Oscar Cubo M edina fi.upm.es)

Transcription

1 . euristics. reedy. LocalSearch. illimbing. Meta-heuristics. Tabu Search. Sim ulated nnealing Oscar ubo M edina (ocubo@ fi.upm.es) ased on Neighbourhood Search euristic derives from the verb heuriskein (ενρισκειν)which m eans to find aster than mathem aticaloptimization (branch & bound,simplex,etc) M eta m eans beyond,in an upper lever Meta-heuristics are strategies that guide the search process The goalis to explore the search space in order to find (near-)optimalsolutions Meta-heuristics are notproblem -specific The basic concepts ofm eta-heuristics perm itan abstract leveldescription They may incorporate mechanism s to avoid getting trapped in confined areas ofthe search space

2 euristic onstructive algorithm s (greedy) Localsearch algorithm s (hil-clim bing ) Meta-heuristic Trajectory methods: escribe a trajectory in the search space during the search process Variable Neighbourhood Search Iterated LocalSearch Sim ulated nnealing Tabu Search Population-based: Perform search processes which describe the evolution ofa setofpoints in the search space volutionary com putation enerate solutions from scratch by adding (to an initia ly em pty partialsolution) com ponents,untilthe solution is com plete greedy algorithm works w in phases.teach phase: You take the bestyou can getrightnow,w ithout regard for future consequences You hope thatby choosing a localoptimum at each step,you wilend up ata globaloptim um Scheduling: processors 9 jobs (,,,0,,,, and minutes) reedy assign to each free processor the longest-running job Scheduling: processors 9 jobs (,,,0,,,, and minutes) reedy assign to each free processor the shortest-running job 0 Total time: m 0 Total time: 0 m

3 Longest-running job: minutes Shortest-running running job: 0 minutes 0 Optimal: minutes 0 0 Iterative algorithm : Startfrom som e initialsolution xplore the neighbourhood ofthe current solution Replace the currentsolution by a better solution Neighbourhood: being X the search space,we define the neighbourhood system N in X as N : x N (x ) ifferentprocedures depending on choice criteria and term ination criteria Stochastic: choose atrandom ilclimbing: only perm itmoves to neighbours thatimprove the current reedy the bestneighbour nxious the first neighbour improving Sidew ays moves alow s moves with sam e fitness s= enerateinitialsolution() while Term ination criteria notm et s =PicktRandom ( N(x) ) If ( f(s ) < f(s) ) then s= s ndif end while

4 itness itness: : itness itness: : itness itness: :

5 itness: : asic principles: Keep only a single state (solution) in m em ory enerate only the neighbours ofthatstate Keep one ofthe neighbours and discard others Key features: No system atic No increm ental Key advantages: Use very little memory (constantam ount) ind solutions in search spaces too large for system atic algorithm s Success ofhil-clim bing depends on shape of landscape Shape oflandscape depends on problem form ulation and fitness function Landscapes for realistic problem s often look like a worst-case scenario NP-hard problem s typica ly have exponential num ber oflocal-m inima ailing on neighbourhood search Propensity to deliver solutions which are only localoptima Solutions depend on the initialsolution pplies localsearch to an initialsolution until itfinds the localoptimum; then itperturbs the solution and itrestartslocalsearch

6 Strategy based on dynam ica ly changing neighborhood structure Steepestdescentw ith m em ory Moves through solution space Uses memory techniques to avoid cycling The overa lapproach is to avoid entrainm entin cycles by forbidding or penalizing moves which take the solution,in the nextiteration,to points in the solution space previously visited (red lover,om puters and Operations Research,98) To avoid revelsalm oves the lastm oves are makedas Tabu hange value ofx from false totrue: [x = false] istabu Sw ap elem entsiand j: [hange j,i] istabu rop iand add j: [add i] and [drop j] are tabu MinimalSpanning Tree: Totalcostofthe links used is a minimum lthe points are connected together onstraints: x + x + x x x Penalty: 0 x 9 x 0 x x x x x 7 8

7 The move considered is the standard edge sw ap move m n edge is listed as tabu ifitwas added within the lasttwo iterations ifitwas deleted within the lasttwo iterations Iteration x + x + x x x Penalty: 0 rop x 9 x 0 x x x x x 7 8 (X) = +00 dd Iteration x + x + x x x Penalty: 0 Tabu x 9 x 0 x (X) = 8 Tabu Iteration x + x + x x x Penalty: 0 Tabu dd x 9 x x 0 (X) = Tabu rop x x x x rop x x dd 7 8 x x 7 Tabu 8 Tabu

8 Iteration x + x + x x x Penalty: 0 Tabu dd x 9 x x 0 (X) = Tabu rop Iteration x + x + x x x Penalty: 0 Tabu x 9 x x 0 (X) = Tabu x x x x x x 7 Tabu 8 Tabu x x 7 Tabu 8 Tabu Pros: Tabu Search yields relatively good solutions to previously intractable problem s Tabu Search provides com parable or superior solutions to other optim ization techniques ons: Tabu Search does notguarantee optim ality Tabu Search is aw kw ard for problem s with continuous variables Tabu Search assum es fastperform ance evaluation The construction oftabu listis heuristic ased on the cooling ofmaterialin a heatbath LocalSearch butwe a low moves resulting in solutions ofworse quality than the currentsolution void from localsolutions

9 s= enerateinitialsolution() t=t(0) while Term ination criteria notmet s =PicktRandom ( N(x) ) If ( f(s ) < f(s) ) then else s= s ccepts as new solution with probability ndif Update(T) end while e f ( s') f ( s) T Initialtem perature Should be suitable high.mostofthe initial moves mustbe accepted (> 0% ) ooling schedule Tem perature is reduced after every m ove Tw o main methods: T T = + β T