Practical Robust Optimization - an introduction - - LNMB

Overview■Motivation for Robust Optimization (RO)◆◆Flaw of using nominal valuesWhy RO can make a difference■■■■■■Robust Optimization MethodologyDeriving tractable Robust CounterpartsPractical issuesPractical exampleAdjustable Robust OptimizationConcluding remarksF A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N2 / 53

Google results ...■■■■■■■”Robust Optimization” – 186,000 hits” ... and logistics” – 30,000 hits” ... and supply chain” – 34,000 hits” ... and production” – 67,000 hits” ... and energy” – 74,000 hits” ... and finance” – 35,000 hits” ... and engineering” – 116,000 hitsF A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N3 / 53

What sets great optimization apart?“The Optimization Edge” (2011) by Steve ShashiharaOne of the eight differentiators for “great optimization” mentioned:Good optimization gives you the best choice based on the data.Great optimization is “robust” and resilient in the face of changeand data errors.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N4 / 53

Uncertainty in optimization problemsOptimization problems often contain uncertain parameters, due to■■■measurement/rounding errorse.g. temperature, current inventoryestimation errorse.g. demand, cost or pricesimplementation errorse.g. length, depth, width, voltageFlaw of using nominal values in optimization problems.RO: find a solution that is robust against this uncertainty in theparameters.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N5 / 53

Flaw of using nominal valuesOptimization based on nominal values often lead to (severe)infeasibilities.Taken from “Flaw of averages” (2009,2012), Sam Savage.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N6 / 53

Flaw of using nominal valuesOptimization based on nominal values often lead to (severe)infeasibilities.Consider the constraint: a T x ≤ b, where■a is uncertain,āis nominal value.■ a = ā+ρζ, and−1 ≤ ζ i ≤ 1.■ζ is uniformly distributed.Assume:this constraint is binding in the optimal nominal solution ¯x(fora = ā).F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N7 / 53

Flaw of using nominal valuesConstraint: a T x = (ā+ρζ) T x = ā T x+ρζ T x ≤ b.Probability of infeasibility= 0.5!F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N8 / 53

Flaw of using nominal valuesOptimization based on nominal values often lead to (severe)infeasibilities.Consider the binding constraints: (a k ) T x ≤ b k ,■a k is uncertain,ā k is nominal value.■ a k = ā k +ρ k ζ k , and−1 ≤ ζ k i ≤ 1.■ζ k are independent and uniformly distributed.k = 1,..,N.Assume: theseN constraints are binding in the optimal solution(fora = ā).Probability of infeasibility= 1− ( 12) N.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N9 / 53

Flaw of using nominal valuesOptimization based on nominal values often lead to (severe)infeasibilities.Consider the constraints: 3.000x 1 −2.999x 2 +... ≤ 1.Suppose:■■■x 1 = x 2 = 1000 is optimalthe number3.000 is uncertainmaximal deviation is1%.Then LHS of constraint can be3.030∗1,000−2.999∗1,000+.... = 31 > 1F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N10 / 53

Flaw of using nominal valuesConsider PILOT4 from NETLIB (1,000 variables, 410 constraints).Constraint # 372:ā T x = −15.79081x 826 −8.598819x 827 −1.88789x 828 −1.362417x 829 −1.526049x 830−0.031883x 849 −28.725555x 850 −10.792065x 851 −0.19004x 852 −2.757176x 853−12.290832x 854 +717.562256x 855 −0.057865x 856 −3.785417x 857 −78.30661x 858−122.163055x 859 −6.46609x 860 −0.48371x 861 −0.615264x 862 −1.353783x 863−84.644257x 864 −122.459045x 865 −43.15593x 866 −1.712592x 870 −0.401597x 871+x 880 −0.96049x 898 −0.946049x 916≥ b = 23.387405Most coefficients are ”ugly reals” (like - 15.79081).Highly unlikely that coefficients are known to this accuracy.Only exception: the coefficient 1 atx 880 - it perhaps reflects thestructure of the problem and might be exact.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N11 / 53

Flaw of using nominal valuesā T x = −15.79081x 826 −8.598819x 827 −1.88789x 828 −1.362417x 829 −1.526049x 830−0.031883x 849 −28.725555x 850 −10.792065x 851 −0.19004x 852 −2.757176x 853−12.290832x 854 +717.562256x 855 −0.057865x 856 −3.785417x 857 −78.30661x 858−122.163055x 859 −6.46609x 860 −0.48371x 861 −0.615264x 862 −1.353783x 863−84.644257x 864 −122.459045x 865 −43.15593x 866 −1.712592x 870 −0.401597x 871+x 880 −0.96049x 898 −0.946049x 916≥ b = 23.387405Suppose: accuracy is 0.1%:(∗)∣ atruei −ā i∣ ∣ ≤ 0.001|āi |.Worst case: the constraint can be violated by as much as 450%:mina true satisfies (∗)(atrue ) T¯x−b = −128.8 ≈ −4.5b.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N12 / 53

Flaw of using nominal valuesAssume “random uncertainty”:a truei = ā i +ǫ i |ā i |, ǫ i ∼ Uniform[−0.001,0.001]Based on 1,000 simulations:V = max[ b−(a true ) T¯x|b|],0.Prob {V >0} Prob{V >150%} Mean(V)0.50 0.18 125%=⇒ The nominal solution is highly “unreliable”F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N13 / 53

Flaw of using nominal valuesAmong 90 NETLIB LP problems:■In 19 problems0.01% - perturbation−→ more than 5%-violations of (someof) the constraints■In 13 of these 19 problems0.01%-perturbation −→ more than 50% violations of theconstraints.■In 6 of these 13 problemsconstraint violaton was over 100%; in one problem even210,000%.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N14 / 53

Why RO can make a difference ...Extreme situation:Take solutionx 2 on the optimal facet, which is much more robust!F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N15 / 53

Why RO can make a difference ...Extreme situation: x 2 −ζx 1 = 1, where−0.1 ≤ ζ ≤ 0.1.Takex 2 on optimal facet, which is (much) more robust!F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N16 / 53

History of Robust Optimization■ Early work by Soyster (1973) and Kouvelis (1997).■■Started with papers in 1997-1998: Ben-Tal, El Ghaoui,Nemirovski.Since 2000 many papers on RO.■ Budget uncertainty set (Bertsimas and Sim, 2004).■ Adjustable Robust Optimization (Ben-Tal et al. 2004).■■Book: Robust Optimization (2009) by Ben-Tal, El Ghaoui, andNemirovski.Practical relevance: many applications!F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N17 / 53

BookImportant chapters: Preface, Chapter 1 and Chapter 14.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N18 / 53

Relation with sensitivity analysis (SA)■■■SA is post analysis (“post mortem”).SA only analyzes infinitesimal changes (shadowprices,reduced costs).SA only analyzes changes in one or a few parameters.Robust Optimization:■■takes data uncertainty into account already at the modellingstage;to ”immunize” solutions against uncertainty.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N19 / 53

Relation with Stochastic Programming“Large” data uncertainty is modelled in a stochastic fashion andthen processed via Stochastic Programming techniques.Disadvantages:■■■Often difficult to specify reliably the distribution of uncertaindata.Expectations/probabilities often difficult to calculate.Feasible region often nonconvex.Robust Optimization:■■does not assume stochastic nature of the uncertain data;remains computationally tractable.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N20 / 53

Basic assumptions in RO1. The optimization variables represent “here and now”decisions.2. The decision maker is fully responsible for decisions made(only) when the actual data is within the uncertainty set.3. The constraints of the uncertain problem in question are“hard”.Later on:[1] will be alleviated by “adjustable robust optimization”;[3] will be alleviated by “globalized robust optimization”.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N21 / 53

Introduction to robust linear optimizationRobust counterpart:(RC)max{c T x : Ax ≤ b, ∀A ∈ U},wherex ∈ R n , andU is a given uncertainty region.We may assume w.l.o.g.:■■■■objective is certainRHS is certainU is closed and convexconstraintwise uncertainty.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N22 / 53

Hence, we focus onwhere■■(a+Bζ) T x ≤ β, ∀ζ ∈ Z,ζ ∈ R L is the primitive uncertain vectorB ∈ R n×L■ Z is the uncertainty region; often|L| ≪ n.Example: factor model in portfolio problems.This is a semi-infinite optimization problem.Goal: obtain computationally tractable reformulation!F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N23 / 53

Tractable robust counterparts(a+Bζ) T x ≤ β, ∀ζ ∈ Z.Uncertainty region Z Robust Counterpart TractabilityBox ‖ζ‖ ∞ ≤ ρ a T x+ρ‖B T x‖ 1 ≤ β LPBall/ellipsoidal ‖ζ‖ 2 ≤ ρ a T x+ρ‖B T x‖ 2 ≤ β CQP⎧⎪⎨ a T x+d T y ≤ βPolyhedral Dζ +d ≥ 0 D⎪⎩T y = −B T x LPy ≥ 0⎧⎪⎨ a T x+d T y ≤ βCone (closed, convex, pointed) Dζ +d ∈ K D⎪⎩T y = −B T xy ∈ K ∗Conic Opt.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N24 / 53

RC for box uncertaintyThis is equivalent to:(a+Bζ) T x ≤ β ∀ζ : ‖ζ‖ ∞ ≤ 1. (1)max (a+Bζ) T x = a T x+ max (B T x) T ζ ≤ β.ζ:‖ζ‖ ∞ ≤1ζ:‖ζ‖ ∞ ≤1We have:maxζ:‖ζ‖ ∞ ≤1 (BT x) T ζ = maxζ:|ζ i |≤1∑(B T x) i ζ i = ∑ ii|(B T x) i | = ‖B T x‖ 1 .Hence, (1) is equivalent to:a T x+‖B T x‖ 1 ≤ β.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N25 / 53

RC for ball uncertaintyThis is equivalent to:(a+Bζ) T x ≤ β ∀ζ : ‖ζ‖ 2 ≤ 1. (2)a T x+ max (B T x) T ζ ≤ β,ζ:‖ζ‖ 2 ≤1Intermezzo: max ζ {d T ζ : ‖ζ‖ 2 ≤ 1} ormax{d T ζ : ζ T ζ ≤ 1}Lagrange: d = λζ, andλ = ‖d‖ 2 .Optimal objective value: dT dλ = ‖d‖ 2.Hence (2) is equivalent to:a T x+‖B T x‖ 2 ≤ β.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N26 / 53

This is equivalent to:Note that by dualityRC for polyhedral uncertainty(a+Bζ) T x ≤ β ∀ζ : Dζ +d ≥ 0.a T x+ maxζ : Dζ+d≥0 (BT x) T ζ ≤ β. (3)max{(B T x) T ζ : Dζ +d ≥ 0} = min{d T y : D T y = −B T x, y ≥ 0}.Hence (3) is equivalent toa T x+miny{d T y : D T y = −B T x, y ≥ 0} ≤ β,ora T x+d T y ≤ β, D T y = −B T x, y ≥ 0.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N27 / 53

Budget uncertainty regionOften used uncertainty region (Bertsimas and Sim, 2004):{ζ : ‖ζ‖ ∞ ≤ 1, number of elements ≠ 0 at most Γ}which is equivalent to (only for linear uncertain constraints):{ζ : ‖ζ‖ ∞ ≤ 1, ‖ζ‖ 1 ≤ Γ}.This is a polyhedral uncertainty set.Hence, (RC) can be reformulated as LP.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N28 / 53

ExampleAccuracy in uncertain data is 0.1%, in the worst case.Robust Counterpart:−15.79081x 826 −8.598819x 827 −1.88789x 828 −1.362417x 829 −1.526049x 830−0.031883x 849 −28.725555x 850 −10.792065x 851 −0.19004x 852 −2.757176x 853−12.290832x 854 +717.562256x 855 −0.057865x 856 −3.785417x 857 −78.30661x 858−122.163055x 859 −6.46609x 860 −0.48371x 861 −0.615264x 862 −1.353783x 863−84.644257x 864 −122.459045x 865 −43.15593x 866 −1.712592x 870 −0.401597x 871+x 880 −0.96049x 898 −0.946049x 916−0.001 ∑ i |ā ix i | ≥ b = 23.387405Robust solution:■does not violate the constraints■ increase in objective value is less than 1%!Remember: nominal F A C U L Tsolution Y O F E C O N O M I Cyields S A N D B U S I Na E S450% S A D M I N I S T Rviolation!A T I O N29 / 53

Planning of Air Traffic ControllersMaster thesis by Dori van Hulst (Quintiq)■■■■■Planning is done 2-3 months ahead.Different “shift types” can be used.Under capacity is dangerous, hence costly.Nominal prolbem is a MIP model.Uncertainty in demand=#planes in air corridors.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N30 / 53

Planning of Air Traffic ControllersUncertainty in demand per day:This is polyhedral uncertainty!F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N31 / 53

Planning of Air Traffic ControllersResults:Costs Under =4.2 Costs Under =1.9Nominal Robust Nominal RobustLabour costs nominal curve 433 762 426 675Labour costs worst case 2054 882 1179 795Average labour costs 1293 759 830 694Average std per day of av. costs 13.2 2.4 6.2 2.2Average under-utilization (hours) 67 6 67 14Percentage of times better 14% 86% 34% 66%F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N32 / 53

Why ellipsoidal uncertainty?1. Resulting Robust Counterpart is tractable (namely CQP).2. Much less conservative than box uncertainty.3. Arises naturally from statistical considerations:Confidence sets in regressionConfidence set for covariance...4. Is a safe approximation for chance constraint.1, 2 and 4 also hold for budget uncertainty!F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N33 / 53

Safe approximation of chance constraintSuppose: ζ i are stochastic and independent, with knownsupport, say[−1,1], and zero mean.Consider: U Ω = {ζ : ‖ζ‖ 2 ≤ Ω} (ball)Ifxsatisfies (4) then:(RC) (a+Bζ) T x ≤ β, ∀ζ ∈ U Ω . (4)Prob ( (a+Bζ) T x ≤ β ) ≥ 1−exp(−Ω 2 /2) =: 1−ǫ.Example: Ω = 7.44 =⇒ ǫ = 10 −12 .Same result forU Ω = {ζ : ‖ζ‖ 2 ≤ Ω, ‖ζ‖ ∞ ≤ 1} (ball-box).F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N34 / 53

Is Robust Optimization too pessimistic?■■■■■■In many cases the constraint is strict, e.g. safety restrictions.User can adapt level of protection.Overly pessimistic solutions may be caused by modelingerrors.Use Globalized Robust Counterparts.Robust solutions often perform well on the average.Chance constraints can be approximated.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N35 / 53

Bridge that was not robust ...F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N36 / 53

Recipe for RO■ Step 0.– Check whether nominal solution is robust.– Check whether SP can solve your problem.■ Step 1.– Determine uncertain parameters.– Determine uncertainty region.■ Step 2.– Derive tractable robust counterpart (exact or approxim.).■ Step 3.– Solve tractable robust counterpart.■ Step 4.– Check robustness of robust solution.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N37 / 53

Multistage problemsSome of the decisionsx j :■■should be made when actual data becomes partially knowncan depend on the corresponding portions of the data.Examples:■■Inventory problem with uncertain demand.e.g. replenishment ordersx t of daytusually can depend onactual demands at days1,...,t−1.Brachytherapy with inaccuracy in positioning needles.e.g. position of needlekcan depend on actual position ofneedles1,...,k −1.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N38 / 53

Multistage problemsConsider the following constraint of a multi-stage problem:where(ARC) (a+Bζ) T x+d T y ≤ β, ∀ζ ∈ Z,■■x is non-adjustabley is adjustable (fixed recourse).ARC stands for Adjustable Robust Counterpart.Hence: y = y(ζ).This leads to an NP-hard problem!F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N39 / 53

Linear Decision Rule(ARC) (a+Bζ) T x+d T y ≤ β, ∀ζ ∈ Z,Linear decision rule y = u+Vζ is often very effective:(AARC) (a+Bζ) T x+d T (u+Vζ) ≤ β, ∀ζ ∈ Z,or equivalently(AARC) a T x+d T u+(B T x+V T d) T ζ ≤ β, ∀ζ ∈ Z.This problem is linear inxandζ; hence previous results apply!AARC stands for Affinely Adjustable Robust Counterpart.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N40 / 53

Example: inventory problems⎧min ∑ Tt=1 c tx t + ∑ Tt=1 h tI t⎪⎨ I t = I t−1 +x t −d t , ∀tI t ≥ 0, ∀t0 ≤ x t ≤ U t , ∀t⎪⎩...Demandd t is uncertain: we assume interval uncertainty.x t andI t are adjustable: linear decision rulesx t = u t +∑t−1i=1v it d i , I t = w t +∑t−1i=1z it d iOne can vary the “information base”!F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N41 / 53

AARC shows often very good behaviour (Ben-Tal et al., 2004):Unc.ty (%) Opt(ARC) Opt(AARC) Opt(RC)10 13531.8 13531.8 (+0.0%) 15033.4 (+11.1%)20 15063.5 15063.5 (+0.0%) 18066.7 (+19.9%)30 16595.3 16595.3 (+0.0%) 21100.0 (+27.1%)40 18127.0 18127.0 (+0.0%) 24300.0 (+34.1%)50 19658.7 19658.7 (+0.0%) 27500.0 (+39.9%)60 21190.5 21190.5 (+0.0%) 30700.0 (+44.9%)70 22722.2 22722.2 (+0.0%) 33960.0 (+49.5%)F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N42 / 53

whereZ 1 ⊂ Z 2 .Example:Globalized robust counterpart(a+Bζ) T x ≤ β +αdist(ζ,Z 1 ) ∀ζ ∈ Z 2 ,■ Z 1 = {ζ : ‖ζ‖ 2 ≤ ρ 1 }, Z 2 = {ζ : ‖ζ‖ ∞ ≤ ρ 2 },ρ 1 ≤ ρ 2 .■ dist(ζ,Z 1 ) = min ζ′ ∈Z 1‖ζ −ζ ′ ‖ 2 .Globalized Robust Counterpart (GRC):{a T x+ρ 2 ‖v‖ 1 +ρ 1 ‖B T x−v‖ 2 ≤ β‖B T x−v‖ 2 ≤ αThis is a system of conic quadratic constraints.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N43 / 53

Advantage of Globalized Robust Counterpart:■■■■Relaxes assumption [3]: ”Constraints of uncertain problemare hard”.Smooth behaviour.Often better ‘average behaviour’.GRC is still tractable for many choices ofZ 1 ,Z 2 , anddistance function.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N44 / 53

Modelling issuesRCs of two equivalent problems may not be equivalent.Example 1:(RP) (2+ζ)x 1 ≤ 1 ∀ζ : |ζ| ≤ 1{(2+ζ)x 1 +s = 1 ∀ζ : |ζ| ≤ 1(RP)s ≥ 0Nominal problems are equivalent, but:Feasible set for(RP): x 1 ≤ 1/3Feasible set for(RP): x 1 = 0.Try to avoid equality constraints (e.g. using elimination)!F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N45 / 53

Example 2:(RP) |x 1 −ζ|+|x 2 −ζ| ≤ 2 ∀ζ : |ζ| ≤ 1or(RP)⎧y 1 +y 2 ≤ 2⎪⎨ y 1 ≥ x 1 −ζ ∀ζ : |ζ| ≤ 1y 1 ≥ ζ −x 1 ∀ζ : |ζ| ≤ 1y 2 ≥ x 2 −ζ ∀ζ : |ζ| ≤ 1⎪⎩y 2 ≥ ζ −x 2 ∀ζ : |ζ| ≤ 1x = (1,−1) is feasible for(RP) but not for(RP)!Be careful with “splitting up” constraints!F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N46 / 53

Example 2:(RP) |x 1 −ζ|+|x 2 −ζ| ≤ 2 ∀ζ : |ζ| ≤ 1Correct reformulation:(RP)⎧x 1 −ζ +x 2 −ζ ≤ 2 ∀ζ : |ζ| ≤ 1⎪⎨x 1 −ζ −(x 2 −ζ) ≤ 2 ∀ζ : |ζ| ≤ 1−(x 1 −ζ)+x 2 −ζ ≤ 2 ∀ζ : |ζ| ≤ 1⎪⎩−(x 1 −ζ)−(x 2 −ζ) ≤ 2 ∀ζ : |ζ| ≤ 1.See Gorissen et al. (2012).F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N47 / 53

General remarks■■■■■RC reformulations also valid for MIP problems.Structure (e.g. network, total unimodularity) often destroyedby RC.Analyze robustness of nominal and robust optimal solution.There are no methods known that can deal with integeradjustable variables.RO methodology implemented in modelling packages asYalmip, ROME, AIMMS.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N48 / 53

Extensions■■■■■Tractable robust counterpart for other uncertainty regions,e.g. defined by separable convex functions.Tractable robust counterpart for problems with certain typesof non-affine uncertain parameters.Example: (1+ζ 2 1)x 1 +ζ 1 ζ 2 x 2 ≤ 5, ∀ζ : ‖ζ‖ 2 ≤ 1.Tractable (approximations of) robust counterpart for (conic)quadratic, SDP and other nonlinear problems.Tractable robust counterpart for certain classes of nonlineardecision rules.Globalized robust counterpart for nonlinear optimizationproblems.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N49 / 53

Application areas of ROHundreds of papers on applications in the following categories:■■■■■■■■■■■Circuit designSignal processing / signal estimationCommunicationControlStructural optimization / material design / shape designMechanical EngineeringOpticsApplied PhysicsChemical engineeringMedical applicationsComputer ScienceF A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N50 / 53

■■■■■■■■■■■■■Machine learningAgricultureWater resources / water management / hydrologyEnergy / EnvironmentVehicle routingInventory / Supply chain managementFacility locationTransportation / civil engineeringRevenue ManagementMaintenanceFinanceDesign of ExperimentsStatisticsF A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N51 / 53

Application: Cancer treatment■ Treatment plan optimization:– tumor should get prescribed dose– healthy organs should be spared.■ Formulated as a large LP probem.■ Uncertainties in location of tumor / organs.e.g. because of breathing■ Robust solution much better than nominal solution.■ See e.g.: Olafsson and Wright (2006).F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N52 / 53

Conclusions■■■■Practical optimization problems often contain uncertainparameters.RO is a tractable way to obtain robust solutions:Provides a natural way for modelling uncertainty.Robust counterpart is often tractable.The basic paradigm of RO is “worst-case”, but also oftenimproves average behaviour.Adjustable RO is an efficient way to solve multistageproblems.■RO implemented in modelling software packages.F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N53 / 53