Multi-objective branch-and-cut algorithm and multi-modal traveling salesman problem

Transcription

1 Multi-objective branch-and-cut algorithm and multi-modal traveling salesman problem Nicolas Jozefowiez 1, Gilbert Laporte 2, Frédéric Semet 3 1. LAAS-CNRS, INSA, Université de Toulouse, Toulouse, France, nicolas.jozefowiez@laas.fr 2. CIRRELT, HEC, Montréal, Canada, gilbert@crt.umontreal.ca 3. LAGIS, Ecole Centrale de Lille, Villeneuve d Ascq, France, frederic.semet@ec-lille.fr 1 / 21

2 Outlines Branch-and-cut algorithm Multi-objective optimization A multi-objective branch-and-cut algorithm The multi-modal traveling salesman problem 2 / 21

3 Branch-and-cut algorithm A method to solve integer programs: min cx Ax b x 0 and integer Branch and bound algorithm Cutting plane method 3 / 21

4 Branch-and-bound algorithm Explicit enumeration Build an exploration tree at each node, branching on a variable Keep the best found feasible solution (the upper bound ub) Implicit enumeration At each node, a lower bound lb is computed A node can be pruned if given the branching choice: 1. the problem is infeasible (pruned by infeasiblity) 2. the solution is feasible (pruned by optimality) 3. lb ub (pruned by bound) 4 / 21

5 Cutting plane method min 1.00x x 2 50x x x 1 2x 2 4 x 1, x 2 0 and integer 5 / 21

6 Cutting plane method min 1.00x x 2 50x x x 1 2x 2 4 x 1, x 2 0 and integer 5 / 21

7 Cutting plane method min 1.00x x 2 50x x x 1 2x 2 4 x 2 4 x 1, x 2 0 and integer 5 / 21

8 Cutting plane method min 1.00x x 2 50x x x 1 2x 2 4 x 2 4 x 1, x 2 0 and integer 5 / 21

9 Cutting plane method min 1.00x x 2 50x x x 1 2x 2 4 x 2 4 3x 1 + 2x 2 15 x 1, x 2 0 and integer 5 / 21

10 Cutting plane method min 1.00x x 2 50x x x 1 2x 2 4 x 2 4 3x 1 + 2x 2 15 x 1, x 2 0 and integer 5 / 21

11 A simple branch-and-cut algorithm STEP 1 (Root of the tree) Generate an initial upper bound ub Define a first sub-problem Insert the sub-problem in a list L STEP 2 (Stopping criterion) If L = then STOP, else choose a sub-problem from L and remove it from L STEP 3 (Sub-problem solution) Solve the sub-problem to obtain the lower bound lb STEP 4 (Constraint generation) if there is no solution or lb ub then Go to STEP 2. else if the solution is integer then ub lb and go to STEP 2. else if violated constraints are identified then Add them to the model and go to STEP 3. else Go to STEP 5. end if STEP 5 (Branching) Branch on variable and introduce new sub-problems in L. Go to STEP 2. 6 / 21

12 Multi-objective optimization problem (P MO) = ( min F (x) = (f 1 (x), f 2 (x),..., f n(x)) s.t. x Ω with: n 2 : number of objectives F = (f 1, f 2,..., f n) : vector of functions to optimize Ω R m : set of feasible solutions x = (x 1, x 2,..., x m) Ω : a feasible solution Y = F (Ω) : objective space y = (y 1, y 2,..., y n) Y avec y i = f i (x) : a point in the objective space 7 / 21

13 Pareto dominance relation A solution x dominates ( ) a solution y if and only if i {1,..., n}, f i (x) f i (y) and i {1,..., n} such that f i (x) < f i (y). f2 A B D C E f1 8 / 21

14 Exact algorithms for MOP n = 2 n 2 Iteration Two-Phase method K-PPM PPM Multi-objective method [Sourd, Spanjaard, 2008] (*) (*) does not work if the aggregated problem is NP-hard a multi-objective branch-and-cut algorithm for multi-objective integer programs Lower bound multi-objective linear program Possibility to use scalar techniques to solve it to optimality (or a subset that can be extended) 9 / 21

15 Adaptations to a multi-objective problem Upper bound = set of non-dominated solutions found during the search Lower bound = set of non-dominated points in the objective space such that all feasible solutions are dominated by these points (1) (2) Upper bound (3) (4) Lower bound 10 / 21

16 A multi-objective branch-and-cut algorithm STEP 1 (Root of the tree) Generate an initial upper bound ub Define a first sub-problem Insert the sub-problem in a list L STEP 2 (Stopping criterion) If L = then STOP, else choose a sub-problem from L and remove it from L STEP 3 (Sub-problem solution) Solve the sub-problem to obtain the lower bound lb STEP 4 (Constraint generation) Try to insert integer solutions from lb in ub if lb = or ub lb then Go to STEP 2. else if violated constraints are identified for {x lb y ub, y x} then Add them to the model and go to STEP 3. else Go to STEP 5. end if STEP 5 (Branching) Branch on variable and introduce new sub-problems in L. Go to STEP / 21

17 The multi-modal traveling salesman problem Data: G = (V, E) : an undirected valuated graph C is a set of colors Each e E has a color k C Goal: Find a Hamiltonian cycle Two objectives: 1. Minimize the total of the cycle 2. Minimize the number of colors appearing on the cycle 12 / 21

18 Integer program Variables x e = u k = ( 1 if e E is used, 0 otherwise. ( 1 if k C is used, 0 otherwise. Constants and notations e E, δ(e) = k C the color of e k C, ζ(k) = {e E δ(e) = k} S V, ω(s) = {e = (i, j) E i S and j V \ S} 13 / 21

19 Integer program Objective functions min min X c ex e e E X u k k C Constraints X x e = 2 i V e ω({i}) X x e 2 S V, 3 S V 3 e ω(s) x e u δ(e) e E x e {0, 1} e E u k {0, 1} k C 14 / 21

20 Valid constraints u k X e ζ(k) x e k C X γi k u k 2 i V k C X λ k (S)u k 2 S V, 3 S V 3 k C with γ k i = 8 >< 0 if e ω({i}), e ζ(k), 1 if!e ω({i}), e ζ(k), >: 2 otherwise. λ k (S) = 8 >< 0 if e ω(s), e ζ(k), 1 if!e ω(s), e ζ(k), >: 2 otherwise. 15 / 21

21 Initial sub-problem : min min X c ex e e E X u k k C X x e = 2 i V e ω({i}) x e u δ(e) e E u k X e ζ(k) x e k C X γi k u k 2 i V k C 0 x e 1 e E 0 u k 1 k C 16 / 21

22 Solve the following problem for different values of ɛ min X c ex e + m X u k e E k C X x e = 2 i V e ω({i}) x e u δ(e) e E u k X x e k C e ζ(k) X γi k u k 2 i V k C X u k ɛ k C 0 x e 1 e E 0 u k 1 k C After founding non-dominated solution for a given ɛ, identify violated constraints and add them 17 / 21

23 ub is the upper bound. Set L tabu and continue TRUE while continue is TRUE do continue FALSE pruned TRUE Set ɛ α with α an integer such that α / L tabu and β / L tabu such that α < β C. while ɛ 0 do Solve the linear program. Let (x, u ) be the optimal solution and l the of the solution and o the number of colors used. if a solution is found then if the solution is feasible and integer or the solution is dominated by ub then if the solution is feasible and integer then Try to add it in ub and update ub if necessary end if L tabu { o... ɛ} else pruned FALSE if constraints violated by (x, u ) are identified then Stock them end if end if else L tabu {1... ɛ} end if Set ɛ α with α an integer such that α / L tabu and β / L tabu such that α < β < o. end while if violated constraints have been found then Add them to the model continue TRUE end if end while

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53 Constraint generation, cutting, and branching Constraint generation Connectivity constraints min-cut problem Call to a CONCORDE function [Padberg & Rinaldi, 1990] Cutting ɛ, the sub-problem is infeasible ɛ, the solution is either feasible or dominated by ub Branching First on the u k variables then on the x e Priority on the variable that is non integral for the most values of ɛ Initial upper bound ɛ-constraint method + MIP/CONCORDE 19 / 21

54 Computational results C V #nodes #u #x #cut #Pareto #Parub #time / 21

55 Conclusions and perspectives Branch-and-cut algorithm able to solve a multi-objective problem in one run Identify new valid constraints variables u k Rules to choose on which variables to branch Progressive partition of the objective space 21 / 21