Bilevel Programming Bilevel Knapsack Algorithms Conclusions Bilevel Knapsack with interdiction constraints Alberto Caprara 1 Margarida Carvalho 2 Andrea Lodi 1 Gerhard Woeginger 3 1 DEI, University of Bologna, Italy 2 Faculdade de Ciˆ encias da Universidade do Porto, Portugal 3 TU Eindhoven, Netherlands 17th Combinatorial Optimization Workshop, Aussois, France January, 2013 Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Computational complexity 1 Bilevel Programming 3 Algorithms Motivation Upper Bound Definition Basic Scheme Challenges ImproveA 2 Bilevel Knapsack CCLW Method Bilevel knapsack with Computational Results interdiction constraints DN Bilevel knapsack DR 4 Conclusions Bilevel knapsack MACH Generalizations Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Motivation 1952 Stackelberg Game: a player, called the leader, takes his decision before decisions of other players, called the followers; Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Motivation 1952 Stackelberg Game: a player, called the leader, takes his decision before decisions of other players, called the followers; 80’s Development policy (e.g. determination of pricing policies); Generalization: multilevel programming - Hierarchical structures; Computational complexity theory; Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Motivation 90’s Algorithms to linear bilevel programming problems; Algorithms to integer linear bilevel programming problems; Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Motivation 90’s Algorithms to linear bilevel programming problems; Algorithms to integer linear bilevel programming problems; Recently Bilevel problem specific algorithms/heuristics; Defence-planning problems (e.g. Transmission networks); Worst-case analyses; Interdiction problems (e.g. sensitivity analysis); Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Definition Minimize x,y L ( x, y ) (1a) subject to ( x, y ) ∈ X (1b) where y solves the follower’s problem Minimize y F ( x, y ) s.t. ( x, y ) ∈ Y (1c) Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Challenges - Example from Moore and Bard (1990) min − x − 10 y P U = R + and P L = R + x,y 4 s.t. x ∈ P U 3 2 where y is optimal to 1 min y y 1 2 3 4 5 6 7 8 s.t 5 x − 4 y ≥ − 6 − x − 2 y ≥ − 10 − 2 x + y ≥ − 15 P U = Z + and P L = Z + 2 x + 10 y ≥ 15 4 y ∈ P L 3 2 1 1 2 3 4 5 6 7 8 Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Challenges - Example from Moore and Bard (1990) min − x − 10 y P U = R + and P L = R + x,y 4 s.t. x ∈ P U 3 2 where y is optimal to 1 min y y 1 2 3 4 5 6 7 8 s.t 5 x − 4 y ≥ − 6 − x − 2 y ≥ − 10 − 2 x + y ≥ − 15 P U = Z + and P L = Z + 2 x + 10 y ≥ 15 4 y ∈ P L 3 2 1 1 2 3 4 5 6 7 8 Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Challenges - Example from Moore and Bard (1990) min − x − 10 y P U = R + and P L = R + x,y 4 OP T = − 18 s.t. x ∈ P U 3 2 where y is optimal to OP T 1 min y y 1 2 3 4 5 6 7 8 s.t 5 x − 4 y ≥ − 6 − x − 2 y ≥ − 10 − 2 x + y ≥ − 15 P U = Z + and P L = Z + 2 x + 10 y ≥ 15 4 y ∈ P L 3 2 1 1 2 3 4 5 6 7 8 Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Challenges - Example from Moore and Bard (1990) min − x − 10 y P U = R + and P L = R + x,y 4 OP T = − 18 s.t. x ∈ P U 3 2 where y is optimal to OP T 1 min y y 1 2 3 4 5 6 7 8 s.t 5 x − 4 y ≥ − 6 − x − 2 y ≥ − 10 − 2 x + y ≥ − 15 P U = Z + and P L = Z + 2 x + 10 y ≥ 15 4 y ∈ P L 3 2 1 1 2 3 4 5 6 7 8 Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Challenges - Example from Moore and Bard (1990) min − x − 10 y P U = R + and P L = R + x,y 4 OP T = − 18 s.t. x ∈ P U 3 2 where y is optimal to OP T 1 min y y 1 2 3 4 5 6 7 8 s.t 5 x − 4 y ≥ − 6 − x − 2 y ≥ − 10 − 2 x + y ≥ − 15 P U = Z + and P L = Z + 2 x + 10 y ≥ 15 4 y ∈ P L 3 2 1 1 2 3 4 5 6 7 8 Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Challenges - Example from Moore and Bard (1990) min − x − 10 y P U = R + and P L = R + x,y 4 OP T = − 18 s.t. x ∈ P U 3 2 where y is optimal to OP T 1 min y y 1 2 3 4 5 6 7 8 s.t 5 x − 4 y ≥ − 6 − x − 2 y ≥ − 10 − 2 x + y ≥ − 15 P U = Z + and P L = Z + 2 x + 10 y ≥ 15 4 OP T = − 22 y ∈ P L 3 OP T 2 1 1 2 3 4 5 6 7 8 Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Bilevel knapsack with interdiction constraints DN n X max (3c) p i y i y ∈ B n i =1 n X s.t. w i y i ≤ C l and (3d) i =1 Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Bilevel knapsack with interdiction constraints DN n X min p i y i (3a) ( x,y ) ∈ B n × B n i =1 n X s. t. (3b) v i x i ≤ C u i =1 where y 1 , . . . , y n solves the follower’s problem n X max (3c) p i y i y ∈ B n i =1 n X s.t. w i y i ≤ C l and (3d) i =1 y i ≤ 1 − x i for 1 ≤ i ≤ n (3e) Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Bilevel knapsack with interdiction constraints DN Applications: n X min p i y i (3a) ( x,y ) ∈ B n × B n i =1 n X s. t. (3b) v i x i ≤ C u i =1 where y 1 , . . . , y n solves the follower’s problem n X max (3c) p i y i y ∈ B n i =1 n X s.t. w i y i ≤ C l and (3d) i =1 y i ≤ 1 − x i for 1 ≤ i ≤ n (3e) Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Bilevel knapsack with interdiction constraints DN Applications: Corporate Strategy; n X min p i y i (3a) ( x,y ) ∈ B n × B n i =1 n X s. t. (3b) v i x i ≤ C u i =1 where y 1 , . . . , y n solves the follower’s problem n X max (3c) p i y i y ∈ B n i =1 n X s.t. w i y i ≤ C l and (3d) i =1 y i ≤ 1 − x i for 1 ≤ i ≤ n (3e) Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Bilevel knapsack with interdiction constraints DN Applications: Corporate Strategy; n X min p i y i (3a) Government control ( x,y ) ∈ B n × B n i =1 for reducing tax evasion. n X s. t. (3b) v i x i ≤ C u i =1 where y 1 , . . . , y n solves the follower’s problem n X max (3c) p i y i y ∈ B n i =1 n X s.t. w i y i ≤ C l and (3d) i =1 y i ≤ 1 − x i for 1 ≤ i ≤ n (3e) Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Bilevel knapsack with interdiction constraints DN Applications: Corporate Strategy; n X min p i y i (3a) Government control ( x,y ) ∈ B n × B n i =1 for reducing tax evasion. n X s. t. (3b) v i x i ≤ C u However, the natural motivation i =1 for looking at these problems is where y 1 , . . . , y n solves the follower’s problem n methodological: knapsack has X max (3c) p i y i y ∈ B n been a fundamental ”playground” i =1 n for understanding single-level X s.t. w i y i ≤ C l and (3d) programming and it is likely that i =1 the same turns out to be true in y i ≤ 1 − x i for 1 ≤ i ≤ n (3e) the bilevel case. Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
Bilevel Programming Bilevel Knapsack Algorithms Conclusions Bilevel knapsack DR The Dempe-Richter variant n � max tx + (4a) a i y i i =1 C ≤ x ≤ C ′ subject to (4b) where y 1 , . . . , y n solves the follower’s problem n n � � max b i y i s.t. b i y i ≤ x (4c) i =1 i =1 y i ∈ { 0 , 1 } for 1 ≤ i ≤ n (4d) Margarida Carvalho margarida.carvalho@dcc.fc.up.pt Bilevel Knapsack with interdiction constraints
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