed objectives assuring the optimal solution to remainfeasible after worst case realization of uncertain problem inputparameters.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionWhat is a Robust Solution?Straightforward RobustnessAn optimal solution is robust if it remains optimal under anyrealization of the input dataAbsolute RobustnessA solution which minimizes the maximum among all scenariosobjective valueRelative RobustnessA solution which minimizes the maximum regret or, among allscenarios, the maximum deviation from optimal solutionYury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionWhat is a Robust Solution?Straightforward RobustnessAn optimal solution is robust if it remains optimal under anyrealization of the input dataAbsolute RobustnessA solution which minimizes the maximum among all scenariosobjective valueRelative RobustnessA solution which minimizes the maximum regret or, among allscenarios, the maximum deviation from optimal solutionYury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionWhat is a Robust Solution?Straightforward RobustnessAn optimal solution is robust if it remains optimal under anyrealization of the input dataAbsolute RobustnessA solution which minimizes the maximum among all scenariosobjective valueRelative RobustnessA solution which minimizes the maximum regret or, among allscenarios, the maximum deviation from optimal solutionYury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionWhat is a Robust Solution?Straightforward RobustnessAn optimal solution is robust if it remains optimal under anyrealization of the input dataAbsolute RobustnessA solution which minimizes the maximum among all scenariosobjective valueRelative RobustnessA solution which minimizes the maximum regret or, among allscenarios, the maximum deviation from optimal solutionYury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionThe Robust Spanning Tree ProblemAssume that with each edge e = ij ∈ E we associate a cost interval[l ij , u ij ], 0 < l ij < u ij , i.e. For each edge ij ∈ E its cost c ij is notfixed and belongs to [l ij , u ij ].The total cost c T of spanning tree T ∈ T G is defined as:c T := ∑ij∈E Tc ij .A scenario s ∈ S is a particular realization of the edge costs takentheir values from associated intervals.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionThe Robust Spanning Tree ProblemAssume that with each edge e = ij ∈ E we associate a cost interval[l ij , u ij ], 0 < l ij < u ij , i.e. For each edge ij ∈ E its cost c ij is notfixed and belongs to [l ij , u ij ].The total cost c T of spanning tree T ∈ T G is defined as:c T := ∑ij∈E Tc ij .A scenario s ∈ S is a particular realization of the edge costs takentheir values from associated intervals.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionThe Robust Spanning Tree ProblemAssume that with each edge e = ij ∈ E we associate a cost interval[l ij , u ij ], 0 < l ij < u ij , i.e. For each edge ij ∈ E its cost c ij is notfixed and belongs to [l ij , u ij ].The total cost c T of spanning tree T ∈ T G is defined as:c T := ∑ij∈E Tc ij .A scenario s ∈ S is a particular realization of the edge costs takentheir values from associated intervals.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionThe Robust Spanning Tree ProblemAssume that with each edge e = ij ∈ E we associate a cost interval[l ij , u ij ], 0 < l ij < u ij , i.e. For each edge ij ∈ E its cost c ij is notfixed and belongs to [l ij , u ij ].The total cost c T of spanning tree T ∈ T G is defined as:c T := ∑ij∈E Tc ij .A scenario s ∈ S is a particular realization of the edge costs takentheir values from associated intervals.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionThe Robust Spanning Tree ProblemFor each spanning tree T in scenario s the deviation for T in scenarios is defined as:dev s T := cs T − minT ′ ∈T cs T ′.Let S be the set of all scenarios. For a given spanning tree T ∈ T Gthe worst-case scenario s T is defined as:s T := arg maxs∈S devs T .Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionThe Robust Spanning Tree ProblemFor each spanning tree T in scenario s the deviation for T in scenarios is defined as:dev s T := cs T − minT ′ ∈T cs T ′.Let S be the set of all scenarios. For a given spanning tree T ∈ T Gthe worst-case scenario s T is defined as:s T := arg maxs∈S devs T .Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionThe Robust Spanning Tree ProblemFor each spanning tree T in scenario s the deviation for T in scenarios is defined as:dev s T := cs T − minT ′ ∈T cs T ′.Let S be the set of all scenarios. For a given spanning tree T ∈ T Gthe worst-case scenario s T is defined as:s T := arg maxs∈S devs T .Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionThe Robust Spanning Tree ProblemFor each spanning tree T in scenario s the deviation for T in scenarios is defined as:dev s T := cs T − minT ′ ∈T cs T ′.Let S be the set of all scenarios. For a given spanning tree T ∈ T Gthe worst-case scenario s T is defined as:s T := arg maxs∈S devs T .Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionThe Robust Spanning Tree ProblemThe robust deviation of T is given as:dev sTT:= csTT− min c sTT ′ ∈TT . ′GA spanning tree T 0 is said to be a robust spanning tree if it has thesmallest robust deviation among all spanning treesT 0 := arg minT ∈T Gdev sTT .Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionThe Robust Spanning Tree ProblemThe robust deviation of T is given as:dev sTT:= csTT− min c sTT ′ ∈TT . ′GA spanning tree T 0 is said to be a robust spanning tree if it has thesmallest robust deviation among all spanning treesT 0 := arg minT ∈T Gdev sTT .Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionThe Robust Spanning Tree ProblemThe robust deviation of T is given as:dev sTT:= csTT− min c sTT ′ ∈TT . ′GA spanning tree T 0 is said to be a robust spanning tree if it has thesmallest robust deviation among all spanning treesT 0 := arg minT ∈T Gdev sTT .Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionThe Robust Spanning Tree ProblemTheoremYaman et al, 2001The scenario in which the costs of every edge on a spanning treeT ∈ T G is at its upper bound and the cost of every other edge is at itslower bound is a worst-case scenario, i.e. c sTij = u ij ∀ ij ∈ E T andc sTij = l ij ∀ ij ∈ E \ E T .Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionThe Robust Spanning Tree Problem: ExampleFigure: Initial graph G, the central tree and the tree which is maximallyYury Nikulin Stability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionThe Robust Spanning Tree ProblemMethodologyMIP, CPLEXYaman et al., 2001Branch and BoundAron and Van Hentenryck, 2002Improved Branch and BoundMontemanni and Gambardella, 20052-Approximation AlgorithmKasperski and Zielinski, 2006Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionThe Robust Spanning Tree ProblemFuture Research Perspectivesdeveloping new fast heuristicsconsidering biobjective modelsfinding approximation algorithmsand many others...Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineOutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusion1 The World of UncertaintyReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling Uncertainty2 Robust Optimization3 Stability Analysis4 ConclusionYury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionStability analysisHistorical OriginJacques Hadamard (December 8, 1865 -October 17, 1963) introduced the stabilitycondition and treated it within the concept ofa well-posed mathematical programmingproblem equally with the conditions ofexistence and uniqueness of the solution.GoalsIt appears to be important to allocate classesof problems in which small changes of inputdata lead to small changes of the result.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionStability analysisHistorical OriginJacques Hadamard (December 8, 1865 -October 17, 1963) introduced the stabilitycondition and treated it within the concept ofa well-posed mathematical programmingproblem equally with the conditions ofexistence and uniqueness of the solution.GoalsIt appears to be important to allocate classesof problems in which small changes of inputdata lead to small changes of the result.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionGeneric Combinatorial Optimization ProblemProblem FormulationE = {e 1 , e 2 , ..., e n }, n > 1, ground set of elementsT ⊆ 2 E , |T | > 1, be a family of non-empty subsets of Efor e ∈ E: c(e) = (c 1 (e), c 2 (e), ..., c m (e)), m ≥ 1C = {c i (e j )} ∈ R m×n+ , where R + = {u ∈ R : u > 0}denote for k ∈ N, N k = {1, 2, ..., k} and let for t ∈ T ,N(t) = {j : e j ∈ t}Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionGeneric Combinatorial Optimization ProblemProblem FormulationA vector criterionf(C, t) = (f 1 (C, t), f 2 (C, t), ..., f m (C, t)),wheref i (C, t) = ∑j∈N(t)c i (e j ), i ∈ N m .DefinitionA solution t is Pareto optimal if and only if there is no solution t ′ suchthat f i (C, t ′ ) ≤ f i (C, t) for all i ∈ N m and at least one strict inequalityholds.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionGeneric Combinatorial Optimization ProblemProblem FormulationA vector criterionf(C, t) = (f 1 (C, t), f 2 (C, t), ..., f m (C, t)),wheref i (C, t) = ∑j∈N(t)c i (e j ), i ∈ N m .DefinitionA solution t is Pareto optimal if and only if there is no solution t ′ suchthat f i (C, t ′ ) ≤ f i (C, t) for all i ∈ N m and at least one strict inequalityholds.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionGeneric Combinatorial Optimization ProblemProblem FormulationIf the sets E and T are fixed, then an instance of m-criteriacombinatorial optimization problem is uniquely determined by thematrix C ∈ R m×n+ .Denote P m (C) the set of all Pareto optimal solutions.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionMeasuring deviation from the optimumRelative ErrorFor t 0 ∈ P m (C 0 ) and C ∈ R m×n+ , we introduced a so-called relativeerror of t 0 :ε P (C, t 0 ) = maxt∈TTransformation in scalar casemin f i (C, t 0 ) − f i (C, t).i∈N m f i (C, t)In the scalar case, i.e. for m=1, the Pareto set transforms into the setof optimal solutions. Therefore the relative error ε P (C, t 0 ) convertsinto:f 1 (C, t 0 ) − min f 1(C, t)t∈Tε P (C, t 0 ) =min f 1(C, t)t∈T.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionMeasuring deviation from the optimumRelative ErrorFor t 0 ∈ P m (C 0 ) and C ∈ R m×n+ , we introduced a so-called relativeerror of t 0 :ε P (C, t 0 ) = maxt∈TTransformation in scalar casemin f i (C, t 0 ) − f i (C, t).i∈N m f i (C, t)In the scalar case, i.e. for m=1, the Pareto set transforms into the setof optimal solutions. Therefore the relative error ε P (C, t 0 ) convertsinto:f 1 (C, t 0 ) − min f 1(C, t)t∈Tε P (C, t 0 ) =min f 1(C, t)t∈T.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionPertubed ProblemPerurbationsFor a given p ∈ [0, q(C 0 )), whereq(C 0 ) = min{c 0 i (e j) : i ∈ N m , j ∈ N n }, we consider a setΩ p (C 0 ) = {C ∈ R m×n+ : |c i (e j ) − c 0 i(e j )| ≤ p, i ∈ N m , j ∈ N n }.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionStability Function DefinitionDefinitionFor a Pareto optimal solution t 0 ∈ P m (C 0 ) and p ∈ [0, q(C 0 )), thevalue of the stability function is defined as follows:S P (t 0 , p) =max ε P(C, t 0 ).C∈Ω p(C 0 )Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionStability Function CalculationLet for t, t ′ ∈ T , t ⊗ t ′ = (t\t ′ ) ∪ (t ′ \t). Thus|t ⊗ t ′ | = |(t\t ′ ) ∪ (t ′ \t)| = |t| + |t ′ | − 2|t ∩ t ′ |.TheoremFor t 0 ∈ P m (C 0 ) and p ∈ [0, q(C 0 )), the stability function can beexpressed by the formula:S P (t 0 , p) = maxt∈Tmin f i (C 0 , t 0 ) − f i (C 0 , t) + p|(t ⊗ t 0 )|i∈N m f i (C 0 ., t) − p|t|Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionStability Function CalculationLet for t, t ′ ∈ T , t ⊗ t ′ = (t\t ′ ) ∪ (t ′ \t). Thus|t ⊗ t ′ | = |(t\t ′ ) ∪ (t ′ \t)| = |t| + |t ′ | − 2|t ∩ t ′ |.TheoremFor t 0 ∈ P m (C 0 ) and p ∈ [0, q(C 0 )), the stability function can beexpressed by the formula:S P (t 0 , p) = maxt∈Tmin f i (C 0 , t 0 ) − f i (C 0 , t) + p|(t ⊗ t 0 )|i∈N m f i (C 0 ., t) − p|t|Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionStability Radius DefinitionDefinitionThe stability radius R S P (t0 ) is defined as:R S P (t0 ) = sup{p ∈ [0, q(C 0 )) : S P (t 0 , p) = 0}.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionStability Radius CalculationTheoremFor t 0 ∈ P m (C 0 ), the stability radius is expressed by the formulaRP(t S 0 ) = min{q(C 0 ), min max f i (C 0 , t) − f i (C 0 , t 0 )t∈T \{t 0 } i∈N m |(t ⊗ t 0 },)|Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

ExampleOutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionExampleConsider the vector traveling salesman problem defined ongraph G = K 4E = {e 1 , e 2 , ..., e 6 }T = {t 1 , t 2 , t 3 },t 1 = {e 1 , e 2 , e 5 , e 6 }, t 2 = {e 1 , e 3 , e 4 , e 6 }, t 3 = {e 2 , e 3 , e 4 , e 5 }1 1 1e e3 3 e3e1 e2 e1 e2 e1e24 4 4e5 e6 e5 e6 e5e62 e 3 2 e 3 2 e 34 4 4Figure: All Hamiltonian cycles in graph K 4Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

ExampleOutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionExampleC 0 =[ 2 2 3 2 1 13 1 1 4 3 3]f(C 0 , t 1 ) = (9, 9), f(C 0 , t 2 ) = (7, 10), f(C 0 , t 3 ) = (6, 11)P 2 (C 0 ) = {t 1 , t 2 , t 3 }Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

ExampleOutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionExample0.80.60.40.2Figure: Stability functions S P(t 1, p), S P(t 2, p), S P(t 3, p)Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionStability AnalysisFuture Research Perspectivesdeveloping heuristics to calculate stability function and stabilityradiusspecifying attainable lower and upper boundsconsidering problems with non-linear objectivesextending the results for problems with various optimalityprinciplesand many others...Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineOutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusion1 The World of UncertaintyReasons for UncertaintyExtreme Consequences of UncertaintyEfficient Use of UncertaintyModeling Uncertainty2 Robust Optimization3 Stability Analysis4 ConclusionYury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionReliable ModelingTake care about initial parameters in your model to make it morerealistic and reliable!... Initial data is the shaky bridge above the precipice separating themodel and the reality. It is necessary to take care of its stability androbustness in order to not stumble on the way to the target...Unknown.Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

OutlineThe World of UncertaintyRobust OptimizationStability AnalysisConclusionQuestions and AnswersThank you for your time and interest!Yury NikulinStability and Robustness: Reliability in the World of Uncertainty

Multicriteria Optimization andSensitivity Analysispresented by16/04/2009 Yury Nikulin, Sensitivity analysis 1

Introduction: Sensitivity analysis vsRobust optimizationSensitivity analysis and stability theoryoriginate from postoptimality analysis, i.e.an optimal solution has been found and we studydifferent aspect of ist optimality under possibleuncertainty of problem parametersRobust optimization deals with constructing a newrobust counterpart model which somehow takesinto account inaccuracy of input data and searches fora solution optimizing a new specific objective16/04/2009 Yury Nikulin, Sensitivity analysis 2

General problem16/04/2009 Yury Nikulin, Sensitivity analysis 3

Dealing with uncertainty16/04/2009 Yury Nikulin, Sensitivity analysis 4

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

Worst-case or maximum regretoptimization: bibliography16/04/2009 Yury Nikulin, Sensitivity analysis 7

Telecommunication and MST16/04/2009 Yury Nikulin, Sensitivity analysis 8

Single commodity reformulation forminimum spanning tree problem16/04/2009 Yury Nikulin, Sensitivity analysis 9

Multicommodity reformulation forminimum spanning tree problem16/04/2009 Yury Nikulin, Sensitivity analysis 10

Multicommodity reformulation forminimum spanning tree problem16/04/2009 Yury Nikulin, Sensitivity analysis 11

Dual problem16/04/2009 Yury Nikulin, Sensitivity analysis 12

Multicommodity Flow Problem.Reformulation for MSTLet A = {( i,j)∈V× V :{ i,j}∈ E}D = ( V , A)− digraph originated by graph G = (V,E)fykijijbe the flow of commodity k on arc(i,j)be an integer variable defining a capacity for the flowof each commodity on arc(i,j)only if that arc is a memberof directed spanning tree defined by the vector y.16/04/2009 Yury Nikulin, Sensitivity analysis 13

Interval data and scenariosG = ( E,Vs ∈ S,cTsij∈[l, u= { t | t = ( V , E), e = ( i,j)∈ E,ijijt]is fixed for any ( i,)},cst=[ lij∑( i,j)∈, uE tijc], 0sij,

RST problem reformulation as mixedinteger programming∑k ≠1∑k ≠1σkijσσ≥∑ke e ke∈Ek∈V, k≠1k1min ux − ( α −α ) −(n-1) μs.t.+ μ ≤ l + ( u −l ) x ∀{ i, j} ∈E(1)kij ij ij ij ij+ μ ≤ l + ( u −l ) x ∀{ i, j} ∈E(2)kji ij ij ij ijαkj∑−α ∀(i, j) ∈A ∀k ∈V \{ 1 }(3)kiσ , α ≥ 0, μ −unrestricted16/04/2009 Yury Nikulin, Sensitivity analysis 15

RST problem reformulation as mixedinteger programming( i,j)∈A≤ (n-1)xf∑ijfij−∑( j,i)∈Aijfji=⎧ n −1if i = 1⎨⎩-1if ∀i∈V\{1}∀{i,j}∈ E(4)(5)≤ (n-1)xfij∑e∈Exf ≥e= n-10, xeij∈{01 , }∀{i,j}∈E(6)(7)16/04/2009 Yury Nikulin, Sensitivity analysis 16

Exact algorithms•Kouvelis and Yu 1997 – first formulation for finite set of scenarios. Ithas been proven that the problem is NP-hard for bounded set S andstrongly NP-hard for unbounde set S.•Yaman, Karasan and Pinar (2001) – basic theoretical backgrounds forthe problem•Aron and Hentenryck (2002) – one more evidence of NP-hardness: theRST problem is related to the well-known central tree problem. Twocases when the RST problem is easily solvable:a) no edge intervals have intersectionsb) graph is complete and all edge costs take their value from the sameinterval•Montemanni and Gambardella (2003) – branch and bound procedurewhich outperforms all exact algorithms known before16/04/2009 Yury Nikulin, Sensitivity analysis 17

Robust Spanning Tree Problem withinterval data16/04/2009 Yury Nikulin, Sensitivity analysis 18

Weak edges and weak treeDefinition. A spanning tree is a weak tree if it is a MST for somerealization s of edge costs. An edge e is a weak edge if it belongsto some weak tree.Theorem.1. A spanning tree t is a weak tree iff it is a MST when the costsof every edge on this tree corresponds to its lower bounds andthe costs of other edges correspond to their upper bounds.2. Edge e is a weak edge iff there exists a MST using edge ewhere the cost of edge e is at its lower bound and the costs ofthe remaining edges are at their upper bounds.16/04/2009 Yury Nikulin, Sensitivity analysis 19

Example: weak edgesweaknon-weak12-57-1553-46-9 8-1022-76-101-3 4-943-8316/04/2009 Yury Nikulin, Sensitivity analysis 20

Strong edgesDefinition. An edge e is a strong edge if it lies on a minimumspanning tree for all realization of edge costs.Theorem.Edge e is a strong edge iff there exists a MST using edge e wherethe cost of edge e is at ist upper bound and the costs of theremaining edges are at their lower bounds.16/04/2009 Yury Nikulin, Sensitivity analysis 21

Example: Strong edgesstrong12-553-422-71-3 4-943-8316/04/2009 Yury Nikulin, Sensitivity analysis 22

RST and weak tree. PreprocessingProposition 1. A RST is a weak tree, i.e. it is a MST for somerealization of edge costs.Proposition 2. A non-weak edge e can be deleted from graph Gwhen solving a RST. (Rule P1)Proposition 3. An edge which lies on some MST for eachscenario (strong edge) can be inserted into the RST. (Rule P2)16/04/2009 Yury Nikulin, Sensitivity analysis 23

CPLEX results on complete graphs16/04/2009 Yury Nikulin, Sensitivity analysis 24

Branch and bound. The search tree16/04/2009 Yury Nikulin, Sensitivity analysis 25

Branching strategyNode d with the smallest value of lb(d),d is not a leafin(d‘)=in(d)out(d‘)=out(d) ∪{e},e∈t(d)\in(d)in(d‘‘)=in(d) ∪{e},out(d‘‘)=out(d),e∈t(d)\in(d)t(d‘‘)=t(d)16/04/2009 Yury Nikulin, Sensitivity analysis 26

Selection of e and calculation of t(d′)16/04/2009 Yury Nikulin, Sensitivity analysis 27

Lower bound16/04/2009 Yury Nikulin, Sensitivity analysis 28

Summary of branch and bound16/04/2009 Yury Nikulin, Sensitivity analysis 29

SA: preprocessing, initial solution andsearch space16/04/2009 Yury Nikulin, Sensitivity analysis 30

SA: neighborhood search moveCase1.16/04/2009 Yury Nikulin, Sensitivity analysis 31

SA: neighborhood search moveCase2.16/04/2009 Yury Nikulin, Sensitivity analysis 32

Acceptance probability16/04/2009 Yury Nikulin, Sensitivity analysis 33

Running time16/04/2009 Yury Nikulin, Sensitivity analysis 34

Efficiency16/04/2009 Yury Nikulin, Sensitivity analysis 35

SA summary• In spite of SA is generally oriented on findinghigh quality solution in long run, we achievedboth good quality and acceptable efficiency• The algorithm can be considered aspromising on large instances when all exactstrategies are time consuming• The algorithm is still open for furtherimprovements16/04/2009 Yury Nikulin, Sensitivity analysis 36

ConclusionsThere is no unique understanding of robustness.There are several approaches, each of them hasown advantages and disadvantages.Which one to choose for a particular problemis a very difficult question which has to be solved bythe decision maker based on his\her preferences,expectations and managing experience!The material for this presentation ic compiled from several different sources:•Mustafa Pinar „Recent Developments in Robust Optimization“•Papers by Dimitras Betsimas et al. , Yaman et al. and Montemanni et al.16/04/2009 Yury Nikulin, Sensitivity analysis 37

16/04/2009 Yury Nikulin, Sensitivity analysis 38

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Heuristics for Central Tree ProblemYury NikulinDepartment of Mathematics, University of Turku

The Central TreeProblemHeuristics for theCTPFinal RemarksThe Central Tree ProblemHeuristics for the Central Tree Problem Yury Nikulin - p. 2/33

Distance MeasureThe Central TreeProblemLet G = (V, E) be a connected graph and letT G := {T | T = (V, E T )} represent the set of all spanning treesof graph G.Heuristics for theCTPFinal RemarksHeuristics for the Central Tree Problem Yury Nikulin - p. 3/33

Distance MeasureThe Central TreeProblemHeuristics for theCTPFinal RemarksLet G = (V, E) be a connected graph and letT G := {T | T = (V, E T )} represent the set of all spanning treesof graph G.Definition. The distance between two spanning trees T 1 and T 2of a graph is defined as a half of cardinality of the symmetricdifference between set of edges E T1 and E T2 :Dist(T 1 , T 2 ) = 1 ˙|T 1 △ T 2 |= 1 1 ∪ T 2 )\(T 1 ∩ T 2 ) |2 2˙|(THeuristics for the Central Tree Problem Yury Nikulin - p. 3/33

Distance MeasureThe Central TreeProblemHeuristics for theCTPFinal RemarksLet G = (V, E) be a connected graph and letT G := {T | T = (V, E T )} represent the set of all spanning treesof graph G.Definition. The distance between two spanning trees T 1 and T 2of a graph is defined as a half of cardinality of the symmetricdifference between set of edges E T1 and E T2 :Dist(T 1 , T 2 ) = 1 ˙|T 1 △ T 2 |= 1 1 ∪ T 2 )\(T 1 ∩ T 2 ) |2 2˙|(TDefinition. The distance between two spanning trees T 1 and T 2of a graph is defined as the number of edges presented in onetree but not in the other:Dist(T 1 , T 2 ) =| T 1 \T 2 |=| T 2 \T 1 | .Heuristics for the Central Tree Problem Yury Nikulin - p. 3/33

The Central Tree ProblemThe Central TreeProblemHeuristics for theCTPRecall that the cospanning tree ¯T of a spanning tree T is theedge complement of T in G, i.e. ¯T = (V, E ¯T), E ¯T = E\E T .Also, if a graph G has n vertices and c connected components,then the rank ρ(G) of graph G is n − c.Final RemarksHeuristics for the Central Tree Problem Yury Nikulin - p. 4/33

The Central Tree ProblemThe Central TreeProblemHeuristics for theCTPFinal RemarksRecall that the cospanning tree ¯T of a spanning tree T is theedge complement of T in G, i.e. ¯T = (V, E ¯T), E ¯T = E\E T .Also, if a graph G has n vertices and c connected components,then the rank ρ(G) of graph G is n − c.Definition. [Deo, 1966]The Central Tree Problem (shortly CTP) consists in finding centraltree T ∗ such that the rank ρ( ¯T ∗ ) of cospanning tree ¯T ∗ isminimal, i.e.ρ( ¯T ∗ ) ≤ ρ( ¯T) ∀ T ∈ T G .Heuristics for the Central Tree Problem Yury Nikulin - p. 4/33

The Central Tree ProblemThe Central TreeProblemHeuristics for theCTPFinal RemarksRecall that the cospanning tree ¯T of a spanning tree T is theedge complement of T in G, i.e. ¯T = (V, E ¯T), E ¯T = E\E T .Also, if a graph G has n vertices and c connected components,then the rank ρ(G) of graph G is n − c.Definition. [Deo, 1966]The Central Tree Problem (shortly CTP) consists in finding centraltree T ∗ such that the rank ρ( ¯T ∗ ) of cospanning tree ¯T ∗ isminimal, i.e.ρ( ¯T ∗ ) ≤ ρ( ¯T) ∀ T ∈ T G .Thus, a central tree is a spanning tree whose edge removalleads to splitting graph G into the maximum number ofcomponents.Heuristics for the Central Tree Problem Yury Nikulin - p. 4/33

The CTP vs. MDTPThe Central TreeProblemThe Maximally Distant Tree Problem (MDTP) consists infinding a pair of extremal trees with the largest distance.Heuristics for theCTPFinal RemarksHeuristics for the Central Tree Problem Yury Nikulin - p. 5/33

The CTP vs. MDTPThe Central TreeProblemHeuristics for theCTPFinal RemarksThe Maximally Distant Tree Problem (MDTP) consists infinding a pair of extremal trees with the largest distance.Definition. The goal of the CTP is to find a spanning tree suchthat the distance to maximally distant tree is minimized:minT ∈T GmaxT ′ ∈T GDist(T, T ′ ) (1)Heuristics for the Central Tree Problem Yury Nikulin - p. 5/33

The CTP vs. MDTPThe Central TreeProblemHeuristics for theCTPFinal RemarksThe Maximally Distant Tree Problem (MDTP) consists infinding a pair of extremal trees with the largest distance.Definition. The goal of the CTP is to find a spanning tree suchthat the distance to maximally distant tree is minimized:minT ∈T GmaxT ′ ∈T GDist(T, T ′ ) (1)Thus, CTP can be considered as dual to MDSTPHeuristics for the Central Tree Problem Yury Nikulin - p. 5/33

Short ExampleThe Central TreeProblemHeuristics for theCTPFinal RemarksFigure 1: Initial graph G and extremal spanning treesHeuristics for the Central Tree Problem Yury Nikulin - p. 6/33

Short ExampleThe Central TreeProblemHeuristics for theCTPFinal RemarksFigure 1: Initial graph G and extremal spanning treesFigure 2: Initial graph G, the central tree and the tree which ismaximally distant to the central treeHeuristics for the Central Tree Problem Yury Nikulin - p. 6/33

Complexity of the CTPThe Central TreeProblemTheorem. [ Bezrukov et al., 1996]The Central Tree Problem is NP-hardHeuristics for theCTPFinal RemarksHeuristics for the Central Tree Problem Yury Nikulin - p. 7/33

Complexity of the CTPThe Central TreeProblemHeuristics for theCTPFinal RemarksTheorem. [ Bezrukov et al., 1996]The Central Tree Problem is NP-hardTo prove this fact the authors used transformation from ExactCover by 3-set problem, which is known to be NP-CompleteHeuristics for the Central Tree Problem Yury Nikulin - p. 7/33

Complexity of the CTPThe Central TreeProblemHeuristics for theCTPFinal RemarksTheorem. [ Bezrukov et al., 1996]The Central Tree Problem is NP-hardTo prove this fact the authors used transformation from ExactCover by 3-set problem, which is known to be NP-CompleteThus, there is no polynomial algorithm for the CTP unlessP = NP .Heuristics for the Central Tree Problem Yury Nikulin - p. 7/33

Complexity of the CTPThe Central TreeProblemHeuristics for theCTPFinal RemarksTheorem. [ Bezrukov et al., 1996]The Central Tree Problem is NP-hardTo prove this fact the authors used transformation from ExactCover by 3-set problem, which is known to be NP-CompleteThus, there is no polynomial algorithm for the CTP unlessP = NP .A non-trivial class of graphs for which the CTP is known tohave polynomial algorithm is a class of complete graphs.Heuristics for the Central Tree Problem Yury Nikulin - p. 7/33

ApplicationsThe Central TreeProblemHeuristics for theCTPFinal RemarksThe CTP can find its application in many real-life models. For example,in computer networking, spanning tree topologies can beefficiently used to model network configurations.Similar network problems have many applications in telecommunications,circuit analysis, facility allocation, and related domainsin electric, gas, and sewer networks.Heuristics for the Central Tree Problem Yury Nikulin - p. 8/33

Equivalence between 0 − 1 RSTP and CTPThe Central TreeProblemThe following theorem establishes equivalence between 0 − 1RSTP and CTP.Heuristics for theCTPFinal RemarksHeuristics for the Central Tree Problem Yury Nikulin - p. 9/33

Equivalence between 0 − 1 RSTP and CTPThe Central TreeProblemHeuristics for theCTPFinal RemarksThe following theorem establishes equivalence between 0 − 1RSTP and CTP.Theorem. [Aron and Van Hentenryck, 2004] A spanning tree T ∈T G is a robust spanning tree of zero-one robust spanning treeproblem if and only if it is a central tree of G.Heuristics for the Central Tree Problem Yury Nikulin - p. 9/33

One more evidence of NP-hardnessThe Central TreeProblemHeuristics for theCTPFinal RemarksAverbakh and Lebedev proved in 2004 that the RSTP isNP-Hard even if the intervals of uncertainty are all equal to[0, 1]. They also proved that the RSTP is polynomially solvableif the number of edges with uncertain lengths is fixed orbounded by the logarithm of a polynomial function of the totalnumber of edges.Heuristics for the Central Tree Problem Yury Nikulin - p. 10/33

Robust deviation for the central treeThe Central TreeProblemHeuristics for theCTPFinal RemarksThus for a central tree T ∗ , the robust deviation is expressed as:dev s T ∗T ∗= maxT ∈T GDist(T ∗ , T).Let T ∗ be a central tree and N T ∗comp be the number ofcomponents of G − E T ∗. Then the robust deviation dev s T ∗T ∗N T ∗comp are related as follows:anddev s T ∗T ∗+ N T ∗comp = |E T ∗| + 1 = |V |Heuristics for the Central Tree Problem Yury Nikulin - p. 11/33

MILP FormulationCorollary from the result by Yaman et al., 2001The Central TreeProblemHeuristics for theCTPmin ∑ij∈Ex ij −∑k∈V, k≠1[α k k − α k 1] − (n − 1)µ (2)Final Remarkssubject toσij k ≥ αj k − αi k ∀{ij} ∈ A ∀k ∈ V \{1} (3)∑σij k + µ ≤ x ij ∀ij ∈ E (4)k≠1∑σji k + µ ≤ x ij ∀ij ∈ E (5)k≠1Heuristics for the Central Tree Problem Yury Nikulin - p. 12/33

MILP FormulationThe Central TreeProblem∑{ij}∈Af ij −∑{ji}∈Af ji ={n-1, if i = 1-1, if i ∈ V \{1}(6)Heuristics for theCTPf ij ≤ (n − 1)x ij ∀ij ∈ E (7)Final Remarksf ji ≤ (n − 1)x ij ∀ij ∈ E (8)∑ij∈Ex ij = n − 1 (9)f, σ, α ≥ 0, µ − unrestricted (10)x ij ∈ {0, 1} ∀ij ∈ E. (11)Heuristics for the Central Tree Problem Yury Nikulin - p. 13/33

Approximation for the CTPThe Central TreeProblemHeuristics for theCTPFinal RemarksFor the CTP, any spanning tree represents an approximated solutionwith approximation ratio 2 in terms of the robust deviation,i.e.∀T ∈ T G dev s TT≤ 2dev s T 0T, 0where T 0 is a central tree.Heuristics for the Central Tree Problem Yury Nikulin - p. 14/33

Case of two disjoint spanning treesThe Central TreeProblemHeuristics for theCTPFinal RemarksTheorem 1. Let T 1 and T 2 be two edge-disjoint spanning trees in the graphG = (V, E). Then dev s T ∗T≥ ⌈ ∗|V |−12⌉, where T ∗ is a central tree.In other words, theorem 1 states that if the graph contains twoedge-disjoint spanning trees, then the central tree has at most|V |−1⌊2⌋ edges in common with the maximally distant tree, or|V |−1equivalently, the robust spanning tree has at most ⌊2⌋edges in common with the corresponding minimum spanningtree in the worst-case scenario associated with T ∗ .Heuristics for the Central Tree Problem Yury Nikulin - p. 15/33

The Central TreeProblemHeuristics for theCTPFinal RemarksHeuristics for the CTPHeuristics for the Central Tree Problem Yury Nikulin - p. 16/33

MINCUTThe Central TreeProblemHeuristics for theCTPFinal RemarksAlgorithm MINCUT(G = (E, V ))Input: A connected graph G = (E, V )Output: A cut of minimum weight1. Let v 1 be any vertex of G and S = {v 1 }2. for i = 2 to |V |3. do v i = arg max {d(v, S)}v∈V \S4. S = S ∪ {v i }5. if |V | = 2 then return δ({v |V | })6. else7. G ′ is obtained from G by contracting {v |V |−1 , v |V | }in G8. C is returned by MINCUT(G ′ = (E ′ , V ′ ))9. Among C and δ(v |V | ) return the smallest cut interms of weightHeuristics for the Central Tree Problem Yury Nikulin - p. 17/33

Greedy Construction HeuristicThe Central TreeProblemHeuristics for theCTPFinal RemarksAlgorithm Greedy Construction Heuristic(G = (E, V ))Input: A connected graph G = (E, V )Output: The edges E 0 of a potential central tree T 01. Initialize G ′ = G, E 0 = {}, LC = [G]2. while |E 0 | < |V | − 13. do Take an arbitrary element C from LC4. Apply MINCUT(C) and return the set of edgesMINCUTSET(C)5. for all edges e ∈ MINCUTSET(C)6. do if E 0 + {e} does not contain cycles7. then E 0 = E 0 + {e}8. else do nothing9. Update LC by removing C and adding thosecomponents of C ′ = C − MINCUTSET(C) whichare not an isolated vertex10. return E 0Heuristics for the Central Tree Problem Yury Nikulin - p. 18/33

Complexity of Greedy strategyThe Central TreeProblemHeuristics for theCTPFinal RemarksWhile running of the greedy strategy we perform the MINCUTprocedure at most |V | − 1 times, so the complexity of our constructionheuristic is O(|E||V | 2 + |V | 3 log|V |). Notice also that theactual running time of the algorithm may also depend on the wayhow the components are ordered in the list LC.Heuristics for the Central Tree Problem Yury Nikulin - p. 18/33

ExampleThe Central TreeProblemHeuristics for theCTPFinal RemarksHeuristics for the Central Tree Problem Yury Nikulin - p. 19/33

ExampleThe Central TreeProblemHeuristics for theCTPFinal RemarksHeuristics for the Central Tree Problem Yury Nikulin - p. 19/33

ExampleThe Central TreeProblemHeuristics for theCTPFinal RemarksHeuristics for the Central Tree Problem Yury Nikulin - p. 19/33

ExampleThe Central TreeProblemHeuristics for theCTPFinal RemarksHeuristics for the Central Tree Problem Yury Nikulin - p. 19/33

ExampleThe Central TreeProblemHeuristics for theCTPFinal RemarksHeuristics for the Central Tree Problem Yury Nikulin - p. 19/33

Open ProblemThe Central TreeProblemHeuristics for theCTPFinal RemarksGiven graph G = (V, E). Is it true that if there existsa spanning tree T 0 and a sequence of MINCUTSETsMCS 1 , MCS 2 , ..., MCS k with |MCS 1 | + |MCS 2 | + |MCS k | =|V | − 1 where k is a number of iterations in GCH, such that T 0can be constructed as a union over all i of all edges e ∈ MCS i ,then T 0 is a central tree? In other words: suppose that when weexecute GCH all the edges from the MINCUTSET are included inthe final tree T 0 ; is it then true that T 0 is always a central tree?Heuristics for the Central Tree Problem Yury Nikulin - p. 20/33

The Central TreeProblemHeuristics for theCTPFinal RemarksFast Local Search ImprovementAlgorithm Fast Local Search Improvement(G = (E, V ), T )Input: A connected graph G = (E, V ), a spanning tree TOutput: An improved potential central tree T 01. Initialize T 0 = T2. Find a spanning tree ˆT which is maximally distant to T by solvingminimum spanning tree problem for the worst-case 0 − 1 scenarioassociated with T3. for all edges e ∈ E ˆT\E T4. do Let V 1 and V 2 be vertex sets of two components of ˆT − e5. if ∃ ẽ ∈ E\{E T ∪ E ˆT} such that one endpoint of ẽ belongs to V 1and the other endpoint belongs to V 2 :6. then choose another edge e ′ ∈ E ˆT\E T7. else Find the fundamental cycle C of T ∪ {e}8. for all edges ē ∈ E C , ē ≠ e9. do Find the fundamental cycle ¯C of ˆT ∪ {ē}10. if ∃ e ∗ ∈ E ¯C, e ∗ ≠ ē such that e ∗ ∈ E T :11. then choose another ē12. else construct a new spanning tree T 0 byreplacing edge ē with e13. break both cycles14. return T 0Heuristics for the Central Tree Problem Yury Nikulin - p. 21/33

Computational ExperimentsThe Central TreeProblemHeuristics for theCTPFinal RemarksLInstance A 30% 30%Instance B 30% 70%Instance C 70% 30%Instance D 70% 70%dTable 1: Instance generating parameters L and d.Heuristics for the Central Tree Problem Yury Nikulin - p. 22/33

Computational ExperimentsThe Central TreeProblemHeuristics for theCTPFinal Remarks|V | 10 15 20 40Instance A 17 24 46 148Instance B 11 21 35 106Instance C 21 50 82 297Instance D 16 27 47 158Table 2: Number of edges.Heuristics for the Central Tree Problem Yury Nikulin - p. 23/33

Computational ExperimentsThe Central TreeProblemHeuristics for theCTPFinal RemarksGCH 10 15 20 40Distance 6 6 11 33Running time (in sec.) 0.10 0.16 0.46 8.46GCH+FLSI 10 15 20 40Distance 5 6 11 31Running time (in sec.) 0.11 0.17 0.48 8.92SLS 10 15 20 40Distance 6 7 13 36Running time (in sec.) 0.32 0.94 30.64 1175.0ES 10 15 20 40Distance 5 6 - -Running time (in sec.) 14.92 486.0 - -Table 3: Instance sample A.Heuristics for the Central Tree Problem Yury Nikulin - p. 24/33

Computational ExperimentsThe Central TreeProblemHeuristics for theCTPFinal RemarksGCH 10 15 20 40Distance 6 12 17 87Running time (in sec.) 0.07 0.34 0.82 38.35GCH+FLSI 10 15 20 40Distance 6 12 17 37Running time (in sec.) 0.07 0.37 0.88 38.72SLS 10 15 20 40Distance 7 13 19 39Running time (in sec.) 0.46 0.62 52.33 2888.0ES 10 15 20 40Distance 6 - - -Running time (in sec.) 195.0 - - -Table 4: Instance sample B.Heuristics for the Central Tree Problem Yury Nikulin - p. 25/33

Computational ExperimentsThe Central TreeProblemHeuristics for theCTPFinal RemarksGCH 10 15 20 40Distance 2 7 12 29Running time (in sec.) 0.02 0.17 0.31 6.17GCH+FLSI 10 15 20 40Distance 2 6 12 26Running time (in sec.) 0.03 0.19 0.32 6.54SLS 10 15 20 40Distance 2 7 12 27Running time (in sec.) 0.07 0.59 3.79 453.9ES 10 15 20 40Distance 2 6 - -Running time (in sec.) 0.28 279.0 - -Table 5: Instance sample C.Heuristics for the Central Tree Problem Yury Nikulin - p. 26/33

Computational ExperimentsThe Central TreeProblemHeuristics for theCTPFinal RemarksGCH 10 15 20 40Distance 6 8 15 34Running time (in sec.) 0.08 0.19 0.47 15.9GCH+FLSI 10 15 20 40Distance 5 8 13 33Running time (in sec.) 0.09 0.24 0.57 16.3SLS 10 15 20 40Distance 6 8 15 38Running time (in sec.) 0.31 2.25 28.46 2742.7ES 10 15 20 40Distance 5 - - -Running time (in sec.) 12.69 - - -Table 6: Instance sample D.Heuristics for the Central Tree Problem Yury Nikulin - p. 27/33

Computational ExperimentsThe Central TreeProblemHeuristics for theCTPFinal Remarks|V | = 10 Instance A Instance B Instance C Instance DGCH 75 90 80 73GCH+FLSI 92 95 94 90Table 7: Average (over 100 runs) performance of GCHand.GCH+FLSI on instances with 10 vertices|V | = 15 Instance A Instance B Instance C Instance DGCH 7 9 8 7GCH+FLSI 9 9 9 8Table 8: Average (over 50 runs) performance of GCH andGCH+FLSI on instances with 15 verticesHeuristics for the Central Tree Problem Yury Nikulin - p. 28/33

Computational ExperimentsThe Central TreeProblemHeuristics for theCTPFinal Remarks|V | 50 100 150 200Instance A 15sec 1h 2h 6hInstance B 40sec 3h 6h 19hInstance C 13sec 50min 2h 6hInstance D 25sec 2h30min 7h 20hTable 9: Average (over 10 runs) running time of GCH+FLSI onlarge instances.Heuristics for the Central Tree Problem Yury Nikulin - p. 29/33

Computational ExperimentsThe Central TreeProblemHeuristics for theCTPFinal Remarks|V | 50 100 150 200GCH 30 59 89 120GCH+FLSI 29 56 81 115Lower Bound 25 50 75 100Running time 6sec 110sec 10min 40minTable 10: Large instances with graphs containing the union oftwo edge-disjoint spanning trees.Heuristics for the Central Tree Problem Yury Nikulin - p. 30/33

The Central TreeProblemHeuristics for theCTPFinal RemarksFinal RemarksHeuristics for the Central Tree Problem Yury Nikulin - p. 31/33

What to do nextThe Central TreeProblemHeuristics for theCTPFinal Remarks■ Find new mechanisms which can be incorporated into thegreedy strategy to make it deterministic and faster.■ Specify classes of graphs for which we can guarantee thequality of the solution found by the greedy constructionheuristic (e.g. being at most a factor of 1.5 from theoptimum). Here we measure in terms of the number ofconnected components in G − E T versus the number ofconnected components in G − E T ∗, where T ∗ is a centraltree■ Develop new approximation algorithms with guaranteedapproximation ratio better than 2 in terms of robust deviation.■ What is the worst case performance of the greedyconstruction heuristic in terms of the number of connectedcomponents in G − E T versus the number of connectedcomponents in G − E T ∗?Heuristics for the Central Tree Problem Yury Nikulin - p. 32/33

ConclusionThe Central TreeProblemHeuristics for theCTPFinal RemarksThe results represent a very first attempt to attack the CTP withheuristics. The proposed approach can be efficiently used tosolve zero-one RSTP. So, it may be considered as a first steptowards general intention to create a powerful methodological arsenalto deal with interval RSTP of larger sizes.Heuristics for the Central Tree Problem Yury Nikulin - p. 33/33

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Sensitivity Analysis in Game TheoryYury NikulinDepartment of Mathematics, University of Turku

ProblemFormulationStability andAccuracyFunctionsProblem FormulationSensitivity Analysis in Game Theory Yury Nikulin - p. 2/29

Game descriptionProblemFormulationStability andAccuracyFunctionsWe consider a coalition game with m ≥ 2 players. Let X i be afinite set of (pure) strategies of the playeri ∈ N m := {1, 2, ..., m}. We assume |X i | = 2 for all i ∈ N mindicating each player has a choice of 2 antagonistic strategiesto play. We also assume X i := {0, 1} for all i ∈ N m , i.e. thechoice of each strategy is encoded by means of Booleanvariables x i . The set of all solutions X can be generallydefined as a subset of Cartesian product over all players oftheir sets of strategiesX ⊆ ∏i∈N mX i = {0, 1} m .Observe that now - formally - X is a set of ordered m-tuples.Sensitivity Analysis in Game Theory Yury Nikulin - p. 3/29

Payoff functionsProblemFormulationStability andAccuracyFunctionsA vector of payoff functions (payoff profile)p(C, x) := (p 1 (C, x), ..., p m (C, x)) Tconsists of individual payoff functions p i (C, x) for each playeri ∈ N m , which are defined as linear functions on the set ofsolutions X:p i (C, x) := C i x.Here C i is i-th row of matrix C = [c ij ] ∈ R m×m+ ,x := (x 1 , x 2 , ..., x m ) T , x i ∈ X i , i ∈ N m .Note that for each player i ∈ N m , the individual payoff p i (C, x)depends on solution x, i.e. on the strategy chosen by player ias well as the strategies chosen by all the other players. Thusa set of payoff profiles P(C, X) is the followingP(C, X) := {p(C, x) : x ∈ X}.Sensitivity Analysis in Game Theory Yury Nikulin - p. 4/29

Forming a coalition, bansProblemFormulationStability andAccuracyFunctionsLet s be a number of coalitions in the game and letJ := (J 1 , J 2 , ..., J s ) be a set of coalitions formed. Let alsoX Jr := ∏j∈J rX j , J r ∈ J, r ∈ N s .be a set of strategy combinations accessible by coalition J r .The set of strategy combinations accessible by coalition J r ,with respect to the set of banned strategies B r , is defined as:X BrJ r:= X Jr \B r , J r ∈ J, r ∈ N s .Thus for a m-player s-coalition game with coalition profile(J 1 , J 2 , ..., J s ) and ban profile B := (B 1 , B 2 , ..., B s ), a set ofsolutions is defined as:X := ∏XJ Brr.r∈N sSensitivity Analysis in Game Theory Yury Nikulin - p. 5/29

Feasible solutionsProblemFormulationStability andAccuracyFunctionsFor any solution x ∗ ∈ X, any coalition J r ∈ J, r ∈ N s , letW Jr (x ∗ ) be a set of all non-banned solutions accessible bycoalition J r from x ∗ by changing strategies of only thoseplayers who are the members of J r :WJ Brr(x ∗ ) := XJ Brr× ∏x ∗ i .i∈N m \J rFor brevity sake, we denote W Jr (x ∗ ) := W BrJ r(x ∗ ).Sensitivity Analysis in Game Theory Yury Nikulin - p. 6/29

EquilibriaProblemFormulationStability andAccuracyFunctionsWe introduce the notion of equilibrium for the m-players-coalition game with matrix C and coalition profileJ = (J 1 , J 2 , ..., J s ). A solution x ∗ ∈ X is called J-equilibrium if∀ r ∈ N s Λ r (C, J r , x ∗ ) = ∅,where{Λ r (C, J r , x ∗ ) := x ∈ W Jr (x ∗ ) :() ()}∀j ∈ J r p j (C, x) ≤ p j (C, x ∗ ) & ∃j ′ ∈ J r p j ′(C, x) < p j ′(C, x ∗ ) .Sensitivity Analysis in Game Theory Yury Nikulin - p. 7/29

Special casesProblemFormulationStability andAccuracyFunctionsObviously, if J = ({1}, {2}, ..., {m}) and B r = ∅ for all r ∈ N m(m coalitions, each coalition has one member, no bans), thenthe concept of J-equilibrium transforms into the well-knowconcept of the Nash equilibrium. On the other hand, ifJ = ({1, 2, ..., m}) and B = (∅) (one coalition that contains allplayers, no bans), then the concept of J-equilibrium transformsinto the well-know concept of the Pareto efficiency or Paretoequilibrium.For the game with matrix C, we denote PE m (C, X) andNE m (C, X) the set of Pareto and Nash equilibria, respectively.Sensitivity Analysis in Game Theory Yury Nikulin - p. 8/29

Original problemProblemFormulationStability andAccuracyFunctionsIt is assumed that the set of strategy profiles X is fixed, i.e wehave coalition and ban profiles J and B fixed, but the originalmatrix of coefficients may vary or be estimated with errors.Moreover, it is assumed that for some originally specifiedmatrix C 0 = {c 0 ij } ∈ Rm×m + we know a J-equilibrium x ∗ whichis an element of the set of all J-equilibria JE(C 0 , X).Sensitivity Analysis in Game Theory Yury Nikulin - p. 9/29

Relative errorProblemFormulationStability andAccuracyFunctionsNamely, in case of J-equilibrium we introduce forx ∗ ∈ JE(C 0 , X) and a given matrix C ∈ R m×m+ the relativeerror of this solution:ε(C, J, x ∗ ) := maxr∈N smax minx∈W J r (x∗ )p i (C, x ∗ ) − p i (C, x).i∈J r p i (C, x)Observe that for an arbitrary C ∈ R m×m+ we haveε(C, J, x ∗ ) ≥ 0. If ε(C, J, x ∗ ) > 0, then x ∗ ∉ JE(C, X) and thispositive value of the relative error may be treated as a measureof inefficiency of the strategy profile x ∗ for the game with matrixC.The equality ε(C, J, x ∗ ) = 0 formulates in general onlynecessary condition for x ∗ to be J-equilibrium, i.e.ε(C, J, x ∗ ) = 0 does not guarantee that x ∗ ∈ JE m (C, X).Sensitivity Analysis in Game Theory Yury Nikulin - p. 10/29

ExampleProblemFormulationStability andAccuracyFunctionsExample 1. Let m = 2, s = 1, J = ({1, 2}) and C 0 =(1 22 1Assume that B = ({(0, 0) T , (1, 1) T }), i.e. X = {x 1 , x 2 },x 1 = (0, 1) T , x 2 = (1, 0) T . Then P(C 0 , X) = {(2, 1) T , (1, 2) T }.If we consider the matrixC =(1 12 1),).then P(C, X) = {(1, 1) T , (1, 2) T }. Evidently, x 2 ∈ PE 2 (C 0 , X)and ε(C 0 , J, x 2 ) = 0, but x 2 ∉ PE 2 (C, X) and ε(C, J, x 2 ) = 0.Sensitivity Analysis in Game Theory Yury Nikulin - p. 11/29

ProblemFormulationStability andAccuracyFunctionsStability and Accuracy FunctionsSensitivity Analysis in Game Theory Yury Nikulin - p. 12/29

Stability FunctionProblemFormulationStability andAccuracyFunctionsFor a given ρ ∈ [0, q(C 0 , X)), whereq(C 0 , X) := min{c 0 ij , i ∈ N m, j ∈ N m }, we consider a setΩ ρ (C 0 ) := {C ∈ R m×m+ : |c ij − c 0 ij| ≤ ρ, i ∈ N m , j ∈ N m }.For a J-equilibrium x ∗ ∈ JE m (C 0 , X) and ρ ∈ [0, q(C 0 , X)), thevalue of the stability function is defined as follows:S(C 0 , J, x ∗ , ρ) :=max ε(C, J,C∈Ω ρ (C 0 ) x∗ ).Sensitivity Analysis in Game Theory Yury Nikulin - p. 13/29

Accuracy FunctionProblemFormulationStability andAccuracyFunctionsIn a similar way, for a given δ ∈ [0, 1), we consider a setΘ δ (C 0 ) := {C ∈ R m×m+ : |c ij − c 0 ij| ≤ δ · c 0 ij, i ∈ N m , j ∈ N m }.For a J-equilibrium x ∗ ∈ JE m (C 0 , X) and δ ∈ [0, 1), the valueof the accuracy function is defined as follows:A(C 0 , J, x ∗ , δ) :=max ε(C, J,C∈Θ δ (C 0 ) x∗ ).It is easy to check that S(C 0 , J, x ∗ , ρ) ≥ 0 for anyρ ∈ [0, q(C 0 , X)) as well as A(C 0 , J, x ∗ , δ) ≥ 0 for eachδ ∈ [0, 1).Sensitivity Analysis in Game Theory Yury Nikulin - p. 14/29

SufficiencyProblemFormulationStability andAccuracyFunctionsDenoteΩ ′ ρ(C 0 ) := {C ∈ R m×m+ : |c ij − c 0 ij| < ρ, i ∈ N m , j ∈ N m }.Proposition 1. For x ∗ ∈ JE m (C 0 , X) and ρ ∈ [0, q(C 0 , X)), we havex ∗ ∈ JE m (C, X) for any C ∈ Ω ′ ρ(C 0 ) if and only ifS(C 0 , J, x ∗ , ρ) = 0.DenoteΘ ′ δ(C 0 ) := {C ∈ R m×m+ : |c ij − c 0 ij| < δ · c 0 ij, i ∈ N m , j ∈ N m }.Proposition 2. For x ∗ ∈ JE m (C 0 , X) and δ ∈ [0, 1), we havex ∗ ∈ JE m (C, X) for any C ∈ Θ ′ δ (C0 ) if and only ifA(C 0 , J, x ∗ , δ) = 0.Sensitivity Analysis in Game Theory Yury Nikulin - p. 15/29

Stability and Accuracy RadiiProblemFormulationStability andAccuracyFunctionsFormally, the stability radius R S (C 0 , J, x ∗ ) and the accuracyradius R A (C 0 , J, x ∗ ) are defined in the following way:R S (C 0 , J, x ∗ ) := sup{ρ ∈ [0, q(C 0 , X)) : S(C 0 , J, x ∗ , ρ) = 0},R A (C 0 , J, x ∗ ) := sup{δ ∈ [0, 1) : A(C 0 , J, x ∗ , δ) = 0}.Sensitivity Analysis in Game Theory Yury Nikulin - p. 16/29

Formula for Stability FunctionProblemFormulationStability andAccuracyFunctionsTheorem 1. For x ∗ ∈ JE m (C 0 , X) and ρ ∈ [0, q(C 0 , X)), the stabilityfunction can be expressed by the formula:S(C 0 , J, x ∗ , ρ) = maxr∈N smax min Ci 0(x∗ − x) + ρ ‖ x ∗ − x ‖ 1x∈W J r (x∗ ) i∈J r Ci 0x − ρ ‖ x ‖ 1.(1)Sensitivity Analysis in Game Theory Yury Nikulin - p. 17/29

Sketch of the proofProblemFormulationStability andAccuracyFunctionsFirst we will prove that S(C 0 , J, x ∗ , ρ) ≤ Γ(C 0 , J, x ∗ , ρ), whereΓ(C 0 , J, x ∗ , ρ) is the right-hand side of (1). Indeed, changingthe order of minimums and maximums, we getS(C 0 , J, x ∗ , ρ) =max ε(C, J,C∈Ω ρ (C 0 ) x∗ ) =maxC∈Ω ρ (C 0 ) maxr∈N smaxr∈N smax min p i (C, x ∗ ) − p i (C, x)x∈W J r (x∗ ) i∈J r p i (C, x)max minx∈W J r (x∗ ) i∈J rmaxC∈Ω ρ (C 0 )C i (x ∗ − x)C i x.≤Sensitivity Analysis in Game Theory Yury Nikulin - p. 18/29

Sketch of the proofProblemFormulationStability andAccuracyFunctionsFor any fixed r ∈ N s and x ∈ W Jr (x ∗ ) and all i ∈ J r themaximum of C i(x ∗ −x)C i xover C ∈ Ω ρ (C 0 ) is attained whenc ∗ ij =⎧⎪⎨⎪⎩c 0 ij − φ, if x∗ j = 0, i ∈ J r, j ∈ N m ;c 0 ij + φ, if x∗ j = 1, i ∈ J r, j ∈ N m ;c 0 ij , if i ∉ J r, j ∈ N m ,where φ ≥ 0 is taken to satisfy C ∗ ∈ Ω ρ (C 0 ). Thenmaxr∈N smaxx∈W J r (x∗ ) mini∈J rC ∗ i (x∗ − x)C ∗ i x =(2)maxr∈N smaxx∈W J r (x∗ ) mini∈J rC 0 i (x∗ − x) + ρ ‖ x ∗ − x ‖ 1C 0 i x − ρ ‖ x ‖ 1= Γ(C 0 , J, x ∗ , ρ).Sensitivity Analysis in Game Theory Yury Nikulin - p. 19/29

Sketch of the proofProblemFormulationStability andAccuracyFunctionsSo, we have that S(C 0 , J, x ∗ , ρ) ≤ Γ(C 0 , J, x ∗ , ρ). Now itremains to show that S(C 0 , J, x ∗ , ρ) ≥ Γ(C 0 , J, x ∗ , ρ). Considera matrix C ∗ with elements defined according to (2) for anyr ∈ N s . ThenS(C 0 , x ∗ , J, ρ) =maxC∈Ω ρ (C 0 ) ε(C, J, x∗ ) ≥ ε(C ∗ , J, x ∗ ) =maxr∈N smaxr∈N smaxx∈W J r (x∗ ) mini∈J rC ∗ i (x∗ − x)C ∗ i x =max min Ci 0(x∗ − x) + ρ ‖ x ∗ − x ‖ 1x∈W J r (x∗ ) i∈J r Ci 0x − ρ ‖ x ‖ 1So, we have that S(C 0 , x ∗ , J, ρ) ≥ Γ(C 0 , x ∗ , J, ρ). Thiscompletes the proof.= Γ(C 0 , J, x ∗ , ρ).Sensitivity Analysis in Game Theory Yury Nikulin - p. 20/29

Formula for Stability RadiusProblemFormulationStability andAccuracyFunctionsTheorem 2. For x ∗ ∈ JE m (C 0 , X), the stability radius can be expressedby the formula:R S (C 0 , J, x ∗ ) = min{q(C 0 , X), minr∈N smin max Ci 0(x∗ − x)}.x∈W J r (x∗ )\{x ∗ } i∈J r ‖ x ∗ − x ‖ 1(3)Sensitivity Analysis in Game Theory Yury Nikulin - p. 21/29

Sketch of the proofProblemFormulationStability andAccuracyFunctionsIf ρ = 0, then S(C 0 , J, x ∗ , 0) = 0. Assume thatS(C 0 , J, x ∗ , ρ) > 0. Using formula (1) specified by theorem 1,we derive that S(C 0 , J, x ∗ , ρ) > 0 if and onlymaxr∈N smax min Ci 0(x∗ − x) + ρ ‖ x ∗ − x ‖ 1x∈W J r (x∗ ) i∈J r Ci 0x − ρ ‖ x ‖ 1> 0.Last inequality holds if and only ifρ ≥ ˜ρ := minr∈N sminx∈W J r (x∗ )\{x ∗ } maxi∈J rC 0 i (x − x∗ )‖ x − x ∗ ‖ 1.Thus, if ˜ρ ≤ q(C 0 , X), then we get that S(C 0 , J, x ∗ , ρ) = 0 onthe interval [0, ˜ρ). Otherwise the stability function is equal tozero on [0, q(C 0 , X)). This ends the proof.Sensitivity Analysis in Game Theory Yury Nikulin - p. 22/29

Independent players, no bansProblemFormulationStability andAccuracyFunctionsFrom theorem 2, we get the following resultsCorollary 1. For a game with m independent players and no bans(J = ({1}, ..., {m}), B r = ∅ for all r ∈ N m ) the stability radius of aNash equilibrium x ∗ ∈ NE m (C 0 , X) can be expressed by the formulaR S (C 0 , J, x ∗ ) = mini∈N mc 0 ii.In other words, corollary 1 states that the stability radius ofx ∗ ∈ NE m (C 0 , X) is defined by elements on the principaldiagonal of C 0 .Sensitivity Analysis in Game Theory Yury Nikulin - p. 23/29

Independent players, no bansProblemFormulationStability andAccuracyFunctionsNotice that, if the game does not accept bans, then the valueof stability (accuracy) radius is determined by principaldiagonal elements, i.e. R S (C 0 , J, x ∗ ) = q(C 0 , X)(R A (C 0 , J, x ∗ ) = 1). It means that in this case the stability andaccuracy functions cannot provide any additional informationabout game solution stability but only duplicate informationderived by means of stability and accuracy radii. That suggeststo use stability and accuracy functions as efficient tools ofstability analysis only for the games with numerous bans.Sensitivity Analysis in Game Theory Yury Nikulin - p. 24/29

Accuracy Function and RadiusProblemFormulationStability andAccuracyFunctionsTheorem 3. For x ∗ ∈ JE m (C 0 , X) and δ ∈ [0, 1), the accuracyfunction can be expressed by the formula:A(C 0 , J, x ∗ , δ) = maxr∈N smax minx∈W J r (x∗ )Ci 0(x∗ − x) + δ ∑ c 0 ij |x∗ j − x j|j∈N mi∈J r (1 − δ)Ci 0x .Theorem 4. For x ∗ ∈ JE m (C 0 , X) and δ ∈ [0, 1), the accuracy radiuscan be expressed by the formula:R A (C 0 , J, x ∗ , δ) = min{1, minr∈N sminx∈W J r (x∗ )\{x ∗ } maxCi 0(x∗ − x)∑i∈J r c 0 ij |x∗ j − x j|j∈N m}.Sensitivity Analysis in Game Theory Yury Nikulin - p. 25/29

ExampleProblemFormulationStability andAccuracyFunctionsLet m = 3, s = 2, J = ({1}, {2, 3}) and⎛1 2 1C 0 ⎜= ⎝ 2 1 21 3 2⎞⎟⎠.Assume also that B = (∅, {(0, 0) T , (1, 1) T }), i.e.X = {x 1 , x 2 , x 3 , x 4 }, x 1 = (0, 0, 1) T , x 2 = (0, 1, 0) T ,x 3 = (1, 0, 1) T , and x 4 = (1, 1, 0) T .Then C 0 x 1 = (1, 2, 2) T , C 0 x 2 = (2, 1, 3) T , C 0 x 3 = (2, 4, 3) T ,C 0 x 4 = (3, 3, 4) T , and JE 3 (C 0 , X) = {x 1 , x 2 }. Using formula(3), we calculate R 3 (C 0 , J, x 1 ) = 1 2 and R3 (C 0 , J, x 2 ) = 1 2 . Ithappens that for both solutions x 1 and x 2 , the stability radius isequal to 1 2 . To get more information about x1 and x 2 , we maycheck the behavior of these solutions when they becomenon-equilibria by means of calculating stability functions oninterval [0, q(C 0 , X)), q(C 0 , X) = 1 using formula (1):Sensitivity Analysis in Game Theory Yury Nikulin - p. 26/29

ExampleProblemFormulationStability andAccuracyFunctions{S(C 0 , J, x 1 , ρ) = max{S(C 0 , J, x 2 , ρ) = maxmax { 0, −1 + ρ2 − 2ρmax{0, −1 + ρ3 − 2ρ}, max{0, min{1 + 2ρ1 − ρ}, max{0,−1 + 2ρ2 − ρ } },,−1 + 2ρ3 − ρwhere ρ ∈ [0, 1).Sensitivity Analysis in Game Theory Yury Nikulin - p. 27/29

ExampleProblemFormulation1.0Stability andAccuracyFunctions0.750.50.250.00.00.250.5rho0.751.0Figure 1: Stability functions S(C 0 , J, x 1 , ρ) and S(C 0 , J, x 2 , ρ)for ρ ∈ [0, 1).Graphics for S(C 0 , J, x 1 , ρ) (continuous line) and S(C 0 , J, x 2 , ρ)(dotted line) are depicted in figure 1.Sensitivity Analysis in Game Theory Yury Nikulin - p. 28/29

Concluding remarksProblemFormulationStability andAccuracyFunctionsThe accuracy and stability functions describe the quality ofequilibrium in the game with uncertain coefficients in payoffs.The definitions of these functions are directly connected withgiven optimality principles. While the stability and accuracyradii determine only maximum values of independentperturbations preserving the property of being equilibrium for agiven solution, the stability and accuracy functions willadditionally give the answer what happens with the solutionafter it becomes non-optimal in terms of quantitative measureof stability reflecting the speed of relative deterioration of thegiven solution.Sensitivity Analysis in Game Theory Yury Nikulin - p. 29/29

presented by28/04/2009 Yury Nikulin, Sensitivity analysis 1

Fuzziness: one more way of dealingwith uncertainty28/04/2009 Yury Nikulin, Sensitivity analysis 2

Membership Function28/04/2009 Yury Nikulin, Sensitivity analysis 3

Crisp and Fuzzy Numbers28/04/2009 Yury Nikulin, Sensitivity analysis 4

Various shapes of membershipfunctions28/04/2009 Yury Nikulin, Sensitivity analysis 5

Geometrical interpretation: Union28/04/2009 Yury Nikulin, Sensitivity analysis 6

Geometrical interpretation:Inversion or complement28/04/2009 Yury Nikulin, Sensitivity analysis 7

Geometrical interpretation:Intersection28/04/2009 Yury Nikulin, Sensitivity analysis 8

Probability vs Possibility28/04/2009 Yury Nikulin, Sensitivity analysis 9

Fuzzy sets28/04/2009 Yury Nikulin, Sensitivity analysis 10

Five-point Fuzzy Number28/04/2009 Yury Nikulin, Sensitivity analysis 11

Comparison of fuzzy numbers:strong comparison rulesμ(x)1λεR28/04/2009 This work has been supported by DFG 12

Comparison of fuzzy numbers:incomparable numbersμ(x)1λεR28/04/2009 This work has been supported by DFG 13

Maximal and Minimalfuzzy numbersμ(x)1λεR28/04/2009 This work has been supported by DFG 14

Fuzzy arithmetics28/04/2009 Yury Nikulin, Sensitivity analysis 15

Fuzzy maximum and minimum28/04/2009 Yury Nikulin, Sensitivity analysis 16

Comparison of fuzzy numbers: weakand strong comparison rules28/04/2009 Yury Nikulin, Sensitivity analysis 17

Statistical data about flight delays28/04/2009 Yury Nikulin, Sensitivity analysis 18

Accumulative delays28/04/2009 Yury Nikulin, Sensitivity analysis 19

Possibility distribution28/04/2009 Yury Nikulin, Sensitivity analysis 20

Nested Interval approximation28/04/2009 Yury Nikulin, Sensitivity analysis 21

Fuzzy Number representing delays28/04/2009 Yury Nikulin, Sensitivity analysis 22

Fuzzy Flight Gate Scheduling28/04/2009 Yury Nikulin, Sensitivity analysis 23

28/04/2009 Yury Nikulin, Sensitivity analysis 24

ConclusionsThere is no unique understanding of dealing with uncertainty.There are several approaches, each of them hasown advantages and disadvantages.Which one to choose for a particular problemis a very difficult question which has to be solved bythe decision maker based on own preferences,expectations and managing experience!28/04/2009 Yury Nikulin, Sensitivity analysis 25

28/04/2009 Yury Nikulin, Sensitivity analysis 26