Tabu Search Strategy Hachemi Bennaceur 5/1/ iroboapp project, 2013

Learning Outcomes 5/1/20152 At the end of the presentation, you will Get an idea about history of TS Understand the concepts used in TS Understand the TS strategy Be aware of the limits of TS and some improvements. Learn how to use TS for the TSP

Plan History General overview of TS Illustration Example Concepts of TS Basic TS Algorithm Improvements Discussion 5/1/20153

History  TS is proposed in 1986 by Fred Glover.  Method developed to solve problems combinatorial (mostly NP-hard)  Advantage of this method compared to others: it allows to overcome the problem of local optima by the use of tabu lists (principle of memory) 5/1/20154

TS: General principle Local search : get stuck in local optimum Basic Idea : continue the search for solutions even when a local optimum is encountered by, ⇒ allowing moves which do not improve the solution ⇒ using the principle of memory to avoid cycle moves. 5/1/20155

Memory: ⇒ it is represented by a TABU list that contains (tabu) moves that are temporarily forbidden ⇒ evolving role during the resolution: diversification (long-term memory) intensification (short-term memory) 5/1/20156 TS: General principle

A tourist is lost in a mountainous region. He knows that a rescue team regularly passes by the point at the lowest altitude in the region. He doesn’t know how to reach the lowest alitude of the region. So when he arrives to an intersection, he must engage in a direction to see if the way up or down. Illustration Example 5/1/20157

Firstly, he starts to go down as he can, by choosing the path of more high lean at each crossing. 5/1/20158

When there is no way leading to down, he decided to follow the path that goes up with the lower lean because he is aware that may be in a local minimum. 5/1/20159

10 However, when he goes up, he goes back down to the point where he was. This strategy does not work. Therefore, he decided to forbid to walk back by storing the direction from which he comes.

5/1/ This strategy allows him to explore other local optimums.

TS: Variables s : the curent solution s’ : the next solution to reach (neighboring solution) 5/1/ f(s) f(s’)

TS: variables 5/1/ N(s) : the space of neighboring solutions N(s)={s’ / s’ is a neighboring solution of s} N(s) m: a move from s to s’.

TS: variables 5/1/ s* the optimal solution minimizing the objective function. s c : the current best solution. S* ScSc

TS: concepts  Bad move to escape local optimum. similar to simulated annealing but TS selects the best bad move in N(s).  Tabu move undesirable move to avoid to return to a visited solution. A move is considered as tabu during k iterations. 5/1/201515

TS: concepts  T denotes the list of tabu moves; the elements of the list are t(s,m). A tabu list with many elements can be too restrictive: it is observed in practice that this enforce the method to ignore interesting solutions. A tabu list containing few elements may be useless and leads to cyclical moves.  a(s,m) : aspiration criterion Perform move m even it is tabu 5/1/201516

Basic TS: Algorithm  Step1 : select an initial solution s in S (set of solutions) s c ← s; k ←0;  Step2 : k ← k+1; compute N(s,k) such that Tabu moves are not considered Aspiration criterion may be applied  Step3 : select the best solution s’ in N(s,k); s ← s’;  Step4 : if f(s) < f(s c ) then s c ←s;  Step5 : update the tabu list T and the aspiration criterions.  Step6 : if the stop condition is not reached go to step 2 5/1/201517

TS for TSP problem  Solution : set of arcs composing a tour - the arcs are all oriented in the same direction chosen arbitrarily.  Move m= : remove the two arcs (a, b) and (c, d) and introducing two new arcs (a, d) and (c, b).  You may use two lists IN OUT containing tabu edges.  Update the lists when the move is performed: - add to OUT the elements (a, d) and (c, b) (edges inserted). - add to IN the elements {a, b} and {c, d} (edges deleted). 5/1/201518

TS for TSP problem Tabu moves :  If the edge (a, b) belongs to the IN list, then it prevents any move that leads to introduce (a, b) in the configuration.  If the edge (a, b) belongs to the list OUT, then it prevents any move that leads to remove (a, b) from the configuration. 5/1/201519

Improvements  Intensification The search is intensified in a neighborhood N(s) of S. A high priority is given to solutions s’ of N(s). 5/1/ N(s)

Improvements  Intensification  Intensify searching in neighborhoods that seem has a chance to lead to solutions close to the optimum.  Make sure that best solutions in these regions are indeed found.  Based-on a recency memory (short-term memory) records the number of consecutive iterations that various solution components have been present in the current solution without interruption. 5/1/201521

Intensification approaches  Approach 1 : Restart the search from the best currently known solution Fix in it the components that seem more attractive.  Approach 2: Restart the search from the best currently known solution Modify the neighborhood in such way to allow more moves ( increase the size of the neighborhood ) 5/1/201522

Improvements  Diversification The search is too local  it is displaced from the current neighborhood N(s) to another one. High priority is given to solutions of another neighborhood than the current one. 5/1/ N(s)

Improvements  Diversification :  A certain number of local optimums are reached without improving the current best solution.  The search is stagnated during a certain number of consecutive iterations  Based-on a frequency memory (long-term memory) records the number of total iterations from the beginning that various solution components have been present in the current solution. 5/1/201524

Diversification approaches Restart the search from a solution that contains components rarely used. Add a term to the objective related to the frequency of the components: - The most common components are penalized - Less frequent components are encouraged 5/1/201525

Conclusion & Discussion  Many parameters to tune  Size of the tabu list?  Size of neighbourhoods? Candidate solutions.  Stopping criterions  Intensification and diversification 5/1/201526