From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial...
Transcript of From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial...
![Page 1: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/1.jpg)
1
1
From Decision Theory to CombinatorialOptimization: Problems and Algorithms in Graphs
Patrice PERNY
Université Paris 6
http://www-poleia.lip6.fr/~perny/
2
Outline
![Page 2: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/2.jpg)
2
1. Examples and motivations
Compromise search in multiobjective optimizationEquity in multiagent assignment problemsRobustness in optimization under uncertainty
Examples and motivations
4
Compromise search in multiobjective(combinatorial) optimization
AugmentedTchebycheffdistance
![Page 3: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/3.jpg)
3
5
Fairness in multiagentassignment/transportation problems
• Paper assignment problems [e.g., Goldsmith and Sloan 07, Wang et al.08]
• Allocation of indivisible goods [e.g. Bouveret and Lang, 05]
• Matching in social networks (e.g. Meetic)
5
5
5
3
3
3
3
3
1
1
1
1
6
Robustness in optimization under uncertainty
1 - Examples and motivations Spanning trees
![Page 4: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/4.jpg)
4
7
Multiobjective combinatorial optimization
8
Some references in MOCO
![Page 5: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/5.jpg)
5
9
The number of Pareto-optimal solutions exponentially grows with the size of the graph (number of nodes)
Pareto-optimal paths: an intractable problem
10
Pareto-optimal spanning trees: an intractable problem
The number of Pareto-optimal solutions exponentially grows with the size of the graph (number of nodes)
![Page 6: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/6.jpg)
6
11
Exploration of Pareto-optimal solutions
12
Preference models for vector optimization
![Page 7: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/7.jpg)
7
13
2. Using decision models in multiobjectivecombinatorial optimization: a research program
….
Knapsack
Flows
Trees
Paths
ChoquetRDUWOWAOWATchebEUSSDLorenzε-ParetoPareto
14
2.1 Lorenz-optimal paths
….
Knapsack
Assign
Trees
Paths
ChoquetRDUWOWAOWATchebEUSSDLorenzε-ParetoPareto
1
![Page 8: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/8.jpg)
8
15
Aim: favouring well-balanced cost distributions
2 – A decision theoretic approach
16
Generalized Lorenz dominance
2 – A decision theoretic approach
• Lorenz dominance refines Pareto dominance• Favours well-balanced solutions (transfer principle)
(11, 9, 10) >L (6, 10, 15) because (11, 21, 30) >P (15, 25, 31)
![Page 9: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/9.jpg)
9
17
L-optimality: complexity issues
4 – Algorithms
18
L-dominance and the Bellman principle
(3, 5) (4, 5) (5, 9) (6, 9)
4 – Algorithms
![Page 10: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/10.jpg)
10
19
A simple label-setting algorithm
5 – Numerical tests
(13, 9)
(11, 11)
[Martins’84]
L= (12,20)
L= (13,22)L= (11,22)
20
Numerical tests for L-optimal paths
5 – Numerical tests
# L-opt time (s)
(random instances, graph density ~ 50%)
![Page 11: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/11.jpg)
11
21
Refining Lorenz dominance
2 – A decision theoretic approach
22
OWA as a measure of inequality
2 – A decision theoretic approach
![Page 12: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/12.jpg)
12
23
2. OWA-optimal assignment/transportation
….
Knapsack
Assign
Trees
Paths
ChoquetRDUWOWAOWATchebEUSSDLorenzε-ParetoPareto
1
2
24
Fair assignment problems
Min OWA
≤ m≤ p
![Page 13: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/13.jpg)
13
25
An example: WS vs OWA in multiagent assignment problems
WS-opt
OWA-opt
WS = 14/5
WS = 16/5
26
LP formulation of OWA-optimization
(Ogryczak, 07)
Lk(y) =
![Page 14: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/14.jpg)
14
27
A mixed-integer LP formulation of the OWA-optimal assignment problem
≤ m≤ p
28
Numerical tests with Cplex for OWA assignment
Times (in seconds) for fair assignment problems with n agents, costs in {1, …, 20}
Times (in seconds) for paper assignment problems with n reviewers, 3n papers
costs in {1, …, 5}, matrix density 20%, max nb of paper per agent = 5.
![Page 15: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/15.jpg)
15
29
2.3 Choquet-optimal spanning trees[Galand, Perny, Spanjaard, 08]
….
Knapsack
Assign
Trees
Paths
ChoquetRDUWOWAOWATchebEUSSDLorenzε-ParetoPareto
1
2
3
30
The Choquet Expected Disutility model
![Page 16: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/16.jpg)
16
31
CED includes multiple models as special cases
32
Compromise search, fairness or uncertainty aversion
![Page 17: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/17.jpg)
17
33
Compromise search, fairness or uncertainty aversion
34
Compromise search, fairness or uncertainty aversion
![Page 18: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/18.jpg)
18
35
Complexity of Choquet optimization
4 – Algorithms
36
Failure of the greedy approach with Choquet
Idem for OWA, WOWA, Yaari’s model, RDU, Lorenz, SSD…
4 – Algorithms
Choquet optimal edge: a (2, 2)
Completion: a ∪ b (5, 3) a ∪ c (3, 5) sub-optimal
b ∪ c is clearly better with (4, 4)
![Page 19: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/19.jpg)
19
37
An important notion: the core of a capacity
5 – Numerical tests
38
Capacity in the core provide default approximations
5 – Numerical tests
Shapley
Max entropy
![Page 20: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/20.jpg)
20
39
A1: Branch and Bound (spanning trees)
Requires a lower bound (must be easily computable)
4 – Algorithms
noyes edge e?
2) Solved in polytime2), 3) p chosen in the core
Improving bounds
40
Numerical tests
5 – Numerical tests
![Page 21: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/21.jpg)
21
41
A2 :The ranking approach for ST
Requires a stopping conditions
4 – Algorithms
42
Stopping condition of the ranking approach
5 – Numerical tests
![Page 22: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/22.jpg)
22
43
Example 1/2
44
Example 2/2
![Page 23: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/23.jpg)
23
45
3. Approximation of Pareto-optimal Knapsacks
….
Knapsack
Flows
Trees
Paths
ChoquetRDUWOWAOWATchebEUSSDLorenzε-ParetoPareto
1
2
3
4
46
3. Approximation of preferred solutions
The case of Pareto dominance
![Page 24: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/24.jpg)
24
47
Approximation = covering of the Pareto set
48
Existence of covering with bounded size (PY00)
![Page 25: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/25.jpg)
25
49
An example using Hansen’s graphs
50
Project selection, product design, team configuration, resource allocation…
[Perny et Spanjaard,
ECAI’08]
Application to biobjective knapsack problems
![Page 26: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/26.jpg)
26
51
Approximation of preferred solutions fordecision models refining Pareto dominance
52
Conclusion (main messages)
![Page 27: From Decision Theory to Combinatorial Optimization ... · From Decision Theory to Combinatorial Optimization: Problems and Algorithms in Graphs Patrice PERNY ... Robustness in optimization](https://reader033.fdocuments.us/reader033/viewer/2022053013/5f1068bf7e708231d448f7bf/html5/thumbnails/27.jpg)
27
53
Still some work to do…
54
Recent publications of our team on this topic
Near Admissible Algorithms for Multiobjective SearchPerny, Patrice; Spanjaard, Olivier; ECAI-08 (2008) pp. 490-494
Search for Choquet-optimal paths under uncertaintyGaland, Lucie; Perny, Patrice, UAI’07, pp. 125-132,
State Space Search for Risk-averse Agents Perny, Patrice; Spanjaard, Olivier; Storme, Louis-Xavier; IJCAI’07, pp. 2353-2358
A decision-theoretic approach to robust optimization in multivalued graphs Perny, Patrice; Spanjaard, Olivier; Storme, Louis-Xavier; Annals of Operations Research (2006) Vol. 147, 1, pp. 317-341
Search for Compromise Solutions in Multiobjective State Space GraphsGaland, Lucie; Perny, Patrice; ECAI’06, pp. 93-97.