Bilevel Optimization, Pricing Problems and Stackelberg Games Martine Labbé Computer Science Department Université Libre de Bruxelles INOCS Team, INRIA Lille Follower Leader CO Workshop - Aussois - January 2017 1
PART I: Bilevel optimization CO Workshop - Aussois - January 2017 2
Bilevel Optimization Problem max f ( x, y ) x,y s.t. ( x, y ) ∈ X y ∈ S ( x ) where S ( x ) = argmax g ( x, y ) y s.t.( x, y ) ∈ Y CO Workshop - Aussois - January 2017 3
Adequate framework for Stackelberg game • Leader: 1st level, • Follower: 2nd level, • Leader takes follower’s optimal reaction into account. (1905 - 1946) CO Workshop - Aussois - January 2017 4
Early papers on bilevel optimization •Falk (1973): Linear min-max problem •Bracken & McGill (1973): First bilevel model, structural properties, military application. CO Workshop - Aussois - January 2017 5
Applications •Economic game theory •Production planning •Revenue management • Security •… CO Workshop - Aussois - January 2017 6
Example: a linear BP y f max f 1 x + f 2 y OS x,y s.t. max g 1 x + g 2 y Y y g s.t.( x, y ) ∈ Y x Inducible region (IR) CO Workshop - Aussois - January 2017 7
Coupling constraints The follower sees only the second level constraints y f Infeasible BP Y X g x f y X Y g x CO Workshop - Aussois - January 2017 8
Multiple second level optima max xy y x ≥ 0 s.t. max y (1 − x ) y f s.t.0 ≤ y ≤ 1 Y x y y “Optimistic” “Pessimistic” f f Y Y x x CO Workshop - Aussois - January 2017 9
Multiple followers independent f ( x, y 1 , . . . , y n ) max x ( x, y 1 , . . . , y n ) ∈ X s.t. y k g ( x, y k ) , k = 1 , . . . , n max s.t.( x, y k ) ∈ Y k CO Workshop - Aussois - January 2017 10
Multiple followers dependent f ( x, y 1 , . . . , y n ) max x ( x, y 1 , . . . , y n ) ∈ X s.t. y k g ( x, y k , y k ) , k = 1 , . . . , n max s.t.( x, y k , y k )) ∈ Y k CO Workshop - Aussois - January 2017 11
Linear BP c 1 x + d 1 y max x } S A 1 x + A 1 y ≤ b 1 s.t. c 2 x + d 2 y, max [ ] y } T s.t. A 2 x + B 2 y ≤ b 2 CO Workshop - Aussois - January 2017 12
Linear BP • Linear BP is strongly NP-hard (Hansen et al. 1992) •MILP is a special case of Linear BP x ∈ { 0 , 1 } ⇔ v = 0 and v = argmax { w : w ≤ x, w ≤ 1 − x, w ≥ 0 } w •IR is not convex and may be disconnected. CO Workshop - Aussois - January 2017 13
Linear BP (Bialas & Karwan(1982), Bard(1983)). • IR is the union of faces of S ∩ T •If Linear BP is feasible, then there exists an optimal solution which is a vertex of S ∩ T. K-th best algorithm CO Workshop - Aussois - January 2017 14
Linear BP- single level reformulation max c 1 x + d 1 y max c 1 x + d 1 y x x s.t. A 1 x + B 1 y ≤ b 1 s.t. A 1 x + B 1 y ≤ b 1 B 2 y ≤ b 2 − A 2 x max d 2 y, λ B 2 = d 2 y s.t. ( λ ) B 2 y ≤ b 2 − A 2 x λ ( B 2 y − b 2 + A 2 x ) = 0 λ ≥ 0 • Branch & Bound (Hansen et al.1992) • Branch & Cut (Audet et al. 2007) CO Workshop - Aussois - January 2017 15
Linear BP - Other approaches • Complementary pivoting (Bialas & Karwan 1984) •Penalty functions (Anandalingam & White 1989) • Reverse convex programming (Al-Khayyal 1992) CO Workshop - Aussois - January 2017 16
Mixed Integer Linear BP P P • -hard (Lodi et al. 2014) 2 •Branch & Bound (Moore & Bard 1990, Fischetti et al. 2016) •Branch & Cut (DeNegre & Ralphs 2009, Fischetti et al. 2016) •Branch & Cut & Price (Han & Zeng 2014) CO Workshop - Aussois - January 2017 17
Some references • L.N. Vincente and P.H. Calamai (1994), Bilevel and multilevel programming : a bibiliography review, J. Global Optim. 5, 291-306. • S. Dempe. Foundations of bilevel programming. In Nonconvex optimization and its applications, volume 61. Kluwer Academic Publishers, 2002. • J. Bard. Practical Bilevel Optimisation: Algorithms and Applications. KluwerAcademic Publishers, 1998 • B. Colson, P. Marcotte, and G. Savard. Bilevel programming: A survey. 4 OR, 3:87-107, 2005. • M. Labbé and A. Violin. Bilevel programming and price setting problems. 4OR, 11:1-30, 2013. CO Workshop - Aussois - January 2017 18
PART II: Pricing problems CO Workshop - Aussois - January 2017 19
Adequate framework for Price Setting Problem max F ( T, x, y ) T ∈ Θ ,x,y s.t. min x,y f ( T, x, y ) s.t.( x, y ) ∈ Π CO Workshop - Aussois - January 2017 20
Applications CO Workshop - Aussois - January 2017 21
Price Setting Problem with linear constraints max Tx T,x,y s.t. TC ≥ f • Π = { x, y : Ax + By ≥ b } is bounded min ( c + T ) x + dy • { ( x, y ) ∈ Π : x = 0 } is nonempty x,y s.t. Ax + By ≥ b CO Workshop - Aussois - January 2017 22
Example: 2 variables in second level max Ty T,x,y s.t. min c 1 x + ( c 2 + T ) y x,y s.t. ( x, y ) ∈ Π CO Workshop - Aussois - January 2017 23
Example: 2 variables in second level CO Workshop - Aussois - January 2017 24
The first level revenue CO Workshop - Aussois - January 2017 25
Network pricing problem • network with toll arcs ( A 1 ) and non toll arcs ( A 2 ) • Costs c a on arcs • Commodities ( o k , d k , n k ) • Routing on cheapest (cost + toll) path • Maximize total revenue CO Workshop - Aussois - January 2017 26
Example 9 2 2 2 0 3 1 4 5 2 10 12 • UB on ( T 1 + T 2 ) = SPL( T = ∞ ) − SPL( T = 0) = 22 − 6 = 16 • T 2 , 3 = 5 , T 4 , 5 = 10 CO Workshop - Aussois - January 2017 27
Example with negative toll arc 2 2 0 0 0 1 2 3 4 T 12 = 4 T 23 = − 2 T 34 = 4 6 CO Workshop - Aussois - January 2017 28
Network pricing problem (Labbé et al., 1998, Roch at al., 2005) • Strongly NP-hard even for only one commodity. • Polynomial for – one commodity if lower level path is known – one commodity if toll arcs with positive flows are known – one single toll arc. • Polynomial algorithm with worst-case guarantee of (log | A 1 | ) / 2 + 1 CO Workshop - Aussois - January 2017 29
Network pricing problem X X n k x k max T a a T ≥ 0 a ∈ A 1 k ∈ K X X X ( c a + T a ) x k min ( a + c a y a ) x,y a ∈ A 1 a ∈ A 2 k ∈ K X X ( x k a + y k ( x k a + y k a ) = b k s.t. a ) − ∀ k, i i a ∈ i + a ∈ i − x k a , y k a ≥ 0 , ∀ k, a CO Workshop - Aussois - January 2017 30
NPP: single level reformulation X n k X T a x k max a T,x,y, λ k ∈ K a ∈ A k X X ( x k a + y k ( x k a + y k a ) = b k s.t. a ) − ∀ k, i i a ∈ i + a ∈ i − λ k i − λ k j ≤ c a + T a ∀ k, a ∈ A 1 , i, j λ k i − λ k j ≤ c a ∀ k, a ∈ A 2 , i, j X X ( c a + T a ) x k c a y a = λ k o k − λ k a + ∀ k d k a ∈ A 1 a ∈ A 2 x k a , y k a ≥ 0 ∀ k, a T a ≥ 0 ∀ a ∈ A 1 CO Workshop - Aussois - January 2017 31 31
Solution approach by Branch & Cut • Formulate NPP as MIP • Tight bound (M ak ,N a ) on tax, if arc used and if arc not used, very effective • Add valid inequalities to strengthen LP relaxation CO Workshop - Aussois - January 2017 32
Product pricing Seller Consumers R k i n k p i R k i is the reservation price of consumer k for product i CO Workshop - Aussois - January 2017 33
Product pricing •PPP is Strongly NP-hard even if reservation price is independent of product (Briest 2006) •PPP is polynomial for one product or one customer. CO Workshop - Aussois - January 2017 34
PPP - bilevel formulation X n k X p i x k max i p ≥ 0 k ∈ K i ∈ I X ( R k i − p i ) x k s.t. max i , k ∈ K x k i ∈ I X x k s.t. i ≤ 1 i ∈ I x k i ≥ 0 CO Workshop - Aussois - January 2017 35
PPP - single level formulation X n k X p i x k max i p ≥ 0 k ∈ K i ∈ I X ( R k i − p i ) x k i ≥ R k s.t. j − p j , j ∈ I, k ∈ K i ∈ I X ( R k i − p i ) x k i ≥ 0 , k ∈ K i ∈ I X x k i ≤ 1 i ∈ I x k i ≥ 0 CO Workshop - Aussois - January 2017 36
PPP: MILP formulation (Heilporn et al., 2010, 2011) •Linearize single level formulation ⇒ MILP • Add new FDI’s ⇒ convex hull for k=1 •FDI’s added ⇒ divides the gap by 2 to 4 CO Workshop - Aussois - January 2017 37
PART III: Stakelberg games CO Workshop - Aussois - January 2017 38
Bimatrix game Follower C D A (2,1) (4,0) 0.5 Leader Mixed Strategy B (1,0) (3,2) 0.5 Leader Follower Pure Strategy A Stackelberg solution to the game (B,D) yielding a payoff of (3.5,1) CO Workshop - Aussois - January 2017 39
Stackelberg vs Nash Player 2 - A Player 2 - B Player 1 - A (2,2) (4,1) Player 1 - B (1,0) (3,1) Nash equilibrium: Player 1-A and Player 2-A => (2,2) Stackelberg solution: Player 1-B and Player 2-B => (3,1) Nash equilibrium may not exist There is always an (optimistic) Stackelberg solution CO Workshop - Aussois - January 2017 40
Stackelberg Games p-Followers Stackelberg Game (Conitzer & Sandblom, 2006) Stackelberg Game Leader Follower Leader Follower (R,C) … Type 1 Type 2 Type 3 Type p Objective of the Game • Reward-maximizing strategy for the Leader. • Follower will best respond to observable Leader’s strategy. CO Workshop - Aussois - January 2017 41
Applications (Tambe et al., USC) CO Workshop - Aussois - January 2017 42
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