Combinatorial Optimization inspired by Uncertainties Arie M.C.A. Koster Operations Research 2018 Brussels, September 14, 2018
Take away message Uncertainties complicates Optimization but understanding the complexity increase helps (and is fun) Case I: developing polyhedral theory further Case II: reformulating to known problems Case III: determining complexity border Joint works with Christina B¨ using, Timo Gersing, Alexandra Grub, Manuel Kutschka, Wlademar Laube, Nils Spiekermann, Martin Tieves Arie M.C.A. Koster – RWTH Aachen University 2 / 38
Outline Case I: Combinatorial Optimization under Uncertainty 1 Case II: Uncertainty-driven Generalizations 2 Case III: Uncertainty-driven novel Combinatorial Optimization 3 Concluding Remarks 4 Arie M.C.A. Koster – RWTH Aachen University 3 / 38
Motivation: Bandwidth Packing Problem Given network topology link dimensioning demands Find routing Observations: single path routing binary decision on single link → 0-1 Knapsack Problem demand values are uncertain Arie M.C.A. Koster – RWTH Aachen University 4 / 38
Motivation: Bandwidth Packing Problem Given network topology link dimensioning demands Find routing Observations: single path routing binary decision on single link → 0-1 Knapsack Problem demand values are uncertain Arie M.C.A. Koster – RWTH Aachen University 5 / 38
Optimization under Uncertainty Robust Optimization according to Ben-Tal and Nemirovski: Uncertain Linear Program An Uncertain Linear Optimization problem (ULO) is a collection of linear optimization problems (instances) � � min { c T x : Ax ≤ b } ( c , A , b ) ∈U where all input data stems from an uncertainty set U ⊂ R n × R m × n × R m . Robust Knapsack Problem � � c T x : { a T x ≤ b , x ∈ { 0 , 1 } n } a ∈U max How to define U ? Arie M.C.A. Koster – RWTH Aachen University 6 / 38
Uncertainty Sets How to define the uncertainty set? Uncertainty set is an ellipsoid, e.g., U = { a ∈ R n : � a − ¯ a � < κ } Uncertainty set is a polyhedron, e.g., U = { a ∈ R n : D · a ≤ d } with D ∈ R k × n , d ∈ R k for some k ∈ N . 6 equivalent: set of discrete scenarios (extreme 5 a 3 points of polyhedron) 4 6 1 1 . 5 4 2 a 2 2 . 5 special case: Γ-Robustness; 3 a 1 � n � a ∈ R n : a i = ¯ � δ i ≤ Γ , δ ∈ { 0 , 1 } n U (Γ) = a i + ˆ a i δ i , i =1 Arie M.C.A. Koster – RWTH Aachen University 7 / 38
Γ-Robust Knapsacks Γ-Robust Knapsack polytope: � � x ∈ { 0 , 1 } | N | : � � a i x i ≤ b ∀ S ⊆ N , | S | ≤ Γ conv a i ¯ a i x i + ˆ i ∈ N i ∈ S Cover inequalities for Knapsack: Extended Cover inequalities: E ( C ) := C ∪ { i : a i ≥ max j ∈ C a j } : Set C with a ( C ) > b : x ( C ) ≤ | C | − 1 x ( E ( C )) ≤ | C | − 1 How to define covers for Γ-robust knapsack? C ⊆ N is a Γ − robust cover : ∃ S ⊆ C with | S | ≤ Γ and ¯ a ( C ) + ˆ a ( S ) > b What about the extension? Arie M.C.A. Koster – RWTH Aachen University 8 / 38
Scenario Extensions Scenario Extension ( C , S ) a cover-pair if S ⊆ C , | S | ≤ Γ, and ¯ a ( C ) + ˆ a ( S ) > b . Extension for cover-pair ( C , S ): � � E ( C , S ) := C ∪ i ∈ N \ C : ¯ a i ≥ max j ∈ C \ S ¯ a j , ¯ a i + ˆ a i ≥ max j ∈ S (¯ a j + ˆ a j ) . Lemma (B¨ using, K., Kutschka (2011)) � x j ≤ | C | − 1 is a valid inequality for all cover-pairs ( C , S ) . j ∈ E ( C , S ) Arie M.C.A. Koster – RWTH Aachen University 9 / 38
Example Scenario Extensions Scenario Extension � � E ( C , S ) := C ∪ i ∈ N : ¯ a i ≥ max j ∈ C \ S ¯ a j , ¯ a i + ˆ a i ≥ max j ∈ S (¯ a j + ˆ a j ) . n = 6 items i 1 2 3 4 5 6 b = 21 capacity a i ¯ 5 5 3 3 4 5 Γ = 2 robustness budget a i ˆ 3 3 3 3 4 1 C = { 1 , 2 , 3 , 4 } robust cover S 1 = { 1 , 2 } and S 2 = { 3 , 4 } build cover-pairs with C = { 1 , 2 , 3 , 4 } extensions E ( C , S 1 ) = C ∪ { 5 } and E ( C , S 2 ) = C ∪ { 6 } � but also x j ≤ 3 = | C | − 1 is valid j ∈ C ∪{ 5 , 6 } does there exist an extension E ( C ) = C ∪ { 5 , 6 } ? Arie M.C.A. Koster – RWTH Aachen University 10 / 38
Union of Extensions Union of Extensions S ( C ) := { S ⊆ C | ( C , S ) is a cover-pair } all cover-pairs with cover C : � E ( C ) := E ( C , S ) . S ∈S ( C ) Theorem (Gersing, 2017) Let C ⊆ N be a Γ − robust cover. Then � x j ≤ | C | − 1 j ∈E ( C ) is a valid inequality for the Γ -robust knapsack. Arie M.C.A. Koster – RWTH Aachen University 11 / 38
Outline Case I: Combinatorial Optimization under Uncertainty 1 Case II: Uncertainty-driven Generalizations 2 Case III: Uncertainty-driven novel Combinatorial Optimization 3 Concluding Remarks 4 Arie M.C.A. Koster – RWTH Aachen University 12 / 38
Energy System schematically Source: ProCom Arie M.C.A. Koster – RWTH Aachen University 13 / 38
Decentralized Energy Case Study Simultaneous production of heat and power in exchange for fuel Source: ProCom Fixed ratio ρ between heat and power generation Heat can be stored for future use, power cannot be stored Heat storage has limited capacity and loss factor Power has to be bought/sold at day-ahead market! Arie M.C.A. Koster – RWTH Aachen University 14 / 38
Lot-Sizing with Storage Deterioration LS-DET: T � min f ( q , z ) + h t u t (1a) t =1 ∀ t ∈ [ T ] s.t. α u t − 1 + q t = u t + d t (1b) U t ≤ u t ≤ U t ∀ t ∈ [ T ] (1c) Qz t ≤ q t ≤ Qz t ∀ t ∈ [ T ] (1d) q t , u t ≥ 0 ∀ t ∈ [ T ] (1e) z t ∈ { 0 , 1 } ∀ t ∈ [ T ] (1f) Lot-Sizing with Complexity Production limitations in general: open Storage limitations if Q = 0 , Q = ∞ , α = 1, f linear: LS-DET ∈ P (Love, 1973; Atamt¨ urk & K¨ u¸ c¨ ukyavuz, 2008) Deterioration of storage if U = 0 , U = ∞ , α = 1: LS-DET ∈ P (Hellion et al., 2012) Concave cost function both cases still in P if 0 < α < 1 (Schmitz, 2016) No backlogging What about uncertain demands? Arie M.C.A. Koster – RWTH Aachen University 15 / 38
Forecast & Actual Heat Demands Heat demands for week 45, 2007 40 forecast actual demands 35 heat demand (MWh) 30 25 20 hours 0 20 40 60 80 100 120 140 160 Forecast error of up to 20% (average: 4.1%) Find solutions that are feasible with high probability ! Arie M.C.A. Koster – RWTH Aachen University 16 / 38
Robust Lot-Sizing Uncertainty Set: U of possible demand realizations ( d t ) t ∈ [ T ] Applying Robust Optimization: α u t − 1 + q t = u t + d t (1b) Impossible to find ( q , z , u ) such that (1b)–(1f) are satisfied ∀ d ∈ U Theorem (folklore) Every (implicit) equality in Ax ≤ b allows for the elimination of a variable involved in the equality. ⇒ In robust optimization, elimination of variable x implies that this variable is moved 2nd stage, i.e., after the uncertain input is known! Arie M.C.A. Koster – RWTH Aachen University 17 / 38
Robust Lot-Sizing with Deterioration RLS-DET: min f ( q , z ) + η (2a) ∀ t ∈ [ T ] , d ∈ U s.t. α u t − 1 ( d ) + q t = u t ( d ) + d t (2b) U ≤ u t ( d ) ≤ U ∀ t ∈ [ T ] , d ∈ U (2c) � h t u t ( d ) η ≥ ∀ d ∈ U (2d) t ∈ [ T ] Qz t ≤ q t ≤ Qz t ∀ t ∈ [ T ] (2e) q t , u t ( d ) ≥ 0 ∀ t ∈ [ T ] (2f) z t ∈ { 0 , 1 } ∀ t ∈ [ T ] (2g) η ≥ 0 (2h) storage u t ( d ) per scenario d ∈ U Arie M.C.A. Koster – RWTH Aachen University 18 / 38
Solving RLS-DET as LS-DET instance Theorem For an uncertainty set U over which a linear function can be optimized in polynomial time, RLS-DET can be polynomially reduced (w.r.t. production ′ thus defined: plans) to an instance of LS-DET with d = d ′ and U = U t − 1 � � d ′ � α t − i � d ′ � d t − i − d i ∀ t ∈ [ T ] t := max (3a) d ∈U i =1 � t � ′ � α t − i � d ′ � U t := U t − max i − d i ∀ t ∈ [ T ] . (3b) d ∈U i =1 Arie M.C.A. Koster – RWTH Aachen University 19 / 38
Robust Lot-Sizing Corollary Given an uncertainty set U over which a linear function can be optimized in polynomial time, RLS-DET is in P (resp., NP -hard) if and only if the corresponding version of LS-DET is in P (resp., NP -hard). Robustness models satisfying precondition: polyhedral uncertainty sets, Γ-robustness discrete scenarios ellipsoidal uncertainty sets Arie M.C.A. Koster – RWTH Aachen University 20 / 38
Running times (96h) Distribution of running times for |U| = 50: 20 RLS-DET ′ LS-DET with d ′ , U 15 time (sec) 10 5 0 0 50 100 150 200 250 instances Speed-up factor between 1.82 and 85.67 with average 29.00 Arie M.C.A. Koster – RWTH Aachen University 21 / 38
Outline Case I: Combinatorial Optimization under Uncertainty 1 Case II: Uncertainty-driven Generalizations 2 Case III: Uncertainty-driven novel Combinatorial Optimization 3 Concluding Remarks 4 Arie M.C.A. Koster – RWTH Aachen University 22 / 38
Fixed vs. Flexgrid Optical Networks Capacity of optical fibre is huge, but limited! Idea: More efficient usage of optical channels 1 Technology: Fixed grid vs. Flexgrid 1 Figure taken from “Innovative Future Optical Transport Network Technologies” by T. Morioka et al., NTT Technical Review, 9 (2011). Arie M.C.A. Koster – RWTH Aachen University 23 / 38
Recommend
More recommend
