EURO Plenary Stackelberg Games Two-player sequential-play game: - - PowerPoint PPT Presentation

▶
euro plenary stackelberg games
SMART_READER_LITE
LIVE PREVIEW

EURO Plenary Stackelberg Games Two-player sequential-play game: - - PowerPoint PPT Presentation

Dec 16, 2023 386 likes •882 views

Ivana LJUBIC ESSEC Business School, Paris OR 2018, Brussels EURO Plenary Stackelberg Games Two-player sequential-play game: LEADER and FOLLOWER LEADER moves before FOLLOWER - first mover advantage Perfect information: both agents

slide-1
SLIDE 1

Ivana LJUBIC ESSEC Business School, Paris OR 2018, Brussels EURO Plenary

slide-2
SLIDE 2

Stackelberg Games

  • Two-player sequential-play game: LEADER and FOLLOWER
  • LEADER moves before FOLLOWER - first mover advantage
  • Perfect information: both agents have perfect knowledge of each others

strategy

  • Rationality: agents act optimally, according to their respective goals
  • LEADER takes FOLLOWERS’s optimal response into account
  • Optimistic vs Pessimistic: when FOLLOWER has multiple optimal responses
slide-3
SLIDE 3

Stackelberg Games

  • Two-player sequential-play game: LEADER and FOLLOWER
  • LEADER moves before FOLLOWER - first mover advantage
  • Perfect information: both agents have perfect knowledge of each others strategy
  • Rationality: agents act optimally, according to their respective goals
  • In any given situation a decision-maker always chooses the action which is the best according

to his/her preferences (a.k.a. rational play).

  • LEADER takes FOLLOWERS’s optimal response into account
  • Optimistic vs Pessimistic: when FOLLOWER has multiple optimal responses

STACKELBERG EQUILIBRIUM: Find the best strategy for LEADER (knowing what will be FOLLOWER‘s best response)

slide-4
SLIDE 4

Stackelberg Games

  • Introduced in economy by v. Stackelberg in

1934

  • 40 years later introduced in Mathematical

Optimization → Bilevel Optimization

slide-5
SLIDE 5

Applications: : Pricing

  • Pricing: operator sets tariffs, and then

customers choose the cheapest alternative

  • Tariff-setting, toll optimization (Labbé et

al., 1998; Brotcorne et al., 2001)

  • Network Design and Pricing (Brotcorne

et al., 2008)

  • Survey (van Hoesel, 2008)

Two competitive agents act in a hierarchical way with different/conflicting

  • bjectives
slide-6
SLIDE 6

Applications: : In Interdiction

source: banderasnews.com

slide-7
SLIDE 7

Applications: : In Interdiction

source: banderasnews.com

  • Monitoring / halting an adversary‘s

activity on a network

  • Maximum-Flow Interdiction
  • Shortest-Path Interdiction
  • Action:
  • Destruction of certain nodes / edges
  • Reduction of capacity / increase of

cost on certain edges

  • The problems are NP-hard! Survey

(Collado and Papp, 2012)

  • Uncertainties:
  • Network characteristics
  • Follower‘s response
slide-8
SLIDE 8

Bilevel Optimization

Follower Both players may involve integer decision variables, functions can be non-linear, non-convex…

slide-9
SLIDE 9

Bilevel Optimization

Follower Both players may involve integer decision variables, functions can be non-linear, non-convex…

1362 references!

slide-10
SLIDE 10

Hierarchy of f bilevel optimization problems

Bilevel Optimization General Case Interdiction-Like Under Uncertainty, Multiobjective, inf dim spaces, … Follower: Convex Follower: Non-Convex Follower: (M)ILP Jeroslow, MP, 1985 NP-hard (LP+LP) Follower: Convex Follower: Non-Convex Follower: (M)ILP … Network Interdiction (LP) Fischetti, Ljubic, Monaci, Sinnl, OR, 2017: Branch&Cut This talk! …

slide-11
SLIDE 11

About our jo journey

  • With sparse MILP formulations, we can now solve to optimality:
  • Covering Facility Location (Cordeau, Furini, Ljubic, 2018): 20M clients
  • Code: https://github.com/fabiofurini/LocationCovering
  • Competitive Facility Location (Ljubic, Moreno, 2017): 80K clients (nonlinear)
  • Facility Location Problems (Fischetti, Ljubic, Sinnl, 2016): 2K x 10K instances
  • Steiner Trees (DIMACS Challenge, 2014): 150k nodes, 600k edges
  • Common to all: Branch-and-Benders-Cut

Is there a way to exploit sparse formulations along with Branch-and-Cut for bilevel optimization?

slide-12
SLIDE 12

Problems addressed today…

  • Interdiction-Like Problems: LEADER ”interdicts” FOLLOWER by removing

some “objects”. Both agents play pure strategies.

  • FOLLOWER solves a combinatorial optimization problem (mostly, an NP-

hard problem!). One could build a payoff matrix (exponential in size!).

  • We propose a generic Branch-and-Interdiction-Cuts framework to

efficiently solve these problems in practice!

  • Assuming monotonicty property for FOLLOWER: interdiction cuts (facet-defining)
  • Computationally outperforming state-of-the-art
  • Draw a connection to some problems in Graph Theory
slide-13
SLIDE 13

Based on a joint work with…

  • M. Fischetti, I. Ljubic, M. Monaci, M. Sinnl: A new general-purpose algorithm for

mixed-integer bilevel linear programs, Operations Research 65(6): 1615-1637, 2017

  • M. Fischetti, I. Ljubic, M. Monaci, M. Sinnl: Interdiction Games and Monotonicity,

with Application to Knapsack Problems, INFORMS Journal on Computing, in print, 2018

  • F. Furini, I. Ljubic, P. San Segundo, S. Martin: The Maximum Clique Interdiction

Game, Optimization Online, 2018

  • F. Furini, I. Ljubic, E. Malaguti, P. Paronuzzi:

On Integer and Bilevel Formulations for the k-Vertex Cut Problem, submitted, 2018

slide-14
SLIDE 14

Branch-and-Interdiction-Cut

A gentle introduction

slide-15
SLIDE 15

In Interdicting Communities in a Network

Critical Nodes: disconnect the network „the most“ Survey: Lalou et al. (2018) Defender-Attacker Game LEADER: eliminates the nodes FOLLOWER: builds communities

slide-16
SLIDE 16

Hamburg Cell: Max-Clique In Interdiction

k=0 k=4

slide-17
SLIDE 17

Hamburg Cell: : Max-Clique In Interdiction

k=8 k=0

slide-18
SLIDE 18

Bilevel In Integer Program

Value Function

slide-19
SLIDE 19

Value Function Reformulation

INTERDICTION: Min-max BLOCKING: Min-num or Min-sum

slide-20
SLIDE 20

Value Function Reformulation

INTERDICTION: Min-max BLOCKING: Min-num or Min-sum

slide-21
SLIDE 21

How to to convexify fy the value function?

slide-22
SLIDE 22

Convexification

slide-23
SLIDE 23

Convexification → Benders-Like Reformulation

slide-24
SLIDE 24

If If the follower satis isfies monotonicity property…

slide-25
SLIDE 25

If If the follower satis isfies monotonicity property…

slide-26
SLIDE 26

Solve Master Problem → Branch-and-Interdiction-Cut

A A Careful Branch-and and-Interdiction-Cut Design

slide-27
SLIDE 27

Solve Master Problem → Branch-and-Interdiction-Cut

A A Careful Branch-and and-Interdiction-Cut Design

slide-28
SLIDE 28

A A Careful Branch-and and-Interdiction-Cut Design

Solve Master Problem → Branch-and-Interdiction-Cut

slide-29
SLIDE 29

Max-Clique-Interdiction on Large-Scale Networks

Furini, Ljubic, Martin, San Segundo (2018) eliminated by preprocessing Max-Clique Solver San Segundo et al. (2016)

slide-30
SLIDE 30

Max-Clique-Interdiction on Large-Scale Networks

#variables Furini, Ljubic, Martin, San Segundo (2018) eliminated by preprocessing Max-Clique Solver San Segundo et al. (2016)

slide-31
SLIDE 31

B&IC In Ingredients

lifting

slide-32
SLIDE 32

Comparison wit ith the state-of

  • f-the-art

MIL ILP bil ilevel solv lver

Generic B&C for Bilevel MILPs (Fischetti, Ljubic, Monaci, Sinnl, 2017) Branch-and- Interdiction-Cut

slide-33
SLIDE 33

Slide “NOT TO BE SHOWN”

B&IC WORKS WELL EVEN IF FOLLOWER HAS MORE DECISION VARIABLES, AS LONG AS MONOTONOCITY HOLDS FOR INTERDICTED VARIABLES

slide-34
SLIDE 34

The result can be fu further generalized

Fischetti, Ljubic, Monaci, Sinnl (2018)

slide-35
SLIDE 35

And what about Graph Theory ry?

slide-36
SLIDE 36

A A weird example…

  • Property: A set of vertices is a vertex cover if and only if its complement is

an independent set

  • Vertex Cover as a Blocking Problem:
  • LEADER: interdicts (removes) the nodes.
  • FOLLOWER: maximizes the size of the largest connected component in the remaining

graph.

  • Find the smallest set of nodes to interdict, so that FOLLOWER‘s optimal response is

at most one.

slide-37
SLIDE 37

The k-Vertex-Cut Problem

Furini, Ljubic, Malaguti, Paronuzzi (2018)

slide-38
SLIDE 38

The k-Vertex-Cut Problem

k=3 Furini, Ljubic, Malaguti, Paronuzzi (2018)

slide-39
SLIDE 39

K-Vertex-Cut

k=3

slide-40
SLIDE 40

K-Vertex-Cut

k=3

slide-41
SLIDE 41

k-Vertex-Cut: Benders-like reformulation

Furini, Ljubic, Malaguti, Paronuzzi (2018)

slide-42
SLIDE 42

k-Vertex-Cut: Benders-like reformulation

Furini, Ljubic, Malaguti, Paronuzzi (2018) Furini et al. (2018)

  • Prev. STATE-OF-

THE-ART Compact model Branch-and- Interdiction-Cut

slide-43
SLIDE 43

Conclusions.

And some directions for the future research.

slide-44
SLIDE 44

Takeaways

  • Bilevel optimization: very difficult!
  • Branch-and-Interdiction-Cuts can work very well in practice:
  • Problem reformulation in the natural space of variables („thinning out“ the heavy MILP

models)

  • Tight „interdiction cuts“ (monotonicity property)
  • Crucial: Problem-dependent (combinatorial) separation strategies, preprocessing,

combinatorial poly-time bounds

  • Many graph theory problems (node-deletion, edge-deletion) could be solved

efficiently, when approached from the bilevel-perspective

slide-45
SLIDE 45

Possible directions for fu future research

  • Bilevel Optimization: a better way of integrating customer behaviour into

decision making models

  • Generalizations of Branch-and-Interdiction-Cuts for:
  • Non-linear follower functions
  • Submodular follower functions
  • Interdiction problems under uncertainty
  • …
  • Extensions to Defender-Attacker-Defender (DAD) Models (trilevel games)
  • Benders-like decomposition for general mixed-integer bilevel optimization
slide-46
SLIDE 46

Thank you for your attention!

References:

  • M. Fischetti, I. Ljubic, M. Monaci, M. Sinnl: A new general-purpose algorithm for mixed-

integer bilevel linear programs, Operations Research 65(6): 1615-1637, 2017 SOLVER: https://msinnl.github.io/pages/bilevel.html

  • M. Fischetti, I. Ljubic, M. Monaci, M. Sinnl: Interdiction Games and Monotonicity, with

Application to Knapsack Problems, INFORMS Journal on Computing, in print, 2018

  • F. Furini, I. Ljubic, P. San Segundo, S. Martin: The Maximum Clique Interdiction Game,

Optimization Online, 2018

  • F. Furini, I. Ljubic, E. Malaguti, P. Paronuzzi:

On Integer and Bilevel Formulations for the k-Vertex Cut Problem, submitted, 2018

slide-47
SLIDE 47

Literature

  • Bastubbe, M., Lübbecke, M.: A branch-and-price algorithm for capacitated hypergraph vertex
  • separation. Technical Report, Optimization Online (2017)
  • L. Brotcorne, M. Labbé, P. Marcotte, and G. Savard. A Bilevel Model for Toll Optimization on a

Multicommodity Transportation Network, Transportation Science, 35(4): 345-358

  • L. Brotcorne, M. Labbé, P. Marcotte, and G. Savard. Joint design and pricing on a network.

Operations Research, 56 (5):1104–1115, 2008

  • Caprara A, Carvalho M, Lodi A, Woeginger GJ (2016) Bilevel knapsack with interdiction
  • constraints. INFORMS Journal on Computing 28(2):319–333
  • R.A.Collado, D. Papp. Network interdiction – models, applications, unexplored directions, Rutcor

Research Report 4-2012, 2012.

  • J.F. Cordeau, F. Furini, I. Ljubic. Benders Decomposition for Very Large Scale Partial Set Covering

and Maximal Covering Problems, submitted, 2018

  • S. Dempe. Bilevel optimization: theory, algorithms and applications, TU Freiberg, ISSN 2512-3750.

Fakultät für Mathematik und Informatik. PREPRINT 2018-11

  • DeNegre S (2011) Interdiction and Discrete Bilevel Linear Programming. Ph.D. thesis, Lehigh

University

  • M. Fischetti, I. Ljubic, M. Sinnl: Redesigning Benders Decomposition for Large Scale Facility

Location, Management Science 63(7): 2146-2162, 2017

slide-48
SLIDE 48

Literature, , cont.

  • R.G. Jeroslow. The polynomial hierarchy and a simple model for competitive analysis.

Mathematical Programming, 32(2):146–164, 1985

  • Kempe, D., Kleinberg, J., Tardos, E.: Influuential nodes in a diffusion model for social networks. In:
  • L. Caires, G.F. Italiano, L. Monteiro, C. Palamidessi, M. Yung (eds.) Automata, Languages and

Programming, pp. 1127-1138. , 2005

  • M. Labbé, P. Marcotte, and G. Savard. A bilevel model of taxation and its application to optimal

highway pricing. Management Science, 44(12):1608–1622, 1998

  • I. Ljubic, E. Moreno: Outer approximation and submodular cuts for maximum capture facility

location problems with random utilities, European Journal of Operational Research 266(1): 46-56, 2018

  • M. Sageman. Understanding Terror Networks. ISBN: 0812238087, University of Pennsylvania

Press, 2005

  • San Segundo P, Lopez A, Pardalos PM. A new exact maximum clique algorithm for large and

massive sparse graphs. Computers & OR 66:81–94, 2016

  • S. van Hoesel. An overview of Stackelberg pricing in networks. European Journal of Operational

Reseach, 189:1393–1492, 2008

  • R.K. Wood. Bilevel Network Interdiction Models: Formulations and Solutions, John Wiley & Sons,

Inc., http://hdl.handle.net/10945/38416, 2010