Robust combinatorial optimization with variable uncertainty Michael Poss Heudiasyc UMR CNRS 7253, Universit´ e de Technologie de Compi` egne 17th Aussois Combinatorial Optimization Workshop M. Poss (Heudiasyc) Variable uncertainty Aussois 1 / 31
Outline Robust optimization 1 Variable budgeted uncertainty 2 Cost uncertainty 3 M. Poss (Heudiasyc) Variable uncertainty Aussois 2 / 31
Outline Robust optimization 1 Variable budgeted uncertainty 2 Cost uncertainty 3 M. Poss (Heudiasyc) Variable uncertainty Aussois 3 / 31
Combinatorial optimization under uncertainty n � min c i x i i =1 n � s.t. a ij x j ≤ b i , i = 1 , . . . , m j =1 x ∈ { 0 , 1 } n Suppose that the parameters ( a , b , c ) are uncertain: They vary over time They must be predicted from historical data They cannot be measured with enough accuracy ... Let’s do something clever (and useful)! M. Poss (Heudiasyc) Variable uncertainty Aussois 4 / 31
How much do we know? Stochastic programming Robust programming ��� � � �� � A lot ⇔ A little M. Poss (Heudiasyc) Variable uncertainty Aussois 5 / 31
How much do we know? Stochastic programming Robust programming ��� � � �� � A lot ⇔ A little Robust pr. Uncertain parameters are merely assumed to belong to an uncertainty set U ⇒ one wishes to optimize some worst-case objective over the uncertainty set Stochastic pr. Uncertain parameters are precisely described by probability distributions ⇒ one wishes to optimize some expectation, variance, Value-at-risk, . . . Intermediary models exist: distributionally robust optimization, ambiguous chance-constrained M. Poss (Heudiasyc) Variable uncertainty Aussois 5 / 31
When do we take decisions? Now All decisions must be taken before the uncertainty is known with precision ⇒ probability constraints, (static) robust optimization Delayed Some decisions may be delayed until the uncertainty is revealed ⇒ multi-stage stochastic programming, adjustable robust optimization M. Poss (Heudiasyc) Variable uncertainty Aussois 6 / 31
Robust combinatorial optimization n � min c i x i i =1 n � s.t. a ij x j ≤ b i , i = 1 , . . . , m , ∀ a i ∈ U i j =1 x ∈ { 0 , 1 } n , The linear relaxation of this problem is tractable if U i is defined by conic constraints: U i = { a i ∈ R n : u a i − v ∈ K } . In particular, polyhedrons and polytopes are nice ( K = R n + ). M. Poss (Heudiasyc) Variable uncertainty Aussois 7 / 31
Feasibility set � j c j α j ≤ b � � a i x i ≤ b , ∀ a ∈ U ⇔ j u ji α j ≥ x j α j ≥ 0 The feasibility set of the constraint is a polyhedron (thus, convex) ! M. Poss (Heudiasyc) Variable uncertainty Aussois 8 / 31
Feasibility set � j c j α j ≤ b � � a i x i ≤ b , ∀ a ∈ U ⇔ j u ji α j ≥ x j α j ≥ 0 The feasibility set of the constraint is a polyhedron (thus, convex) ! A (very) popular polyhedral uncertainty set is (Bertsimas and Sim, 2004): � � � U Γ := a ∈ R n : a i = a i + δ i ˆ a i , − 1 ≤ δ i ≤ 1 , | δ i | ≤ Γ . Main reasons for popularity: Nice computational properties for MIP and combinatorial problems. Intuitive interpretation. Probabilistic interpretation. M. Poss (Heudiasyc) Variable uncertainty Aussois 8 / 31
Consider a vehicle routing problem with uncertain travel times t and time limit T . For simplicity, we suppose t ∈ U Γ for Γ = 5. M. Poss (Heudiasyc) Variable uncertainty Aussois 9 / 31
Consider a vehicle routing problem with uncertain travel times t and time limit T . For simplicity, we suppose t ∈ U Γ for Γ = 5. Consider two robust-feasible routes x 1 and x 2 : � x 1 � 1 = 3 and � x 2 � 1 = 10. M. Poss (Heudiasyc) Variable uncertainty Aussois 9 / 31
Consider a vehicle routing problem with uncertain travel times t and time limit T . For simplicity, we suppose t ∈ U Γ for Γ = 5. Consider two robust-feasible routes x 1 and x 2 : � x 1 � 1 = 3 and � x 2 � 1 = 10. Because x 1 and x 2 are robust-feasible: � � t i ≤ T , t i ≤ T , ∀ a ∈ U Γ , ∀ a ∈ U Γ and i : x 1 i : x 2 i =1 i =1 M. Poss (Heudiasyc) Variable uncertainty Aussois 9 / 31
Consider a vehicle routing problem with uncertain travel times t and time limit T . For simplicity, we suppose t ∈ U Γ for Γ = 5. Consider two robust-feasible routes x 1 and x 2 : � x 1 � 1 = 3 and � x 2 � 1 = 10. Because x 1 and x 2 are robust-feasible: � � t i ≤ T , t i ≤ T , ∀ a ∈ U Γ , ∀ a ∈ U Γ and i : x 1 i : x 2 i =1 i =1 which becomes � � t i ≤ T , t i ≤ T , ∀ a ∈ U Γ and i : x 1 i : x 2 i =1 i =1 M. Poss (Heudiasyc) Variable uncertainty Aussois 9 / 31
Consider a vehicle routing problem with uncertain travel times t and time limit T . For simplicity, we suppose t ∈ U Γ for Γ = 5. Consider two robust-feasible routes x 1 and x 2 : � x 1 � 1 = 3 and � x 2 � 1 = 10. Because x 1 and x 2 are robust-feasible: � � t i ≤ T , t i ≤ T , ∀ a ∈ U Γ , ∀ a ∈ U Γ and i : x 1 i : x 2 i =1 i =1 which becomes � � t i ≤ T , t i ≤ T , ∀ a ∈ U Γ and i : x 1 i : x 2 i =1 i =1 For any probability distribution for t : � t i > T = 0 . P i : x 1 i =1 M. Poss (Heudiasyc) Variable uncertainty Aussois 9 / 31
Consider a vehicle routing problem with uncertain travel times t and time limit T . For simplicity, we suppose t ∈ U Γ for Γ = 5. Consider two robust-feasible routes x 1 and x 2 : � x 1 � 1 = 3 and � x 2 � 1 = 10. Because x 1 and x 2 are robust-feasible: � � t i ≤ T , t i ≤ T , ∀ a ∈ U Γ , ∀ a ∈ U Γ and i : x 1 i : x 2 i =1 i =1 which becomes � � t i ≤ T , t i ≤ T , ∀ a ∈ U Γ and i : x 1 i : x 2 i =1 i =1 For any probability distribution for t : � t i > T = 0 . P i : x 1 i =1 If x 2 is not robust-feasible for Γ = 10, there exists probability distributions: � t i > T > 0 P i : x 2 i =1 M. Poss (Heudiasyc) Variable uncertainty Aussois 9 / 31
Outline Robust optimization 1 Variable budgeted uncertainty 2 Cost uncertainty 3 M. Poss (Heudiasyc) Variable uncertainty Aussois 10 / 31
Robust optimization and probabilistic constraint Let ˜ a i be random variables and ǫ > 0. The chance constraint �� � P ˜ a i x i > b ≤ ǫ (1) leads to very difficult optimization problems in general. M. Poss (Heudiasyc) Variable uncertainty Aussois 11 / 31
Robust optimization and probabilistic constraint Let ˜ a i be random variables and ǫ > 0. The chance constraint �� � P ˜ a i x i > b ≤ ǫ (1) leads to very difficult optimization problems in general. In some situations, we know that (1) can be approximated by � a i x i ≤ b ∀ a ∈ U (2) for a properly chosen U . These approximations are conservative: any x feasible for (2) is feasible for (1). We must balance conservatism and protection cost ⇒ devise good protection sets U . M. Poss (Heudiasyc) Variable uncertainty Aussois 11 / 31
Robust optimization and probabilistic constraint What about U Γ ? M. Poss (Heudiasyc) Variable uncertainty Aussois 12 / 31
Robust optimization and probabilistic constraint What about U Γ ? Let ˜ a i be random variables independently and symmetrically distributed in [ a i − ˆ a i , a i + ˆ a i ]. Bertsimas and Sim (2004) prove that if a vector x satisfies the robust constraint � ∀ a ∈ U Γ , a i x i ≤ b then it satisfies also the probabilistic constraint � � − Γ 2 �� � a i x i > b ˜ ≤ exp . P 2 n M. Poss (Heudiasyc) Variable uncertainty Aussois 12 / 31
Something is wrong ... From � � − Γ 2 �� � P ˜ a i x i > b ≤ exp , 2 n we see that choosing Γ = ( − 2 ln( ǫ )) 1 / 2 n 1 / 2 yields �� � P ˜ a i x i > b ≤ ǫ. M. Poss (Heudiasyc) Variable uncertainty Aussois 13 / 31
Something is wrong ... From � � − Γ 2 �� � P ˜ a i x i > b ≤ exp , 2 n we see that choosing Γ = ( − 2 ln( ǫ )) 1 / 2 n 1 / 2 yields �� � P ˜ a i x i > b ≤ ǫ. For many problems, � x � 1 < n 1 / 2 for optimal (or feasible) vectors x (network design, assignment, ...) ⇒ Γ > n 1 / 2 already for ǫ = 0 . 5 ⇒ for these problems, protecting with probability 0 . 5 yields protection with probability 0! ⇒ overprotection ! M. Poss (Heudiasyc) Variable uncertainty Aussois 13 / 31
Multifunctions It is easy to see that the bound from Bertsimas and Sim can be adapted to � � Γ 2 �� � P ˜ a i x i > b ≤ exp − . 2 � x � 1 M. Poss (Heudiasyc) Variable uncertainty Aussois 14 / 31
Multifunctions It is easy to see that the bound from Bertsimas and Sim can be adapted to � � Γ 2 �� � P a i x i > b ˜ ≤ exp − . 2 � x � 1 Γ can be reduced when x is small Let’s use multifunctions ! M. Poss (Heudiasyc) Variable uncertainty Aussois 14 / 31
Recommend
More recommend
